Longitudinal and Panel Data: Analysis and Applications for the Social Sciences. Edward W. Frees

Logudal ad Pael Daa: Aalss ad Applcaos for he Socal Sceces b Edward W. Frees

Logudal ad Pael Daa: Aalss ad Applcaos for he Socal Sceces Bref Table of Coes Chaper. Iroduco PART I - LINEAR MODELS Chaper. Chaper 3. Chaper 4. Chaper 5. Chaper 6. Chaper 7. Chaper 8. Fxed Effecs Models Models wh Radom Effecs Predco ad Baesa Iferece Mullevel Models Radom Regressors Modelg Issues Damc Models PART II - NONLINEAR MODELS Chaper 9. Bar Depede Varables Chaper 0. Geeralzed Lear Models Chaper. Caegorcal Depede Varables ad Survval Models Appedx A. Elemes of Marx Algebra Appedx B. Normal Dsrbuo Appedx C. Lkelhood-Based Iferece Appedx D. Kalma Fler Appedx E. Smbols ad Noao Appedx F. Seleced Logudal ad Pael Daa Ses Appedx G. Refereces Ths draf s parall fuded b he Fors Healh Isurace Professorshp of Acuaral Scece. 003 b Edward W. Frees. All rghs reserved To appear: Cambrdge Uvers Press 004

Logudal ad Pael Daa: Aalss ad Applcaos for he Socal Sceces Table of Coes Ocober 003 Table of Coes Preface v. Iroduco. Wha are logudal ad pael daa? -. Beefs ad drawbacks of logudal daa -4.3 Logudal daa models -9.4 Hsorcal oes -3 PART I - LINEAR MODELS. Fxed Effecs Models. Basc fxed effecs model -. Explorg logudal daa -5.3 Esmao ad ferece -0.4 Model specfcao ad dagoscs -4.4. Poolg es -4.4. Added varable plos -5.4.3 Ifluece dagoscs -6.4.4 Cross-secoal correlao -7.4.5 Heeroscedasc -8.5 Model exesos -9.5. Seral correlao -0.5. Subjec-specfc slopes -.5.3 Robus esmao of sadard errors - Furher readg -3 Appedx A - Leas squares esmao -4 A. Basc Fxed Effecs Model Ordar Leas Squares Esmao -4 A. Fxed Effecs Model Geeralzed Leas Squares Esmao -4 A.3 Dagosc Sascs -5 A.4 Cross-secoal Correlao -6. Exercses ad Exesos -7 3. Models wh Radom Effecs 3. Error compoes / radom erceps model 3-3. Example: Icome ax pames 3-7 3.3 Mxed effecs models 3-3.3. Lear mxed effecs model 3-3.3. Mxed lear model 3-5 3.4 Iferece for regresso coeffces 3-6

3.5 Varace compoe esmao 3-0 3.5. Maxmum lkelhood esmao 3-0 3.5. Resrced maxmum lkelhood 3-3.5.3 MIVQUE esmaors 3-3 Furher readg 3-5 Appedx 3A REML calculaos 3-6 3A. Idepedece Of Resduals Ad Leas Squares Esmaors 3-6 3A. Resrced Lkelhoods 3-6 3A.3 Lkelhood Rao Tess Ad REML 3-8 3. Exercses ad Exesos 3-30 4. Predco ad Baesa Iferece 4. Predco for oe-wa ANOVA models 4-4. Bes lear ubased predcors BLUP 4-4 4.3 Mxed model predcors 4-7 4.3. Lear mxed effecs model 4-7 4.3. Lear combaos of global parameers ad subjecspecfc effecs 4-7 4.3.3 BLUP resduals 4-8 4.3.4 Predcg fuure observaos 4-9 4.4 Example: Forecasg Wscos loer sales 4-0 4.4. Sources ad characerscs of daa 4-4.4. I-sample model specfcao 4-3 4.4.3 Ou-of-sample model specfcao 4-5 4.4.4 Forecass 4-6 4.5 Baesa ferece 4-7 4.6 Credbl heor 4-0 4.6. Credbl heor models 4-4.6. Credbl heor raemakg 4- Furher readg 4-5 Appedx 4A Lear ubased predco 4-6 4A. Mmum Mea Square Predcor 4-6 4A. Bes Lear Ubased Predcor 4-6 4A.3 BLUP Varace 4-7 4. Exercses ad Exesos 4-8 5. Mullevel Models 5. Cross-secoal mullevel models 5-5.. Two-level models 5-5.. Mulple level models 5-5 5..3 Mulple level modelg oher felds 5-6 5. Logudal mullevel models 5-6 5.. Two-level models 5-6 5.. Mulple level models 5-0 5.3 Predco 5-5.4 Tesg varace compoes 5-3 Furher readg 5-5 Appedx 5A Hgh order mullevel models 5-6 5. Exercses ad Exesos 5-9

6. Radom Regressors 6. Sochasc regressors o-logudal segs 6-6.. Edogeous sochasc regressors 6-6.. Weak ad srog exogee 6-3 6..3 Causal effecs 6-4 6..4 Isrumeal varable esmao 6-5 6. Sochasc regressors logudal segs 6-7 6.. Logudal daa models whou heerogee erms 6-7 6.. Logudal daa models wh heerogee erms ad srcl exogeous regressors 6-8 6.3 Logudal daa models wh heerogee erms ad sequeall exogeous regressors 6-6.4 Mulvarae resposes 6-6 6.4. Mulvarae regressos 6-6 6.4. Seemgl urelaed regressos 6-7 6.4.3 Smulaeous equaos models 6-8 6.4.4 Ssems of equaos wh error compoes 6-0 6.5 Smulaeous equao models wh lae varables 6-6.5. Cross-secoal models 6-3 6.5. Logudal daa applcaos 6-6 Furher readg 6-9 Appedx 6A Lear projecos 6-30 7. Modelg Issues 7. Heerogee 7-7. Comparg fxed ad radom effecs esmaors 7-4 7.. A specal case 7-7 7.. Geeral case 7-9 7.3 Omed varables 7-7.3. Models of omed varables 7-3 7.3. Augmeed regresso esmao 7-4 7.4 Samplg selecv bas aro 7-7 7.4. Icomplee ad roag paels 7-7 7.4. Uplaed orespose 7-8 7.4.3 No-gorable mssg daa 7-7. Exercses ad Exesos 7-4 8. Damc Models 8. Iroduco 8-8. Seral correlao models 8-3 8.. Covarace srucures 8-3 8.. Nosaoar srucures 8-4 8..3 Couous me correlao models 8-5 8.3 Cross-secoal correlaos ad me-seres cross-seco models 8.4 Tme-varg coeffces 8-9 8.4. The model 8-9 8.4. Esmao 8-0 8-7

8.4.4 Forecasg 8-8.5 Kalma fler approach 8-3 8.5. Traso equaos 8-4 8.5. Observao se 8-5 8.5.3 Measureme equaos 8-5 8.5.4 Ial codos 8-6 8.5.5 Kalma fler algorhm 8-6 8.6 Example: Capal asse prcg model 8-8 Appedx 8A Iferece for he me-varg coeffce model 8-3 8A. The Model 8-3 8A. Esmao 8-3 8A.3 Predco 8-5 PART II - NONLINEAR MODELS 9. Bar Depede Varables 9. Homogeeous models 9-9.. Logsc ad prob regresso models 9-9.. Iferece for logsc ad prob regresso models 9-5 9..3 Example: Icome ax pames ad ax preparers 9-7 9. Radom effecs models 9-9 9.3 Fxed effecs models 9-3 9.4 Margal models ad GEE 9-6 Furher readg 9-0 Appedx 9A Lkelhood calculaos 9-0 9A. Cossec Of Lkelhood Esmaors 9-9A. Compug Codoal Maxmum Lkelhood Esmaors 9-9. Exercses ad Exesos 9-3 0. Geeralzed Lear Models 0. Homogeeous models 0-0.. Lear expoeal famles of dsrbuos 0-0.. Lk fucos 0-0..3 Esmao 0-3 0. Example: Tor flgs 0-5 0.3 Margal models ad GEE 0-8 0.4 Radom effecs models 0-0.5 Fxed effecs models 0-6 0.5. Maxmum lkelhood esmao for caocal lks 0-6 0.5. Codoal maxmum lkelhood esmao for caocal lks 0-7 0.5.3 Posso dsrbuo 0-8 0.6 Baesa ferece 0-9 Furher readg 0- Appedx 0A Expoeal famles of dsrbuos 0-3 0A. Mome Geerag Fucos 0-3 0A. Suffcec 0-4 0A.3 Cojugae Dsrbuos 0-4

0A.4 Margal Dsrbuos 0-5 0. Exercses ad Exesos 0-7. Caegorcal Depede Varables ad Survval Models. Homogeeous models -.. Sascal ferece -.. Geeralzed log -..3 Mulomal codoal log -4..4 Radom ul erpreao -6..5 Nesed log -7..6 Geeralzed exreme value dsrbuo -8. Mulomal log models wh radom effecs -8.3 Traso Markov models -0.4 Survval models -8 Appedx A. Codoal lkelhood esmao for - mulomal log models wh radom effecs APPENDICES Appedx A. Elemes of Marx Algebra A- A. Basc Defos A- A. Basc Operaos A- A.3 Furher Defos A- A.4 Marx Decomposos A- A.5 Paroed Marces A-3 A.6 Kroecker Drec Producs A-4 Appedx B. Normal Dsrbuo A-5 Appedx C. Lkelhood-Based Iferece A-6 C. Characerscs of Lkelhood Fucos A-6 C. Maxmum Lkelhood Esmaors A-6 C.3 Ieraed Reweghed Leas Squares A-8 C.4 Profle Lkelhood A-8 C.5 Quas-Lkelhood A-8 C.6 Esmag Equaos A-9 C.7 Hpohess Tess A- C.8 Iformao Crera A- C.9 Goodess of F Sascs A-3 Appedx D. Kalma Fler A-4 D. Basc Sae Space Model A-4 D. Kalma Fler Algorhm A-4 D.3 Lkelhood Equaos A-5 D.4 Exeded Sae Space Model ad Mxed Lear Models A-5 D.5 Lkelhood Equaos for Mxed Lear Models A-6 Appedx E. Smbols ad Noao A-8 Appedx F. Seleced Logudal ad Pael Daa A-4 Ses Appedx G. Refereces A-8 Idex A-40

Preface Ieded Audece ad Level Ths ex focuses o models ad daa ha arse from repeaed measuremes ake from a cross-seco of subjecs. These models ad daa have foud subsave applcaos ma dscples wh he bologcal ad socal sceces. The breadh ad scope of applcaos appears o be creasg over me. However hs wdespread eres has spawed a hodgepodge of erms; ma dffere erms are used o descrbe he same cocep. To llusrae eve he subjec le akes o dffere meags dffere leraures; somemes hs opc s referred o as logudal daa ad somemes as pael daa. To welcome readers from a vare of dscples I use he cumbersome e more clusve descrpor logudal ad pael daa. Ths ex s prmarl oreed o applcaos he socal sceces. Thus he daa ses cosdered here are from dffere areas of socal scece cludg busess ecoomcs educao ad socolog. The mehods roduced o ex are oreed owards hadlg observaoal daa coras o daa arsg from expermeal suaos ha are he orm he bologcal sceces. Eve wh hs socal scece oreao oe of m goals wrg hs ex s o roduce mehodolog ha has bee developed he sascal ad bologcal sceces as well as he socal sceces. Tha s mpora mehodologcal corbuos have bee made each of hese areas; m goal s o shesze he resuls ha are mpora for aalzg socal scece daa regardless of her orgs. Because ma erms ad oaos ha appear hs book are also foud he bologcal sceces where pael daa aalss s kow as logudal daa aalss hs book ma also appeal o researchers eresed he bologcal sceces. Despe s for-ear hsor ad wdespread usage a surve of he leraure shows ha he qual of applcaos s ueve. Perhaps hs s because logudal ad pael daa aalss has developed separae felds of qur; wha s wdel kow ad acceped oe feld s gve lle promece a relaed feld. To provde a reame ha s accessble o researchers from a vare of dscples hs ex roduces he subjec usg relavel sophscaed quaave ools cludg regresso ad lear model heor. Kowledge of calculus as well as marx algebra s also assumed. For Chaper 8 o damc models a me seres course would also be useful. Wh hs level of prerequse mahemacs ad sascs I hope ha he ex s accessble o quaavel oreed graduae socal scece sudes who are m prmar audece. To help sudes work hrough he maeral he ex feaures several aalcal ad emprcal exercses. Moreover dealed appedces o dffere mahemacal ad sascal supporg opcs should help sudes develop her kowledge of he opc as he work he exercses. I also hope ha he exbook sle such as he boxed procedures ad a orgazed se of smbols ad oao wll appeal o appled researchers ha would lke a referece ex o logudal ad pael daa modelg. Orgazao The begg chaper ses he sage for he book. Chaper roduces logudal ad pael daa as repeaed observaos from a subjec ad ces examples from ma dscples whch logudal daa aalss s used. Ths chaper oules mpora beefs of logudal daa aalss cludg he abl o hadle he heerogee ad damc feaures of he daa. The chaper also ackowledges some mpora drawbacks of hs scefc mehodolog parcularl he problem of aro. Furhermore Chaper provdes a overvew of he several pes of models used o hadle logudal daa; hese models are cosdered greaer deal

subseque chapers. Ths chaper should be read a he begg ad ed of oe s roduco o logudal daa aalss. Whe dscussg heerogee he coex of logudal daa aalss we mea ha observaos from dffere subjecs ed o be dssmlar whe compared o observaos from he same subjec ha ed o be smlar. Oe wa of modelg heerogee s o use fxed parameers ha var b dvdual; hs formulao s kow as a fxed effecs model ad s descrbed Chaper. A useful pedagogc feaure of fxed effecs models s ha he ca be roduced usg sadard lear model heor. Lear model ad regresso heor s wdel kow amog research aalss; wh hs sold foudao fxed effecs models provde a desrable foudao for roducg logudal daa models. Ths ex s wre assumg ha readers are famlar wh lear model ad regresso heor a he level of for example Draper ad Smh 995 or Greee 993. Chaper provdes a overvew of lear models wh a heav emphass o aalss of covarace echques ha are useful for logudal ad pael daa aalss. Moreover he Chaper fxed effecs models provde a sold framework for roducg ma graphcal ad dagosc echques. Aoher wa of modelg heerogee s o use parameers ha var b dvdual e ha are represeed as radom quaes; hese quaes are kow as radom effecs ad are descrbed Chaper 3. Because models wh radom effecs geerall clude fxed effecs o accou for he mea models ha corporae boh fxed ad radom quaes are kow as lear mxed effecs models. Jus as a fxed effecs model ca be hough of he lear model coex a lear mxed effecs model ca be expressed as a specal case of he mxed lear model. Because mxed lear model heor s o as wdel kow as regresso Chaper 3 provdes more deals o he esmao ad oher fereal aspecs ha he correspodg developme Chaper. Sll he good ews for appled researchers s ha b wrg lear mxed effecs models as mxed lear models wdel avalable sascal sofware ca be used o aalze lear mxed effecs models. B appealg o lear model ad mxed lear model heor Chapers ad 3 we wll be able o hadle ma applcaos of logudal ad pael daa models. Sll he specal srucure of logudal daa rases addoal ferece quesos ad ssues ha are o commol addressed he sadard roducos o lear model ad mxed lear model heor. Oe such se of quesos deals wh he problem of esmag radom quaes kow as predco. Chaper 4 roduces he predco problem he logudal daa coex ad shows how o esmae resduals codoal meas ad fuure values of a process. Chaper 4 also shows how o use Baesa ferece as a alerave mehod for predco. To provde addoal movao ad uo for Chapers 3 ad 4 Chaper 5 roduces mullevel modelg. Mullevel models are wdel used educaoal sceces ad developmeal pscholog where oe assumes ha complex ssems ca be modeled herarchcall; ha s modelg oe level a a me each level codoal o lower levels. Ma mullevel models ca be wre as lear mxed effecs models; hus he ferece properes of esmao ad predco ha we develop Chapers 3 ad 4 ca be appled drecl o he Chaper 5 mullevel models. Chaper 6 reurs o he basc lear mxed effecs model bu ow adops a ecoomerc perspecve. I parcular hs chaper cosders suaos where he explaaor varables are sochasc ad ma be flueced b he respose varable. I such crcumsaces he explaaor varables are kow as edogeous. Dffcules assocaed wh edogeous explaaor varables ad mehods for addressg hese dffcules are well kow for cross-secoal daa. Because o all readers wll be famlar wh he releva ecoomerc leraure Chaper 6 revews hese dffcules ad mehods. Moreover Chaper 6 descrbes he more rece leraure o smlar suaos for logudal daa. Chaper 7 aalzes several ssues ha are specfc o a logudal or pael daa sud. Oe ssue s he choce of he represeao o model heerogee. The ma choces clude

fxed effecs radom effecs ad seral correlao models. Chaper 7. revews mpora defcao ssues whe rg o decde upo he approprae model for heerogee. Oe ssue s he comparso of fxed ad radom effecs models ha has receved subsaal aeo he ecoomercs leraure. As descrbed Chaper 7 hs comparso volves eresg dscussos of he omed varables problem. Brefl we wll see ha me-vara omed varables ca be capured hrough he parameers used o represe heerogee hus hadlg wo problems a he same me. Chaper 7 cocludes wh a dscusso of samplg ad selecv bas. Pael daa surves wh repeaed observaos o a subjec are parcularl suscepble o a pe of selecv problem kow as aro where dvduals leave a pael surve. Logudal ad pael daa applcaos are pcall log he cross-seco ad shor he me dmeso. Hece he developme of hese mehods sem prmarl from regresso-pe mehodologes such as lear model ad mxed lear model heor. Chapers ad 3 roduce some damc aspecs such as seral correlao where he prmar movao s o provde mproved parameer esmaors. For ma mpora applcaos he damc aspec s he prmar focus o a acllar cosderao. Furher for some daa ses he emporal dmeso s log hus provdg opporues o model he damc aspec deal. For hese suaos logudal daa mehods are closer spr o mulvarae me seres aalss ha o cross-secoal regresso aalss. Chaper 8 roduces damc models where he me dmeso s of prmar mporace. Chapers hrough 8 are devoed o aalzg daa ha ma be represeed usg models ha are lear he parameers cludg lear ad mxed lear models. I coras Chapers 9 hrough are devoed o aalzg daa ha ca be represeed usg olear models. The colleco of olear models s vas. To provde a coceraed dscusso ha relaes o he applcaos oreao of hs book we focus o models where he dsrbuo of he respose cao be reasoabl approxmaed b a ormal dsrbuo ad alerave dsrbuos mus be cosdered. We beg Chaper 9 wh a dscusso of modelg resposes ha are dchoomous; we call hese bar depede varable models. Because o all readers wh a backgroud regresso heor have bee exposed o bar depede models such as logsc regresso Chaper 9 begs wh a roducor seco uder he headg of homogeeous models; hese are smpl he usual cross-secoal models whou heerogee parameers. The Chaper 9 roduces he ssues assocaed wh radom ad fxed effecs models o accommodae he heerogee. Uforuael radom effecs model esmaors are dffcul o compue ad he usual fxed effecs model esmaors have udesrable properes. Thus Chaper 9 roduces a alerave modelg sraeg ha s wdel used bologcal sceces based o a so-called margal model. Ths model emplos geeralzed esmag equao GEE or geeralzed mehod of momes GMM esmaors ha are smple o compue ad have desrable properes. Chaper 0 exeds ha Chaper 9 dscusso o geeralzed lear models GLMs. Ths class of models hadles he ormal-based models of Chaper hrough 8 he bar models of Chaper 9 as well as addoal mpora appled models. Chaper 0 focuses o cou daa hrough he Posso dsrbuo alhough he geeral argumes ca also be used for oher dsrbuos. Lke Chaper 9 we beg wh he homogeeous case o provde a revew for readers ha have o bee roduced o GLM. The ex seco s o margal models ha are parcularl useful for applcaos. Chaper 0 follows wh a roduco o radom ad fxed effecs models. Usg he Posso dsrbuo as a bass Chaper exeds he dscusso o mulomal models. These models are parcularl useful ecoomc choce models ha have see broad applcaos he markeg research leraures. Chaper provdes a bref overvew of he ecoomc bass for hese choce models ad he shows how o appl hese o radom effecs mulomal models.

Sascal Sofware M goal wrg hs ex s o reach a broad group of researchers. Thus o avod excludg large segmes of dvduals I have chose o o egrae a specfc sascal sofware package o he ex. Noeheless because of he applcaos oreao s crcal ha he mehodolog preseed be easl accomplshed usg readl avalable packages. For he course augh a he Uvers of Wscos I use he sascal package SAS. Alhough ma of m sudes op o use alerave packages such as STATA ad R. I ecourage free choce! I m md hs s he aalog of a exsece heorem. If a procedure s mpora ad ca be readl accomplshed b oe package he s or wll soo be avalable hrough s compeors. O he book web se hp://research.bus.wsc.edu/jfrees/book/pdaabook.hm users wll fd roues wre SAS for he mehods advocaed he ex hus provg ha he are readl avalable o appled researchers. Roues wre for STATA ad R are also avalable o he web se. For more formao o SAS STATA ad R vs her web ses: hp://www.sas.com hp://www.saa.com hp://www.r-projec.org Refereces Codes I keepg wh m goal of reachg a broad group of researchers I have aemped o egrae corbuos from dffere felds ha regularl sud logudal ad pael daa echques. To hs ed Appedx G coas he refereces ha are subdvded o sx secos. Ths subdvso s maaed o emphasze he breadh of logudal ad pael daa aalss ad he mpac ha has made o several scefc felds. I refer o hese secos usg he followg codg scheme: B Bologcal Sceces Logudal Daa E Ecoomercs Pael Daa EP Educaoal Scece ad Pscholog O Oher Socal Sceces S Sascal Logudal Daa G Geeral Sascs For example I use Nema ad Sco 948E o refer o a arcle wre b Nema ad Sco publshed 948 ha appears he Ecoomercs Pael Daa poro of he refereces. Approach Ths book grew ou of lecure oes for a course offered a he Uvers of Wscos. The pedagogc approach of he mauscrp evolved from he course. Each chaper cosss of a roduco o he ma deas words ad he as mahemacal expressos. The coceps uderlg he mahemacal expressos are he reforced wh emprcal examples; hese daa are avalable o he reader a he Wscos book web se. Mos chapers coclude wh exercses ha are prmarl aalc; some are desged o reforce basc coceps for mahemacall ovce readers. Ohers are desged for mahemacall sophscaed readers ad cosue exesos of he heor preseed he ma bod of he ex. The begg chapers -5 also clude emprcal exercses ha allow readers o develop her daa aalss sklls a logudal daa coex. Seleced soluos o he exercses are also avalable from he auhor. Readers wll fd ha he ex becomes more mahemacall challegg as progresses. Chapers 3 descrbe he fudameals of logudal daa aalss ad are prerequses for he remader of he ex. Chaper 4 s prerequse readg for Chapers 5 ad 8. Chaper 6 coas

mpora elemes ecessar for readg Chaper 7. As descrbed above a me seres aalss course would also be useful for maserg Chaper 8 parcularl he Seco 8.5 Kalma fler approach. Chaper 9 begs he seco o olear modelg. Ol Chapers -3 are ecessar backgroud for he seco. However because deals wh olear models he requse level of mahemacal sascs s hgher ha Chapers -3. Chapers 0 ad coue he developme of hese models. I do o assume pror backgroud o olear models. Thus Chapers 9- he frs seco roduces he chaper opc a o-logudal coex ha I call a homogeeous model. Despe he emphass placed o applcaos ad erpreaos I have o shed from usg mahemacs o express he deals of logudal ad pael daa models. There are ma sudes wh excelle rag mahemacs ad sascs ha eed o see he foudaos of logudal ad pael daa models. Furher here are ow a umber of exs ad summar arcles ha are ow avalable ad ced hroughou he ex ha place a heaver emphass o applcaos. However applcaos-oreed exs ed o be feld-specfc; sudg ol from such a source ca mea ha a ecoomcs sude wll be uaware of mpora developmes educaoal sceces ad vce versa. M hope s ha ma srucors wll chose o use hs ex as a echcal suppleme o a applcaos-oreed ex from her ow feld. The sudes m course come from he wde vare of backgrouds mahemacal sascs. To develop logudal ad pael daa aalss ools ad acheve a commo se of oao mos chapers coa a shor appedx ha develops mahemacal resuls ced he chaper. Furher here are four appedces a he ed of he ex ha expad mahemacal developmes used hroughou he ex. A ffh appedx o smbols ad oao furher summarzes he se of oao used hroughou he ex. The sxh appedx provdes a bref descrpo of seleced logudal ad pael daa ses ha are used several dscples hroughou he world. Ackowledgemes Ths ex was revewed b several geeraos of logudal ad pael daa classes here a he Uvers of Wscos. The sudes m classes corbued a remedous amou of pu o he ex; her pu drove he ex s developme far more ha he realze. I have ejoed workg wh several colleagues o logudal ad pael daa problems over he ears. Ther corbuos are refleced drecl hroughou he ex. Moreover I have beefed from dealed revews b: Aocha Arborg Mousum Baerjee Jee-Seo Km Yueh- Chua Kug ad Georgos Pels. Savg he mos mpora for las I hak m faml for her suppor. Te housad haks o m moher Mar m wfe Derdre our sos Naha ad Adam ad our source of amuseme Luck our dog.

Chaper. Iroduco / - 003 b Edward W. Frees. All rghs reserved Chaper. Iroduco Absrac. Ths chaper roduces he ma ke feaures of he daa ad models used he aalss of logudal ad pael daa. Here logudal ad pael daa are defed ad a dcao of her wdespread usage s gve. The chaper dscusses he beefs of hese daa; hese clude opporues o sud damc relaoshps whle udersadg or a leas accoug for cross-secoal heerogee. Desgg a logudal sud does o come whou a prce; parcular logudal daa sudes are sesve o he problem of aro ha s uplaed ex from a sud. Ths book focuses o models ha are approprae for he aalss of logudal ad pael daa; hs roducor chaper oules he se of models ha wll be cosdered subseque chapers.. Wha are logudal ad pael daa? Sascal modelg Sascs s abou daa. I s he dscple cocered wh he colleco summarzao ad aalss of daa o make saemes abou our world. Whe aalss collec daa he are reall collecg formao ha s quafed ha s rasformed o a umercal scale. There are ma well-udersood rules for reducg daa usg eher umercal or graphcal summar measures. These summar measures ca he be lked o a heorecal represeao or model of he daa. Wh a model ha s calbraed b daa saemes abou he world ca be made. As users we def a basc e ha we measure b collecg formao o a umercal scale. Ths basc e s our u of aalss also kow as he research u or observaoal u. I he socal sceces he u of aalss s pcall a perso frm or govermeal u alhough oher applcaos ca ad do arse. Oher erms used for he observaoal u clude dvdual from he ecoomercs leraure as well as subjec from he bosascs leraure. Regresso aalss ad me seres aalss are wo mpora appled sascal mehods used o aalze daa. Regresso aalss s a specal pe of mulvarae aalss where several measuremes are ake from each subjec. We def oe measureme as a respose or depede varable; he eres s makg saemes abou hs measureme corollg for he oher varables. Wh regresso aalss s cusomar o aalze daa from a cross-seco of subjecs. I coras wh me seres aalss we def oe or more subjecs ad observe hem over me. Ths allows us o sud relaoshps over me he so-called damc aspec of a problem. To emplo me seres mehods we geerall resrc ourselves o a lmed umber of subjecs ha have ma observaos over me. Defg logudal ad pael daa Logudal daa aalss represes a marrage of regresso ad me seres aalss. As wh ma regresso daa ses logudal daa are composed of a cross-seco of subjecs. Ulke regresso daa wh logudal daa we observe subjecs over me. Ulke me seres

- / Chaper. Iroduco daa wh logudal daa we observe ma subjecs. Observg a broad cross-seco of subjecs over me allows us o sud damc as well as cross-secoal aspecs of a problem. The descrpor pael daa comes from surves of dvduals. I hs coex a pael s a group of dvduals surveed repeaedl over me. Hsorcall pael daa mehodolog wh ecoomcs had bee largel developed hrough labor ecoomcs applcaos. Now ecoomc applcaos of pael daa mehods are o cofed o surve or labor ecoomcs problems ad he erpreao of he descrpor pael aalss s much broader. Hece we wll use he erms logudal daa ad pael daa erchageabl alhough for smplc we ofe use ol he former erm. Example. - Dvorce raes Fgure. shows he 965 dvorce raes versus AFDC Ad o Famles wh Depede Chldre pames for he ff saes. For hs example each sae represes a observaoal u he dvorce rae s he respose of eres ad he level of AFDC pame represes a varable ha ma corbue formao o our udersadg of dvorce raes. The daa are observaoal; hus s o approprae o argue for a causal relaoshp bewee welfare pames AFDC ad dvorce raes whou addoal ecoomc or socologcal heor. Noeheless her relao s mpora o labor ecoomss ad polcmakers. Fgure. shows a egave relao; he correspodg correlao coeffce s -0.37. Some argue ha hs egave relao s couer-uve ha oe would expec a posve relao bewee welfare pames ad dvorce raes; saes wh desrable ecoomc clmaes ejo boh a low dvorce rae ad low welfare pames. Ohers argue ha hs egave relaoshp s uvel plausble; wealh saes ca afford hgh welfare pames ad produce a culural ad ecoomc clmae coducve o low dvorce raes. DIVORCE 6 5 4 3 0 0 40 60 80 00 0 40 60 80 00 0 Fgure.. Plo of 965 Dvorce versus AFDC Pames Source: US Sascal Absracs Aoher plo o dsplaed here shows a smlar egave relao for 975; he correspodg correlao s -0.45. Furher a plo wh boh he 965 ad 975 daa dsplas a egave relao bewee dvorce raes ad AFDC pames. AFDC

Chaper. Iroduco / -3 DIVORCE 0 8 6 4 0 0 00 00 300 400 AFDC Fgure.. Plo of Dvorce versus AFDC Pames 965 ad 975 Fgure. shows boh he 965 ad 975 daa; a le coecs he wo observaos wh each sae. The le represes a chage over me damc o a cross-secoal relaoshp. Each le dsplas a posve relaoshp ha s as welfare pames crease so do dvorce raes for each sae. Aga we do o fer drecos of causal from hs dspla. The po s ha he damc relao bewee dvorce ad welfare pames wh a sae dffers dramacall from he cross-secoal relaoshp bewee saes. Some oao Models of logudal daa are somemes dffereaed from regresso ad me seres hrough her double subscrps. Wh hs oao we ma dsgush amog resposes b subjec ad me. To hs ed defe o be he respose for he h subjec durg he h me perod. A logudal daa se cosss of observaos of he h subjec over... T me perods for each of... subjecs. Thus we observe: K frs subjec - { T } secod subjec - { K } T...... K. h subjec - { } I Example. mos saes have T observaos ad are depced graphcall Fgure. b a le coecg he wo observaos. Some saes have ol T observao ad are depced graphcall b a ope crcle plog smbol. For ma daa ses s useful o le he umber of observaos deped o he subjec; T deoes he umber of observaos for he h subjec. Ths suao s kow as he ubalaced daa case. I oher daa ses each subjec has he same umber of observaos; hs s kow as he balaced daa case. Tradoall much of he ecoomercs leraure has focused o he balaced daa case. We wll cosder he more broadl applcable ubalaced daa case. T

-4 / Chaper. Iroduco Prevalece of logudal ad pael daa aalss Logudal ad pael daabases ad models have ake a mpora role he leraure. The are wdel used he socal scece leraure where pael daa are also kow as pooled cross-secoal me seres ad he aural sceces where pael daa are referred o as logudal daa. To llusrae a dex of busess ad ecoomc jourals ABI/INFORM lss 70 arcles 00 ad 00 ha use pael daa mehods. Aoher dex of scefc jourals he ISI Web of Scece lss 8 arcles 00 ad 00 ha use logudal daa mehods. Ad hese are ol he applcaos ha were cosdered ovave eough o be publshed scholarl revews! Logudal daa mehods have also developed because mpora daabases have become avalable o emprcal researchers. Wh ecoomcs wo mpora surves ha rack dvduals over repeaed surves clude he Pael Surve of Icome Damcs PSID ad he Naoal Logudal Surve of Labor Marke Experece NLS. I coras he Cosumer Prce Surve CPS s aoher surve coduced repeaedl over me. However he CPS s geerall o regarded as a pael surve because dvduals are o racked over me. For sudg frm-level behavor daabases such as Compusa ad CRSP Uvers of Chcago s Ceer for Research o Secur Prces have bee avalable for over hr ears. More recel he Naoal Assocao of Isurace Commssoers NAIC has made surace compa facal saemes avalable elecrocall. Wh he rapd pace of sofware developme wh he daabase dusr s eas o acpae he developme of ma more daabases ha would beef from logudal daa aalss. To llusrae wh he markeg area produc codes are scaed whe cusomers check ou of a sore ad are rasferred o a ceral daabase. These so-called scaer daa represe e aoher source of daa formao ha ma ell markeg researchers abou purchasg decsos of buers over me or he effcec of a sore s promooal effors. Appedx F summarzes logudal ad pael daa ses used worldwde.. Beefs ad drawbacks of logudal daa There are several advaages of logudal daa compared wh eher purel crosssecoal or purel me seres daa. I hs roducor chaper we focus o wo mpora advaages: he abl o sud damc relaoshps ad o model he dffereces or heerogee amog subjecs. Of course logudal daa are more complex ha purel crosssecoal or mes seres daa ad so here s a prce workg wh hem. The mos mpora drawback s he dffcul desgg he samplg scheme o reduce he problem of subjecs leavg he sud pror o s compleo kow as aro. Damc relaoshps Fgure. shows he 965 dvorce rae versus welfare pames. Because hese are daa from a sgle po me he are sad o represe a sac relaoshp. To llusrae we mgh summarze he daa b fg a le usg he mehod of leas squares. Ierpreg he slope of hs le we esmae a decrease of 0.95% dvorce raes for each $00 crease AFDC pames. I coras Fgure. shows chages dvorce raes for each sae based o chages welfare pames from 965 o 975. Usg leas squares he overall slope represes a crease of.9% dvorce raes for each $00 crease AFDC pames. From 965 o 975 welfare pames creased a average of $59 omal erms ad dvorce raes creased.5%. Now he slope represes a pcal me chage dvorce raes per $00 u me chage welfare pames; hece represes a damc relaoshp. Perhaps he example mgh be more ecoomcall meagful f welfare pames were real dollars ad perhaps o for example deflaed b he Cosumer Prce Idex.

Chaper. Iroduco / -5 Noeheless he daa srogl reforce he oo ha damc relaos ca provde a ver dffere message ha cross-secoal relaos. Damc relaoshps ca ol be suded wh repeaed observaos ad we have o hk carefull abou how we defe our subjec whe cosderg damcs. To llusrae suppose ha we are lookg a he eve of dvorce o dvduals. B lookg a a cross-seco of dvduals we ca esmae dvorce raes. B lookg a cross-secos repeaed over me whou rackg dvduals we ca esmae dvorce raes over me ad hus sud hs pe of damc moveme. However ol b rackg repeaed observaos o a sample of dvduals ca we sud he durao of marrage or me ul dvorce aoher damc eve of eres. Hsorcal approach Earl pael daa sudes used he followg sraeg o aalze pooled cross-secoal daa: Esmae cross-secoal parameers usg regresso. Use me seres mehods o model he regresso parameer esmaors reag esmaors as kow wh cera. Alhough useful some coexs hs approach s adequae ohers such as Example.. Here he slope esmaed from 965 daa s 0.95%. Smlarl he slope esmaed from 975 daa urs ou o be.0%. Exrapolag hese egave esmaors from dffere cross-secos elds ver dffere resuls from he damc esmae a posve.9%. Thel ad Goldberger 96E provde a earl dscusso of he advaages of esmag he cross-secoal ad me seres aspecs smulaeousl. Damc relaoshps ad me seres aalss Whe sudg damc relaoshps uvarae me seres aalss s a well-developed mehodolog. However hs mehodolog does o accou for relaoshps amog dffere subjecs. I coras mulvarae me seres aalss does accou for relaoshps amog a lmed umber of dffere subjecs. Wheher uvarae or mulvarae a mpora lmao of me seres aalss s ha requres several geerall a leas hr observaos o make relable fereces. For a aual ecoomc seres wh hr observaos usg me seres aalss meas ha we are usg he same model o represe a ecoomc ssem over a perod of hr ears. Ma problems of eres lack hs degree of sabl; we would lke alerave sascal mehodologes ha do o mpose such srog assumpos. Logudal daa as repeaed me seres Wh logudal daa we use several repeaed observaos of ma subjecs over dffere me perods. Repeaed observaos from he same subjec ed o be correlaed. Oe wa o represe hs correlao s hrough damc paers. A model ha we use s: E + ε... T..... Here ε represes he devao of he respose from s mea; hs devao ma clude damc paers. Iuvel f here s a damc paer ha s commo amog subjecs he b observg hs paer over ma subjecs we hope o esmae he paer wh fewer me seres observaos ha requred of coveoal me seres mehods. For ma daa ses of eres subjecs do o have decal meas. As a frs order approxmao a lear combao of kow explaaor varables such as E α + x β serves as a useful specfcao of he mea fuco. Here x s a vecor of explaaor or depede varables.

-6 / Chaper. Iroduco Logudal daa as repeaed cross-secoal sudes Logudal daa ma be reaed as a repeaed cross-seco b gorg he formao abou dvduals ha s racked over me. As meoed above here are ma mpora repeaed surves such as he CPS where subjecs are o racked over me. Such surves are useful for udersadg aggregae chages a varable such as he dvorce rae over me. However f he eres s sudg he me-varg effecs of ecoomc demographc or socologcal characerscs of a dvdual o dvorce he rackg dvduals over me s much more formave ha a repeaed cross-seco. Heerogee B rackg subjecs over me we ma model subjec behavor. I ma daa ses of eres subjecs are ulke oe aoher ha s he are heerogeeous. I repeaed crosssecoal regresso aalss we use models such as α + x β + ε ad ascrbe he uqueess of subjecs o he dsurbace erm ε. I coras wh logudal daa we have a opporu o model hs uqueess. A basc logudal daa model ha corporaes heerogee amog subjecs s based o E α + x β... T..... I cross-secoal sudes where T he parameers of hs model are udefable. However logudal daa we have a suffce umber of observaos o esmae β ad α... α. Allowg for subjec-specfc parameers such as α provdes a mpora mechasm for corollg heerogee of dvduals. Models ha corporae heerogee erms such as equao. wll be called heerogeeous models. Models whou such erms wll be called homogeeous models. We ma also erpre heerogee o mea ha observaos from he same subjec ed o be smlar compared o observaos from dffere subjecs. Based o hs erpreao heerogee ca be modeled b examg he sources of correlao amog repeaed observaos from a subjec. Tha s for ma daa ses we acpae fdg a posve correlao whe examg {... T }. As oed above oe possble explaao s he damc paer amog he observaos. Aoher possble explaao s ha he respose shares a commo e uobserved subjec-specfc parameer ha duces a posve correlao. There are wo dsc approaches for modelg he quaes ha represe heerogee amog subjecs {α }. Chaper explores oe approach where {α } are reaed as fxed e ukow parameers o be esmaed. I hs case equao. s kow as a fxed effecs model. Chaper 3 roduces he secod approach where {α } are reaed as ex-ae draws from a ukow populao ad hus are radom varables. I hs case equao. ma be expressed as E α α + x β. Ths s kow as a radom effecs formulao. Heerogee bas Falure o clude heerogee quaes he model ma roduce serous bas o he model esmaors. To llusrae suppose ha a daa aals msakel uses he fuco E α + x β whe equao. s he rue fuco. Ths s a example of heerogee bas or a problem wh aggregao wh daa. Smlarl oe could have dffere heerogeeous slopes

Chaper. Iroduco / -7 or dffere erceps ad slopes E α + x β E α + x β. Omed varables Icorporag heerogee quaes o logudal daa models are ofe movaed b he cocer ha mpora varables have bee omed from he model. To llusrae cosder he rue model α + x β + z γ + ε. Assume ha we do o have avalable he varables represeed b he vecor z ; hese omed varables are also sad o be lurkg. If hese omed varables do o deped o me he s sll possble o ge relable esmaors of oher model parameers such as hose cluded he vecor β. Oe sraeg s o cosder he devaos of a respose from s me seres average. Ths elds he derved model: * - α + x β + z γ + ε - α + x β + z γ + ε * x - x β + ε - ε x β + ε *. T ad smlarl for x adε. Here we use he respose me seres average T Thus ordar leas square esmaors based o regressg he devaos x o he devaos elds a desrable esmaor of β. Ths sraeg demosraes how logudal daa ca mgae he problem of omed varable bas. For sraeges ha rel o purel cross-secoal daa s well kow ha correlaos of lurkg varables z wh he model explaaor varables x duce bas whe esmag β. If he lurkg varable s me-vara he s perfecl collear wh he subjec-specfc varables α. Thus esmao sraeges ha accou for subjecs-specfc parameers also accou for me-vara omed varables. Furher because of he collear bewee subjec-specfc varables ad me-vara omed varables we ma erpre he subjec-specfc quaes α as proxes for omed varables. Chaper 7 descrbes sraeges for dealg wh omed varable bas. Effcec of esmaors A logudal daa desg ma eld more effce esmaors ha esmaors based o a comparable amou of daa from alerave desgs. To llusrae suppose ha he eres s assessg he average chage a respose over me such as he dvorce rae. Thus le deoe he dfferece bewee dvorce raes bewee wo me perods. I a repeaed cross-secoal sud such as he CPS we would calculae he relabl of hs sasc assumg depedece amog cross-secos o ge Var Var + Var. However a pael surve ha racks dvduals over me we have Var Var + Var Cov. The covarace erm s geerall posve because observaos from he same subjec ed o be posvel correlaed. Thus oher hgs beg equal a pael surve desg elds more effce esmaors ha a repeaed cross-seco desg. Oe mehod of accoug for hs posve correlao amog same-subjec observaos s hrough he heerogee erms α. I ma daa ses roducg subjec-specfc varables α

-8 / Chaper. Iroduco also accous for a large poro of he varabl. Accoug for hs varao reduces he mea square error ad sadard errors assocaed wh parameer esmaors. Thus we are more effce parameer esmao ha he case whou subjec-specfc varables α. I s also possble o corporae subjec-vara parameers ofe deoed b λ o accou for perod emporal varao. For ma daa ses hs does o accou for he same amou of varabl as {α }. Wh small umbers of me perods s sraghforward o use me dumm bar varables o corporae subjec-vara parameers. Oher hgs equal sadard errors become smaller ad effcec mproves as he umber of observaos creases. For some suaos a researcher ma oba more formao b samplg each subjec repeaedl. Thus some advocae ha a advaage of logudal daa s ha we geerall have more observaos due o he repeaed samplg ad greaer effcec of esmaors compared o a purel cross-secoal regresso desg. The dager of hs phlosoph s ha geerall observaos from he same subjec are relaed. Thus alhough more formao s obaed b repeaed samplg researchers eed o be cauous assessg he amou of addoal formao gaed. Correlao ad causao For ma sascal sudes aalss are happ o descrbe assocaos amog varables. Ths s parcularl rue of forecasg sudes where he goal s o predc he fuure. However for oher aalses researchers are eresed assessg causal relaoshps amog varables. Logudal ad pael daa are somemes oued as provdg evdece of causal effecs. Jus as wh a sascal mehodolog logudal daa models ad of hemselves are o eough o esablsh causal relaoshps amog varables. However logudal daa ca be more useful ha purel cross-secoal daa esablshg causal. To llusrae cosder he hree gredes ecessar for esablshg causal ake from he socolog leraure see for example Too 000: A sascall sgfca relaoshp s requred. The assocao bewee wo varables mus o be due o aoher omed varable. The causal varable mus precede he oher varable me. Logudal daa are based o measuremes ake over me ad hus address he hrd requreme of a emporal orderg of eves. Moreover as descrbed above logudal daa models provde addoal sraeges for accommodag omed varables ha are o avalable purel cross-secoal daa. Observaoal daa are o from carefull corolled expermes where radom allocaos are made amog groups. Causal ferece s o drecl accomplshed whe usg observaoal daa ad ol sascal models. Raher oe hks abou he daa ad sascal models as provdg releva emprcal evdece a cha of reasog abou causal mechasms. Alhough logudal daa provde sroger evdece ha purel cross-secoal daa mos of he work esablshg causal saemes should be based o he heor of he subsave feld from whch he daa are derved. Chaper 6 dscusses hs ssue greaer deal. Drawbacks: Aro Logudal daa samplg desg offers ma beefs compared o purel crosssecoal or purel me-seres desgs. However because he samplg srucure s more complex ca also fal suble was. The mos commo falure of logudal daa ses o mee sadard samplg desg assumpos s hrough dffcules ha resul from aro. I hs coex aro refers o a gradual eroso of resposes b subjecs. Because we follow he same subjecs over me orespose pcall creases hrough me. To llusrae cosder he

Chaper. Iroduco / -9 US Pael Sud of Icome Damcs PSID. I he frs ear 968 he orespose rae was 4%. However b 985 he orespose rae grew o abou 50%. Aro ca be a problem because ma resul a seleco bas. Seleco bas poeall occurs whe a rule oher ha smple radom or srafed samplg s used o selec observaoal us. Examples of seleco bas ofe cocer edogeous decsos b ages o jo a labor pool or parcpae a socal program. To llusrae suppose ha we are sudg a solvec measure of a sample of surace frms. If he frm becomes bakrup or evolves o aoher pe of facal dsress he we ma o be able o exame facal sascs assocaed wh he frm. Noeheless hs s exacl he suao whch we would acpae observg low values of he solvec measure. The respose of eres s relaed o our opporu o observe he subjec a pe of seleco bas. Chaper 7 dscusses he aro problem greaer deal..3 Logudal daa models Whe examg he beefs ad drawbacks of logudal daa modelg s also useful o cosder he pes of ferece ha are based o logudal daa models as well as he vare of modelg approaches. The pe of applcao uder cosderao flueces he choce of ferece ad modelg approaches. Tpes of ferece For ma logudal daa applcaos he prmar movao for he aalss s o lear abou he effec ha a exogeous explaaor varable has o a respose corollg for oher varables cludg omed varables. Users are eresed wheher esmaors of parameer coeffces coaed he vecor β dffer a sascall sgfca fasho from zero. Ths s also he prmar movao for mos sudes ha volve regresso aalss; hs s o surprsg gve ha ma models of logudal daa are specal cases of regresso models. Because logudal daa are colleced over me he also provde us wh a abl o predc fuure values of a respose for a specfc subjec. Chaper 4 cosders hs pe of ferece kow as forecasg. The focus of Chaper 4 s o he esmao of radom varables kow as predco. Because fuure values of a respose are o he aals radom varables forecasg s a specal case of predco. Aoher specal case volves suaos where we would lke o predc he expeced value of a fuure respose from a specfc subjec codoal o lae uobserved characerscs assocaed wh he subjec. For example hs codoal expeced value s kow surace heor as a credbl premum a qua ha s useful prcg of surace coracs. Socal scece sascal modelg Sascal models are mahemacal dealzaos cosruced o represe he behavor of daa. Whe a sascal model s cosruced desged o represe a daa se wh lle regard o he uderlg fucoal feld from whch he daa emaaes we ma hk of he model as esseall daa drve. For example we mgh exame a daa se of he form x x ad pos a regresso model o capure he assocao bewee x ad. We wll call hs pe of model a samplg based model or followg he ecoomercs leraure sa ha he model arses from he daa geerag process. I mos cases however we wll kow somehg abou he us of measureme of x ad ad acpae a pe of relaoshp bewee x ad based o kowledge of he fucoal feld from whch hese varables arse. To coue our example a face coex suppose ha x represes a reur from a marke dex ad ha represes a sock reur from a dvdual

-0 / Chaper. Iroduco secur. I hs case facal ecoomcs heor suggess a lear regresso relaoshp of o x. I he ecoomcs leraure Goldberger 97E defes a srucural model o be a sascal model ha represes causal relaoshps as opposed o relaoshps ha smpl capure sascal assocaos. Chaper 6 furher develops he dea of causal ferece. If a samplg based model adequael represes sascal assocaos our daa he wh boher wh a exra laer of heor whe cosderg sascal models? I he coex of bar depede varables Mask 99E offers hree movaos: erpreao precso ad exrapolao. Ierpreao s mpora because he prmar purpose of ma sascal aalses s o assess relaoshps geeraed b heor from a scefc feld. A samplg based model ma o have suffce srucure o make hs assessme hus falg he prmar movao for he aalss. Srucural models ulze addoal formao from a uderlg fucoal feld. If hs formao s ulzed correcl he some sese he srucural model should provde a beer represeao ha a model whou hs formao. Wh a properl ulzed srucural model we acpae geg more precse esmaes of model parameers ad oher characerscs. I praccal erms hs mproved precso ca be measured erms of smaller sadard errors. A leas he coex of bar depede varables Mask 99E feels ha exrapolao s he mos compellg movao for combg heor from a fucoal feld wh a samplg based model. I a me seres coex exrapolao meas forecasg; hs s geerall he ma mpeus for a aalss. I a regresso coex exrapolao meas ferece abou resposes for ses of predcor varables ousde of hose realzed he sample. Parcularl for publc polc aalss he goal of a sascal aalss s o fer he lkel behavor of daa ousde of hose realzed. Modelg ssues Ths chaper has porraed logudal daa modelg as a specal pe of regresso modelg. However he bomercs leraure logudal daa models have her roos mulvarae aalss. Uder hs framework we vew he resposes from a dvdual as a K T. Wh he bomercs framework he frs applcaos are referred o as growh curve models. These classc examples use he hegh of chldre as he respose o exame he chages hegh ad growh over me; see Chaper 5. Wh he ecoomercs leraure Chamberla 98E 984E exploed he mulvarae srucure. The mulvarae aalss approach s mos effecve wh balaced daa a equall spaced me pos. However compared o he regresso approach here are several lmaos of he mulvarae approach. These clude: I s harder o aalze mssg daa aro ad dffere accrual paers. Because here s o explc allowace for me s harder o forecas ad predc a me pos bewee hose colleced erpolao. vecor of resposes ha s Eve wh he regresso approach for logudal daa modelg here are sll a umber of ssues ha eed o be resolved choosg a model. We have alread roduced he ssue of modelg heerogee. Recall ha here are wo mpora pes of models of heerogee fxed ad radom effecs models he subjecs of Chapers ad 3. Aoher mpora ssue s he srucure for modelg he damcs; hs s he subjec of Chaper 8. We have descrbed mposg a seral correlao o he dsurbace erms. Aoher approach descrbed Seco 8. volves usg lagged edogeous resposes o accou for emporal paers. These models are mpora ecoomercs because he are more suable for srucural modelg where here s a greaer e bewee ecoomc heor ad sascal modelg

Chaper. Iroduco / - ha models ha are based exclusvel o feaures of he daa. Whe he umber of me observaos per subjec T s small he smple correlao srucures of he dsurbaces erms provde a adequae f for ma daa ses. However as T creases we have greaer opporues o model he damc srucure. The Kalma fler descrbed Seco 8.5 provdes a compuaoal echque ha allows he aals o hadle a broad vare of complex damc paers. Ma of he logudal daa applcaos ha appear he leraure are based o lear model heor. Hece hs ex s predomal Chapers hrough 8 devoed o developg lear logudal daa models. However olear models represe a area of rece developme where examples of her mporace o sascal pracce appear wh greaer frequec. The phrase olear models hs coex refers o saces where he dsrbuo of he respose cao be reasoabl approxmaed usg a ormal curve. Some examples of hs occur whe he respose s bar or oher pes of cou daa such as he umber of accdes a sae ad whe he respose s from a ver heav aled dsrbuo such as wh surace clams. Chapers 9 hrough roduce echques from hs buddg leraure o hadle hese pes of olear models. Tpes of applcaos A sascal model s ulmael useful ol f provdes a useful approxmao o real daa. Table. oules he daa ses used hs ex o uderscore he mporace of logudal daa modelg.

- / Chaper. Iroduco Daa Tle Table.. Several Illusrave Logudal Daa Ses Subjec Fle U of Aalss Descrpo Area Name Arle Face Arle Subjecs are 9 arles over T ears: 970-980. N87 observaos. Bod Maur Face Bodma Subjecs are 38 frms over T0 ears: 980-989. N380 observaos. Capal Srucure Charable Corbuos Face Capal Subjecs are 36 Japaese frms over T5 ears: 984-998. N545 observaos. Accoug Char Subjecs are 47 axpaers over T0 ears; 979-988. N470 observaos. Dvorce Socolog Dvorce Subjecs are 5 saes over T4 ears: 965 975 985 ad 995. N04 observaos. Elecrc Ules Ecoomcs Elecrc Subjecs are 68 elecrc ules over T mohs. N86 observaos. Group Term Lfe Daa Isurace Glfe Subjecs are 06 cred uos over T7 ears. N74 observaos. Housg Prces Real esae Hprce Subjecs are 36 meropola sascal areas MSAs over T9 ears: 986-994. N34 observaos. Loer Sales Markeg Loer Subjecs are 50 posal code areas over T 40 weeks. Medcare Hospal Coss Proper ad Labl Isurace Sude Acheveme Socal Isurace Medcare Subjecs are 54 saes over T6 ears: 990-995. N34 observaos. Isurace Pdemad Subjecs are coures over T7 ears: 987-993. N54 observaos. Educao Sude Subjecs are 400 sudes from 0 schools are observed over T4 grades 3-6. N0 observaos. Tax Preparers Accoug Taxprep Subjecs are 43 axpaers over T5 ears: 98 984-988. N5 observaos. Tor Flgs Isurace Tflg Subjecs are 9 saes over T6 ears: 984-989. N4 observaos. Worker s Compesao Isurace Workerc Subjecs are occupao classes over T7 ears. N847 observaos. Exame characerscs of arles o deerme oal operag coss. Exame he maur of deb srucure erms of corporae facal characerscs. Exame chages capal srucure before ad afer he marke crash for dffere pes of cross holdg srucures. Exame characerscs of axpaers o deerme facors ha fluece he amou of charable gvg. Assess socoecoomc varables ha affec he dvorce rae. Exame he average cos of ules erms of he prce of labor fuel ad capal. Forecas group erm lfe surace clams of Florda cred uos. Exame aual housg prces erms of MSA demographc ad ecoomc dces. Exame effecs of area ecoomc ad demographc characerscs o loer sales. Forecas Medcare hospal coss b sae based o ulzao raes ad pas hsor. Exame he demad for proper ad labl surace erms of aoal ecoomc ad rsk averso characerscs. Exame sude mah acheveme based o sude ad school demographc ad socoecoomc characerscs. Exame characerscs of axpaers o deerme he demad for a professoal ax preparer. Exame demographc ad legal characerscs of saes ha fluece he umber of or flgs. Forecas worker s compesao clams b occupao class.

Chaper. Iroduco / -3.4 Hsorcal oes The erm pael sud was coed a markeg coex whe Lazarsfeld ad Fske 938O cosdered he effec of rado adversg o produc sales. Tradoall hearg rado adversemes was hough o crease he lkelhood of purchasg a produc. Lazarsfeld ad Fske cosdered wheher hose ha bough he produc would be more lkel o hear he adverseme hus posg a reverse he dreco of causal. The proposed repeaedl ervewg a se of people he pael o clarf he ssue. Bales ad Nesselroade 979EP race he hsor of logudal daa ad mehods wh a emphass o chldhood developme ad pscholog. The descrbe logudal research as cossg of a vare of mehods coeced b he dea ha he e uder vesgao s observed repeaedl as exss ad evolves over me. Moreover he race he eed for logudal research o a leas as earl as he eeeh ceur. Too 000EP ces Egel s 857 budge surve examg how he amou of moe spe o food chages as a fuco of come as perhaps he earles example of a sud volvg repeaed measuremes from he same se of subjecs. As oed Seco. earl pael daa sudes pooled cross-secoal daa were aalzed b esmag cross-secoal parameers usg regresso ad usg me seres mehods o model he regresso parameer esmaes reag he esmaes as kow wh cera. Delma 989O dscusses hs approach more deal ad provdes examples. Earl applcaos ecoomcs of he basc fxed effecs model clude Kuh 959E Johso 960E Mudlak 96E ad Hoch 96E. Chaper roduces hs ad relaed models deal. Balesra ad Nerlove 966E ad Wallace ad Hussa 969E roduced he radom effecs error compoes model he model wh {α } as radom varables. Chaper 3 roduces hs ad relaed models deal. Wshar 938B Rao 959S 965B Pohoff ad Ro 964B were amog he frs corbuos he bomercs leraure o use mulvarae aalss for aalzg growh curves. Specfcall he cosdered he problem of fg polomal growh curves of seral measuremes from a group of subjecs. Chaper 5 coas examples of growh curve aalss. Ths approach o aalzg logudal daa was exeded b Grzzle ad Alle 969B who roduced covaraes or explaaor varables o he aalss. Lard ad Ware 98B made he oher mpora raso from mulvarae aalss o regresso modelg. The roduce he wo-sage model ha allows for boh fxed ad radom effecs. Chaper 3 cosders hs modelg approach.

Chaper. Fxed Effecs Models / - Chaper. Fxed Effecs Models 003 b Edward W. Frees. All rghs reserved Absrac. Ths chaper roduces he aalss of logudal ad pael daa usg he geeral lear model framework. Here logudal daa modelg s cas as a regresso problem b usg fxed parameers o represe he heerogee; oradom quaes ha accou for he heerogee are kow as fxed effecs. I hs wa deas of model represeao ad daa explorao are roduced usg regresso aalss a oolk ha s wdel kow. Aalss of covarace from he geeral lear model easl hadles he ma parameers ha are eeded o represe he heerogee. Alhough logudal ad pael daa ca be aalzed usg regresso echques s also mpora o emphasze he specal feaures of hese daa. Specfcall he chaper emphaszes he wde cross-seco ad he shor me-seres of ma logudal ad pael daa ses as well as he specal model specfcao ad dagosc ools eeded o hadle hese feaures.. Basc fxed effecs model Daa Suppose ha we are eresed explag hospal coss for each sae erms of measures of ulzao such as he umber of dscharged paes ad he average hospal sa per dscharge. Here we cosder he sae o be he u of observao or subjec. We dffereae amog saes wh he dex where ma rage from o ad s he umber of subjecs. Each sae s observed T mes ad we use he dex o dffereae he observao mes. Wh hese dces le deoe he respose of he h subjec a he h me po. Assocaed wh each respose s a se of explaaor varables or covaraes. For example for sae hospal coss hese explaaor varables clude he umber of dscharged paes ad he average hospal sa per dscharge. I geeral we assume here are K explaaor varables x x x K ha ma var b subjec ad me. We acheve a more compac oaoal form b expressg he K explaaor varables as a K colum vecor x x x. M x K To save space s cusomar o use he alerae expresso x x x x K where he prme meas raspose. You wll fd ha some sources prefer o use a superscrp T for raspose. Here T wll refer o he umber of me replcaos. Thus he daa for he h subjec cosss of: { x L x K } x L M x { } T T K T

- / Chaper. Fxed Effecs Models ha ca be expressed more compacl as { x } M. { x } T T Uless specfed oherwse we allow he umber of resposes o var b subjec dcaed wh he oao T. Ths s kow as he ubalaced case. We use he oao T max{t T T } o be he maxmal umber of resposes for a subjec. Recall from Seco. ha he case T T for each s called he balaced case. Basc models To aalze relaoshps amog varables he relaoshps bewee he respose ad he explaaor varables are summarzed hrough he regresso fuco E α + β x + β x +... + β K x K. ha s lear he parameers α β β k. For applcaos where he explaaor varables are oradom he ol resrco of equao. s ha we beleve ha he varables eer learl. As we wll see Chaper 6 for applcaos where he explaaor varables are radom we ma erpre he expecao equao. as codoal o he observed explaaor varables. We focus aeo o assumpos ha cocer he observable varables {x... x K }. Assumpos of he Observables Represeao of he Lear Regresso Model F. E α + β x + β x +... + β K x K. F. {x... x K } are osochasc varables. F3. Var σ. F4. { } are depede radom varables. The observables represeao s based o he dea of codoal lear expecaos see Goldberger 99 for addoal backgroud. Oe ca movae assumpo F b hkg of x... x K as a draw from a populao where he mea of he codoal dsrbuo of gve { x... x K } s lear he explaaor varables. Iferece abou he dsrbuo of s codoal o he observed explaaor varables so ha we ma rea { x... x K } as osochasc varables. Whe cosderg pes of samplg mechasms for hkg of x... x K as a draw from a populao s covee o hk of a srafed radom samplg scheme where values of {x... x K } are reaed as he sraa. Tha s for each value of {x... x K } we draw a radom sample of resposes from a populao. Ths samplg scheme also provdes movao for assumpo F4 he depedece amog resposes. To llusrae whe drawg from a daabase of frms o udersad sock reur performace oe ca choose large frms measured b asse sze focus o a dusr measured b sadard dusral classfcao ad so forh. You ma o selec frms wh he larges sock reur performace because hs s srafg based o he respose o he explaaor varables. A ffh assumpo ha s ofe mplcl requred he lear regresso model s: F5. { } s ormall dsrbued. Ths assumpo s o requred for all sascal ferece procedures because ceral lm heorems provde approxmae ormal for ma sascs of eres. However formal jusfcao for some such as -sascs do requre hs addoal assumpo.

Chaper. Fxed Effecs Models / -3 I coras o he observables represeao he classcal formulao of he lear regresso model focuses aeo o he errors he regresso defed as ε α + β x + β x +... + β K x K. Assumpos of he Error Represeao of he Lear Regresso Model E. α + β x + β x +... + β K x K + ε where E ε 0. E. {x... x K } are osochasc varables. E3. Var ε σ. E4. { ε } are depede radom varables. The error represeao s based o he Gaussa heor of errors see Sgler 986 for a hsorcal backgroud. As descrbed above he lear regresso fuco corporaes he addoal kowledge from depede varables hrough he relao E α + β x + β x +... + β K x K. Oher uobserved varables ha fluece he measureme of are ecapsulaed he error erm ε whch s also kow as he dsurbace erm. The depedece of errors F4 ca be movaed b assumg ha {ε } are realzed hrough a smple radom sample from a ukow populao of errors. Assumpos E-E4 are equvale o assumpos F-F4. The error represeao provdes a useful sprgboard for movag goodess of f measures. However a drawback of he error represeao s ha draws he aeo from he observable quaes x... x K o a uobservable qua {ε }. To llusrae he samplg bass vewg {ε } as a smple radom sample s o drecl verfable because oe cao drecl observe he sample {ε }. Moreover he assumpo of addve errors E wll be roublesome whe we cosder olear regresso models Par II. Our reame focuses o he observable represeao Assumpos F-F4. I assumpo F he slope parameers β β β K are assocaed wh he K explaaor varables. For a more compac expresso we summarze he parameers as a colum vecor of dmeso K deoed b β β M. β K Wh hs oao we ma re-wre assumpo F as E α + x β. because of he relao x β β x + β x +... + β K x K. We call he represeao equao. cross-secoal because alhough relaes he explaaor varables o he respose does o use he formao he repeaed measuremes o a subjec. Because also does o clude subjec-specfc heerogeeous erms we also refer o he equao. represeao as par of a homogeeous model. Our frs represeao ha uses he formao he repeaed measuremes o a subjec s E α + x β..3 Equao.3 ad assumpos F-F4 comprse he basc fxed effecs model. Ulke equao. equao.3 he ercep erms α are allowed o var b subjec. Parameers of eres The parameers {β j } are commo o each subjec ad are called global or populao parameers. The parameers {α } var b subjec ad are kow as dvdual or subjec-specfc

-4 / Chaper. Fxed Effecs Models parameers. I ma applcaos we wll see ha populao parameers capure broad relaoshps of eres ad hece are he parameers of eres. The subjec-specfc parameers accou for he dffere feaures of subjecs o broad populao paers. Hece he are ofe of secodar eres ad are called usace parameers. As we saw Seco.3 he subjec-specfc parameers represe our frs devce ha helps corol for he heerogee amog subjecs. We wll see ha esmaors of hese parameers use formao he repeaed measuremes o a subjec. Coversel he parameers {α } are o-esmable cross-secoal regresso models whou repeaed observaos. Tha s wh T he model α + β x + β x +... + β K x K + ε has more parameers +K ha observaos ad hus we cao def all he parameers. Tpcall he dsurbace erm ε cludes he formao α cross-secoal regresso models. A mpora advaage of logudal daa models whe compared o cross-secoal regresso models s he abl o separae he effecs of {α } from he dsurbace erms { ε }. B separag ou subjec-specfc effecs our esmaes of he varabl become more precse ad we acheve more accurae fereces. Subjec ad me heerogee We wll argue ha he subjec-specfc parameer α capures much of he me-cosa formao he resposes. However he basc fxed effecs model assumes ha { } are depede erms ad parcular ha here s: o seral correlao correlao over me ad o coemporaeous correlao correlao across subjecs. Thus o specal relaoshps bewee subjecs ad me perods are assumed. B erchagg he roles of ad we ma cosder he fuco E λ + x β..4 Here he parameer λ s a me-specfc varable ha does o deped o subjecs. For mos logudal daa applcaos he umber of subjecs subsaall exceeds he maxmal umber of me perods T. Furher geerall he heerogee amog subjecs explas a greaer proporo of varabl ha he heerogee amog me perods. Thus we beg wh he basc fuco E α + x β. Ths model allows explc parameerzao of he subjec-specfc heerogee. Boh fucos equaos.3 ad.4 are based o radoal oe-wa aalss of covarace models. For hs reaso he basc fxed effecs model s also called he oe-wa fxed effecs model. B usg bar dumm varables for he me dmeso we ca corporae me-specfc parameers o he populao parameers. I hs wa s sraghforward o cosder he fuco E α + λ + x β..5 Equao.5 wh assumpos F-F4 s called he wo-wa fxed effecs model. Example. Urba wages Glaeser ad Maré 00 vesgaed he effecs of deermas o wages wh he goal of udersadg wh workers ces ear more ha her o-urba couerpars. The examed wo-wa fxed effecs models usg daa from he Naoal Logudal Surve of Youh NLSY; he also used daa from he Pael Sud of Icome Damcs PSID o assess

Chaper. Fxed Effecs Models / -5 he robusess of her resuls o aoher sample. For he NLSY daa he examed 5405 male heads of households over he ears 983-993 cossg of a oal of N 4094 observaos. The depede varable was logarhmc hourl wage. The prmar explaaor varable of eres was a 3-level caegorcal varable ha measures he c sze whch workers resde. To capure hs varable wo bar dumm varables were used: a varable o dcae wheher he worker resdes a large c wh more ha oe-half mllo resdes a dese meropola area ad a varable o dcae wheher he worker resdes a meropola area ha does o coa a large c a o-dese meropola area. The omed caegor s o-meropola area. Several oher corol varables were cluded o capure effecs of a worker s experece occupao educao ad race. Whe cludg me dumm varables here were K 30 explaaor varables he repored regressos.. Explorg logudal daa Wh explore? The models ha we use o represe real are smplfed approxmaos. As saed b George Box 979G All models are wrog bu some are useful. The fereces ha we draw b examg a model calbraed wh a daa se depeds o he daa characerscs; we expec a reasoable proxm bewee he model assumpos ad he daa. To assess hs proxm we explore he ma mpora feaures of he daa. B daa explorao we mea summarzg he daa eher umercall or graphcall whou referece o a model. Daa explorao provdes hs of he approprae model. To draw relable fereces from he modelg procedure s mpora ha he daa be cogrue wh he model. Furher explorg he daa also alers us o a uusual observaos or subjecs. Because sadard ferece echques descrbed are geerall o-robus o uusual feaures s mpora o def hese feaures earl he modelg process. Daa explorao also provdes a mpora commucao devce. Because daa explorao echques are o model depede he ma be beer udersood ha model depede ferece echques. Thus he ca be used o commucae feaures of a daa se ofe supplemeg model based fereces. Daa explorao echques Logudal daa aalss s closel lked o mulvarae aalss ad regresso aalss. Thus he daa explorao echques developed hese felds are applcable o logudal daa ad wll o be developed here. The reader ma cosul Tuke 977G for he orgal source o exploraor daa aalss. To summarze he followg s a ls of commol used daa explorao echques ha wll be demosraed hroughou hs book: Exame graphcall he dsrbuo of ad each x hrough hsograms des esmaes boxplos ad so o. Exame umercall he dsrbuo of ad each x hrough sascs such as meas medas sadard devaos mmums maxmums ad so o. Exame he relaoshp bewee ad each x hrough correlaos ad scaer plos. Furher summar sascs ad graphs b me perod ma be useful for deecg emporal paers. Three daa explorao echques ha are specfc o logudal daa are mulple me seres plos scaer plos wh smbols ad 3 added varable plos. Because hese echques are specfc o logudal daa aalss he are less wdel kow ad descrbed below. Aoher wa o exame daa s hrough dagosc echques descrbed Seco.4.

-6 / Chaper. Fxed Effecs Models I coras o daa explorao echques dagosc echques are performed afer he f of a prelmar model. Mulple me seres plos A mulple me seres plo s a plo of a varable geerall he respose versus me. Wh he coex of logudal daa we serall over me coec observaos over a commo subjec. Ths graph helps deec paers he respose b subjec ad over me def uusual observaos ad/or subjecs ad 3 vsualze he heerogee. Scaer plos wh smbols I he coex of regresso a plo of he respose versus a explaaor varable x j helps us o assess he relaoshp bewee hese varables. I he coex of logudal daa s ofe useful o add a plog smbol o he scaer plo o def he subjec. Ths allows us o see he relaoshp bewee he respose ad explaaor varable e accou for he varg erceps. Furher f here s a separao he explaaor varable such as creasg over me he we ca serall coec he observaos. I hs case we ma o requre a separae plog smbol for each subjec. Basc added varable plo A basc added varable plo s a scaer plo of { } versus { xj xj }. Here ad xj are averages of { } ad {x j } over me. A added varable plo s a sadard regresso dagosc echque ha s descrbed furher deal Seco.4. Alhough he basc added varable plo ca be vewed as a specal case of he more geeral dagosc echque ca also be movaed whou referece o a model. Tha s ma logudal daa ses he subjecspecfc parameers accou for a large poro of he varabl. Ths plo allows us o vsualze he relaoshp bewee ad each x whou forcg our ee o adjus for he heerogee of he subjec-specfc erceps. Example: Medcare hospal coss We cosder T6 ears 990-995 of daa for pae hospal charges ha are covered b he Medcare program. The daa were obaed from he Healh Care Facg Admsrao. To llusrae 995 he oal covered charges were $57.8 bllos for welve mllo dscharges. For hs aalss we use sae as he subjec or rsk class. Thus we cosder 54 saes ha clude he 50 saes he Uo he Dsrc of Columba Vrg Islads Puero Rco ad a uspecfed oher caegor. The respose varable of eres s he sever compoe covered clams per dscharge whch we label as CCPD. The varable CCPD s of eres o acuares because he Medcare program remburses hospals o a per-sa bass. Also ma maaged care plas remburse hospals o a per-sa bass. Because CCPD vares over sae ad me boh he sae ad me YEAR 6 are poeall mpora explaaor varables. We do o assume a pror ha frequec s depede of sever. Thus umber of dscharges NUM_DSCHG s aoher poeal explaaor varable. We also vesgae he mporace of aoher compoe of hospal ulzao AVE_DAYS defed o be he average hospal sa per dscharge das. Table. summarzes hese basc varables b ear. Here we see ha boh clams ad umber of dscharges crease over me whereas he average hospal sa decreases. The sadard devaos ad exreme values dcae ha here s subsaal varabl amog saes.

Chaper. Fxed Effecs Models / -7 TABLE.. Summar Sascs of Covered Clams Per Dscharge Number of Dscharges ad Average Hospal Sa b Year. Varable Tme Mea Meda Sadard Mmum Maxmum Perod Devao Covered Clams 990 8503 799 467 39 6485 per Dscharge 99 9473 93 7 966 7637 CCPD 99 0443 0055 304 334 984 Number of Dscharges 993 60 0667 360 438 994 53 0955 3346 4355 500 995 797 7 378 5058 03 Toal 0483 007 33 966 500 990 97.73 4.59 0.99 0.53 849.37 99 03.4 4.69 0.38 0.5 885.9 NUM_DSCHG 99 0.89 43.5 8.9 0.65 908.59 housads 993.5 43.67 9.8 0.97 894. 994 8.87 50.08 6.78.6 905.6 995.5 5.70 9.46.06 90.48 Average Hospal Sa Toal 0.73 44.8 6.7 0.5 908.59 990 9.05 8.53.08 6.33 7.48 99 9.8 8.57 7.3 6.4 60.5 AVE_DAYS 99 8.6 8.36.86 5.83 6.35 993 8.5 8.. 5.83 7.4 994 7.90 7.56.73 5.38 4.39 995 7.34 7.4.44 5..80 Toal 8.54 8.07 3.47 5. 60.5 Noes: The varable CCPD s dollars of clam per dscharge. Each ear summarzes 54 saes. The oal summarzes 6*54 34 observaos. Source: Ceer for Medcare ad Medcad Servces Fgure. llusraes he mulple me seres plo. Here we see ha o ol are overall clams creasg bu also ha clams crease for each sae. Dffere levels of hospals coss amog saes are also appare; we call hs feaure heerogee. Furher Fgure. dcaes ha here s greaer varabl amog saes ha over me. Fgure. llusraes he scaer plo wh smbols. Ths s a plo of CCPD versus umber of dscharges coecg observaos over me. Ths plo shows a posve overall relaoshp bewee CCPD ad he umber of dscharges. Lke CCPD we see a subsaal sae varao of dffere umbers of dscharges. Also lke CCPD he umber of dscharges creases over me so ha for each sae here s a posve relaoshp bewee CCPD ad umber of dscharges. The slope s hgher for hose saes wh smaller umber of dscharges. Ths plo also suggess ha he umber of dscharges lagged b oe ear s a mpora predcor of CCPD. Fgure.3 s a scaer plo of CCPD versus average oal das coecg observaos over me. Ths plo demosraes he uusual aure of he secod observao for he 54 h sae. We also see evdece of hs po hrough he maxmum sasc of he average hospal sa Table.. Ths po does o appear o follow he same paer as he res of our daa ad urs ou o have a large mpac o our fed models. Fgure.4 llusraes he basc added varable plo. Ths plo porras CCPD versus ear afer excludg he secod observao for he 54 h sae. I Fgure.4 we have corolled for he sae facor ha we observed o be a mpora source of varao. Fgure.4 shows ha he rae of crease of CCPD over me s approxmael cosse amog saes e here exss mpora varaos. The rae of crease s subsaall larger for he 3 s sae New Jerse.

-8 / Chaper. Fxed Effecs Models CCPD 000 0000 8000 6000 4000 000 0000 8000 6000 4000 000 990 99 99 993 994 995 Tme Perod Fgure. Mulple Tme Seres Plo of CCPD. Covered clams per dscharge CCPD are ploed over T6 ears 990-995. The le segmes coec saes; hus we see ha CCPD creases for almos ever sae over me. CCPD 000 0000 8000 6000 4000 000 0000 8000 6000 4000 000 0 00 00 300 400 500 600 700 800 900 000 Number of Dscharges Thousads Fgure. Scaer Plo of CCPD versus Number of Dscharges. The le segmes coec observaos wh a sae over 990-995. We see a subsaal sae varao of umbers of dscharges. There s a posve relaoshp bewee CCPD ad umber of dscharges for each sae. Slopes are hgher for hose saes wh smaller umber of dscharges.

Chaper. Fxed Effecs Models / -9 CCPD 000 0000 8000 6000 4000 000 0000 8000 6000 4000 000 0 0 0 30 40 50 60 70 Average Hospal Sa Fgure.3 Scaer Plo of CCPD versus Average Hospal Sa. The le segmes coec saes over 990-995. Ths fgure demosraes ha he secod observao for he 54 h sae s uusual. Resduals from CCPD 4000 3000 000 000 0-000 -000-3000 -4000-5000 -6000-3 - - 0 3 Resduals from YEAR Fgure.4 Added Varable Plo of CCPD versus Year. Here we have corolled for he sae facor. I hs fgure he secod observao for he 54 h sae has bee excluded. We see ha he rae of crease of CCPD over me s approxmael cosse amog saes e here exss mpora varaos. The rae of crease s subsaall large for he 3 s sae New Jerse.

-0 / Chaper. Fxed Effecs Models Trells plo A echque for graphcal dspla ha has recel become popular he sascal leraure s a rells plo. Ths graphcal echque akes s ame from a rells whch s a srucure of ope lacework. Whe vewg a house or garde oe pcall hks of a rells as beg used o suppor creepg plas such as ves. We wll use hs lace srucure ad refer o a rells plo as cossg of oe or more paels arraged a recagular arra. Graphs ha coa mulple versos of a basc graphcal form each verso porrag a varao of he basc heme promoe comparsos ad assessmes of chage. B repeag a basc graphcal form we promoe he process of commucao. Trells plos have bee advocaed b Clevelad 993G Becker Clevelad ad Shu 996G Veables ad Rple 999G ad b Phero ad Baes 000S. Tufe 997G saes ha usg small mulples graphcal dsplas acheves he same desrable effecs as usg parallel srucure wrg. Parallel srucure wrg s successful because allows readers o def a seece relaoshp ol oce ad he focus o he meag of each dvdual seece eleme such as a word phrase or clause. Parallel srucure helps acheve ecoom of expresso ad draw ogeher relaed deas for comparso ad coras. Smlarl small mulples graphs allow us o vsualze complex relaoshps across dffere groups ad over me. Fgure.5 llusraes he use of small mulples. I each pael he plo porraed s decal excep ha s based o a dffere sae; hs use of parallel srucure allows us o demosrae he creasg covered clams per dscharge CCPD for each sae. Moreover b orgazg he saes b average CCPD we ca see he overall level of CCPD for each sae as well as varaos he slope rae of crease. Ths plo was produced usg he sascal package R. 0000 AK MO MI AL LA DE NY IL AZ CT TX FL PA NJ CA HI NV DC 5000 0000 5000 KY MN VT IN UT ME NE GA TN OH NC KS RI VA MA NH SC CO 0000 CCPD 5000 0000 5000 PR UN VI MD ID SD IA WY ND WV MS MT OR AR WA NM WI OK 0000 5000 0000 5000 YEAR Fgure.5 Trells Plo of CCPD versus Year. Each of he 54 paels represes a plo of CCPD versus YEAR 990-995 he horzoal axs s suppressed. Sae 3 correspods o New Jerse.

Chaper. Fxed Effecs Models / -.3 Esmao ad ferece Leas squares esmao Reurg o our model equao.3 we ow cosder esmao of he regresso coeffces β ad α ad he he varace parameer σ. B he Gauss-Markov heorem he bes lear ubased esmaors of β ad α are he ordar leas squares OLS esmaors. These are gve b T T b x x x x x x.6 where b b b b K ad a b..7 x The dervaos of hese esmaors are Appedx A.. Noe o Reader: We ow beg o use marx oao exesvel. You ma wsh o revew hs se of oao Appedx A focusg o he defos ad basc operaos A.-3 before proceedg. Sascal ad ecoomerc packages are wdel avalable ad hus users wll rarel have o code he leas squares esmaor expressos. Noeheless he expressos equaos.6 ad.7 offer several valuable sghs. Frs we oe ha here are +K ukow regresso coeffces equao.3 for he {α } parameers ad K for he β parameers. Usg sadard regresso roues hs calls for he verso of a +K +K marx. However he calculao of he ordar leas squares esmaors equao.6 requres verso of ol a K K marx. Ths s a sadard feaure of aalss of covarace models reag he subjec defer as a caegorcal explaaor varable kow as a facor. Secod he OLS esmaor of β ca also be expressed as a weghed average of subjecspecfc esmaors. Specfcall suppose ha all parameers are subjec-specfc so ha he regresso fuco s E α + x β. The roue calculaos show ha he ordar leas squares esmaor of β urs ou o be T T b x x x x x x. Now defe a wegh marx T W x x x x so ha a smpler expresso for b s b W x x express he esmaor of β as T. Wh hs wegh we ca b W Wb.8 a marx weghed average of subjec-specfc parameer esmaes. To help erpre equao.8 cosder Fgure.. Here we see ha he respose CCPD s posvel relaed o umber of dscharges for each sae. Thus because each subjec-specfc coeffce s posve we expec he weghed average of coeffces o also be posve.

- / Chaper. Fxed Effecs Models For a hrd sgh from equaos.6 ad.7 cosder aoher weghg vecor W x x. W Wh hs vecor aoher expresso for equao.6 s T W b..9 From hs we see ha he regresso coeffces b are lear combaos of he resposes. B he lear f he resposes are ormall dsrbued assumpo F5 he so are he regresso coeffces b. Fourh regresso coeffces assocaed wh me-cosa varables cao be esmaed usg equao.6. Specfcall suppose ha he jh varable does o deped o me so ha x j x j. The elemes he jh row ad colum of T x x x x are decall zero so ha he marx s o verble. Thus regresso coeffces cao be calculaed usg equao.6 ad fac are o esmable whe oe of he explaaor varables s me-cosa. Oher properes of esmaors Boh a ad b have he usual fe sample properes of ordar leas squares regresso esmaors. I parcular he are ubased esmaors. Furher b he Gauss-Markov heorem he are mmum varace amog he class of ubased esmaors. If he resposes are ormall dsrbued assumpo F5 he so are a ad b. Furher usg equao.9 s eas o check ha he varace of b urs ou o be Var b σ W..0 ANOVA able ad sadard errors The esmaor of he varace parameer σ follows from he cusomar regresso seup. Tha s s covee o frs defe resduals ad he aalss of varace ANOVA able. From hs we ge a esmaor of σ as well as sadard errors for he regresso coeffce esmaors. To hs ed defe he resduals as e - a + x b he dfferece bewee he observed ad fed values. I ANOVA ermolog he sum of squared resduals s called he error sum of squares ad deoed b Error SS e. The mea square error s our esmaor of σ deoed b Error SS s Error MS.. N + K The correspodg posve square roo s he resdual sadard devao deoed b s. Here recall ha T + T + + T N s he oal umber of observaos. These calculaos are summarzed Table..

Chaper. Fxed Effecs Models / -3 Table.. ANOVA Table Source Sum of Squares df Mea Square Regresso Regresso SS -+K Regresso MS Error Error SS N - +K Error MS Toal Toal SS N- To complee he defos of he expressos Table. we have Toal SS ad Regresso SS a + x b. Furher he mea square quaes are he sum of square quaes dvded b her respecve degrees of freedom df. The ANOVA able calculaos are ofe repored hrough he goodess of f sasc called he coeffce of deermao Regresso SS R Toal SS or he verso adjused for degrees of freedom Error SS / N + K R a. Toal SS / N A mpora fuco of he resdual sadard devao s o esmae sadard errors assocaed wh parameer esmaors. Usg he ANOVA able ad equao.0 he esmaed 678 varace marx of he vecor of regresso coeffces s Var b s W. Thus he sadard error for he jh regresso coeffce b j s h se b j s j dagoal eleme of W. Sadard errors are he bass for he -raos arguabl he mos mpora or a leas mos wdel ced sascs appled regresso aalss. To llusrae he -rao for he jh regresso coeffce b j s b j b j se b s j h j dagoal eleme of W Assumg he resposes are ormall dsrbued b j has a -dsrbuo wh N-+K degrees of freedom. Example Medcare hospal coss Coued To llusrae we reur o he Medcare example. Fgures.-.4 suggesed ha he sae caegorcal varable s mpora. Furher NUM_DSCH AVE_DAYS ad YEAR are also poeall mpora. From Fgure.4 we oed ha he crease CCPD s hgher for New Jerse ha oher saes. Thus we also cluded a specal eraco varable YEAR*STATE3 ha allowed us o represe he uusuall large me slope for he 3 s sae New Jerse. Thus we esmae he fuco E CCPD α + β YEAR + β AVE_DAYS + β 3 NUM_DSCH + β 4 YEAR *STATE3.. b j.

-4 / Chaper. Fxed Effecs Models The fed model appears Dspla. usg he sascal package SAS. Dspla. SAS OUTPUT Geeral Lear Models Procedure Depede Varable: CCPD Sum of Mea Source DF Squares Square F Value Pr > F Model 57 35850685.0 5766775. 03.94 0.000 Error 65 7484379. 8038.4 Correced Toal 3 333790564. R-Square C.V. Roo MSE CCPD Mea 0.9777 5.0466 59.4505 050.344 T for H0: Pr > T Sd Error of Parameer Esmae Parameer0 Esmae YEAR 70.88403 6.5 0.000 6.8388 AVE_DAYS 36.9007 6.3 0.000 57.9789849 NUM_DCHG 0.75477 4.8 0.000.5769 YR_3 6.456077 9.8 0.000 8.6088909 Example. Urba wages Coued To llusrae Table.3 summarzes hree regresso models repored b Glaeser ad Maré 00 her vesgao of deermas of hourl wages. The wo homogeeous models do o clude worker-specfc erceps whereas hese are cluded he fxed effecs model. For he homogeeous model whou corols he ol wo explaaor varables are he bar varables for dcag wheher a worker resdes a dese or o-dese meropola area. The omed caegor hs regresso s o-meropola area so we erpre he 0.63 coeffce o mea ha workers dese meropola areas o average ear 0.63 log dollars or 6.3% more ha her o-meropola area couerpars. Smlarl hose o-dese meropola areas ear 7.5% more ha her o-meropola area couerpars. Wages ma also be flueced b a worker s experece occupao educao ad race ad here s o guaraee ha hese characerscs are dsrbued uforml over dffere c szes. Thus a regresso model wh hese corols s also repored Table.3. Table.3 shows ha workers ces parcularl dese ces sll receve more ha workers o-urba areas eve whe corollg for a worker s experece occupao educao ad race. Glaeser ad Maré offer addoal explaaos as o wh workers ces ear more ha her o-urba couerpars cludg hgher cos of lvg ad urba dsamees he also dscou hese explaaos as of less mporace. The do pos a omed abl bas ha s suggesg ha he abl of a worker s a mpora wage deerma ha should be corolled for. The sugges ha hgher abl workers ma flock o ces because f ces speed he flow of formao he hs mgh be more valuable o workers who have hgh huma capal. Furher ces ma be ceers of cosumpo ha caer o he rch. Abl s a dffcul arbue o measure he exame a prox he Armed Forces Qualfcao Tes ad fd ha s o useful. However f oe reas abl as me-cosa he s effecs o wages wll be capure he me-cosa worker-specfc ercep α. Table.3 repors o he fxed effecs regresso ha cludes a worker-specfc ercep. Here we see ha he parameer esmaes for c premums have bee subsaall reduced alhough he are sll sascall sgfcal. Oe erpreao s ha a worker wll receve a 0.9% 7% hgher wage for workg a dese o-dese c whe compared o a o-meropola worker eve whe corollg for a worker s experece occupao educao ad race ad omed me-cosa arbues such as

Chaper. Fxed Effecs Models / -5 abl. Seco 7. wll prese a much more dealed dscusso of hs omed varable erpreao. Table.3 Regresso coeffce esmaors of several hourl wage models Varable Homogeous model whou corols Homogeeous model wh corols Two-wa fxed effecs model Dese meropola premum 0.63 0.45 0.0 0.09 0.0 No-dese meropola premum 0.75 0.47 0.0 0.070 0.0 Coeffce of deermao R.4 33. 38. Adjused R.4 33.0 8.4 Source: Glaeser ad Maré 00. Sadard errors are pareheses. Large sample properes of esmaors I pcal regresso suaos resposes are a bes ol approxmael ormall dsrbued. Noeheless hpohess ess ad predcos are based o regresso coeffces ha are approxmael ormall dsrbued. Ths premse s reasoable because of he Ceral Lm Theorem whch roughl saes ha weghed sums of depede radom varables are approxmael ormall dsrbued ad ha he approxmao mproves as he umber of radom varables creases. Regresso coeffces ca be expressed as weghed sums of resposes. Thus f sample szes are large he we ma assume approxmae ormal of regresso coeffce esmaors. I he logudal daa coex large samples meas ha eher he umber of subjecs ad/or he umber of observaos per subjec T becomes large. I hs chaper we dscuss he case where becomes large alhough T remas fxed. Ths ca be movaed b he fac ha ma daa applcaos he umber of subjecs s large relave o he umber of me perods observed. Chaper 8 wll dscuss he problem of large T. As he umber of subjecs becomes large e T remas fxed mos of he properes of b are reaed from he sadard regresso suaos. To llusrae we have ha b s a weakl cosse esmae of β. Specfcall weak cossec meas approachg covergece probabl. Ths s a drec resul of he ubasedess ad a assumpo ha Σ W grows whou boud. Furher uder mld regular codos we have a ceral lm heorem for he slope esmaor. Tha s b s approxmael ormall dsrbued eve hough he resposes ma o be. The suao for he esmaors of subjec-specfc erceps α s dramacall dffere. To llusrae he leas squares esmaor of α s o cosse as becomes large. Furher f he resposes are o ormall dsrbued he a s o eve approxmael ormal. Iuvel hs s because we assume ha he umber of observaos per subjec T s a bouded umber. As grows he umber of parameers grows a suao called fe dmesoal usace parameers he leraure; see for example Nema ad Sco 948E for a classc reame. Whe he umber of parameers grows wh sample sze he usual large sample properes of esmaors ma o be vald. Seco 7. ad Chaper 9 wll dscuss hs ssue furher.

-6 / Chaper. Fxed Effecs Models.4 Model specfcao ad dagoscs Iferece based o a fed sascal model ofe ma be crczed because he feaures of he daa are o cogruece wh he model assumpos. Dagosc echques are procedures ha exame hs cogruece. Because we somemes use dscoveres abou model adequaces o mprove he model specfcao hs group of procedures s also called model specfcao or ms-specfcao ess or procedures. The broad dsco bewee dagoscs ad he Seco. daa explorao echques s ha he former are performed afer a prelmar model f whereas he laer are doe before fg models wh daa. Whe a aals fs a grea umber of models o a daa se hs leads o dffcules kow as daa soopg. Tha s wh several explaaor varables oe ca geerae a large umber of lear models ad a fe umber of olear models. B searchg over ma models s possble o overf a model so ha sadard errors are smaller ha he should be ad sgfca relaoshps appear sgfca. There are wdel dffere phlosophes espoused he leraure for model specfcao. Oe ed of he specrum beleves ha daa soopg s a problem edemc o all daa aalss. Propoes of hs phlosoph beleve ha a model should be full specfed before examg he daa; hs wa fereces draw from he daa are o mgaed from daa soopg. The oher ed of he specrum argues ha fereces from a model are urelable f he daa are o accordace wh model assumpos. Propoes of hs phlosoph argue ha a model summarzes mpora characerscs of he daa ad ha he bes model should be sough hrough a seres of specfcao ess. These dscos are wdel dscussed he appled sascal modelg leraure. We prese here several specfcao ess ad procedures ha ca be used o descrbe how well he daa fs he model. Resuls from he specfcao ess ad procedures ca he be used o eher re-specf he model or erpre model resuls accordg o oe s belefs model fg..4.. Poolg es A poolg es also kow as a es for heerogee exames wheher or o he erceps ake o a commo value sa α. A mpora advaage of logudal daa models compared o cross-secoal regresso models s ha we ca allow for heerogee amog subjecs geerall hrough subjec-specfc parameers. Thus a mpora frs procedure s o jusf he eed for subjec-specfc effecs. The ull hpohess of homogee ca be expressed as H 0 : α α... α α. Tesg hs ull hpohess s smpl a specal case of he geeral lear hpohess ad ca be hadled drecl as such. Here s oe wa o perform a paral F- Chow es. Procedure for he poolg es. Ru he full model wh E α + x β o ge Error SS ad s.. Ru he reduced model wh E α + x β o ge Error SS reduced. Error SS Error SS reduced 3. Compue he paral F-sasc F-rao s 4. Rejec H 0 f F-rao exceeds a percele from a F-dsrbuo wh umeraor degrees of freedom df - ad deomaor degrees of freedom df N - +K. The percele s oe mus he sgfcace level of he es.. Ths s a exac es he sese ha does o requre large sample szes e does requre ormal of he resposes assumpo F5. Sudes have show ha he F-es s o sesve o deparures from ormal see for example Laard 973G. Furher oe ha f he

Chaper. Fxed Effecs Models / -7 deomaor degrees of freedom df s large he we ma approxmae he dsrbuo b a chsquare dsrbuo wh - degrees of freedom. Example Medcare hospal coss Coued To es for heerogee Medcare Hospal cos he eres s esg he ull hpohess H 0 : α α... α 5. From Dspla. we have Error SS 7484379. ad s 8038.4. Fg he pooled cross-secoal regresso fuco wh commo effecs α E CCPD α + β YEAR + β AVE_DAYS + β 3 NUM_DSCH + β 4 YEAR *STATE3 elds Error SS reduced 373593.9. Thus he es sasc s: 373593.9 7484379. F-rao 54. 7. 54 8038.4 For a F-dsrbuo wh df 53 ad df 33-54+465 he assocaed p-value s less ha 0.000. Ths provdes srog evdece for he case for reag subjec-specfc parameers α he model specfcao..4.. Added varable plos A added varable plo also kow as a paral regresso plo s a sadard graphcal devce used regresso aalss; see for example Cook ad Wesberg 98G. I allows oe o vew he relaoshp bewee a respose ad a explaaor varable afer corollg for he lear effecs of oher explaaor varables. Thus added varable plos allow aalss o vsualze he relaoshp bewee ad each x whou forcg he ee o adjus for he dffereces duced b he oher explaaor varables. The Seco. basc added varable plo s a specal case of he followg procedure ha ca be used for addoal regresso varables. To produce a added varable plo oe frs selecs a explaaor varable sa x j ad he follows he procedure below. Procedure for producg a added varable plo. Ru a regresso of o he oher explaaor varables omg x j ad calculae he resduals from hs regresso. Call hese resduals e.. Ru a regresso of x j o he oher explaaor varables omg x j ad calculae he resduals from hs regresso. Call hese resduals e. 3. Produce a plo of e versus e. Ths s a added varable plo. Correlaos ad added varable plos To help erpre added varable plos use equao. o express he dsurbace erm as ε α + β x +... + β K x K. Tha s we ma hk of he error as he respose afer corollg for he lear effecs of he explaaor varables. The resdual e s a approxmao of he error erpreed o be he respose afer corollg for he effecs of explaaor varables. Smlarl we ma erpre e o be he jh explaaor varable afer corollg for he effecs of oher explaaor varables. Thus we erpre he added varable plo as a graph of he relaoshp bewee ad x j afer corollg for he effecs of oher explaaor varables.

-8 / Chaper. Fxed Effecs Models As wh all scaer plos he added varable plo ca be summarzed umercall hrough a correlao coeffce ha we wll deoe b corre e. I s relaed o he -sasc of x j b j from he full regresso equao cludg x j hrough he followg expresso: b j corr e e. b j + N + K Here +K s he umber of regresso coeffces he full regresso equao ad N s he umber of observaos. Thus he -sasc from he full regresso equao ca be used o deerme he correlao coeffce of he added varable plo whou rug he hree-sep procedure. However ulke correlao coeffces he added varable plo allows us o vsualze poeal olear relaoshps bewee ad x j..4.3. Ifluece dagoscs Tradoal fluece dagoscs are mpora because he allow a aals o udersad he mpac of dvdual observaos o he esmaed model. Tha s fluece sascs allow aalss o perform a pe of sesv aalss; oe ca calbrae he effec of dvdual observaos o regresso coeffces. Cook s dsace s a dagosc sasc ha s wdel used regresso aalss ad s revewed Appedx A.3. For he Chaper fxed effecs logudal daa models observao level dagosc sascs are of less eres because he effec of uusual observaos s absorbed b subjec-specfc parameers. Of greaer eres s he mpac ha a ere subjec has o he populao parameers. To assess he mpac ha a subjec has o esmaed regresso coeffces we use he sasc B b b b W b b / K. Here b s he ordar leas squares esmaor b calculaed wh he h subjec omed. Thus B b measures he dsace bewee regresso coeffces calculaed wh ad whou he h subjec. I hs wa we ca assess he effec of he h subjec. The logudal daa fluece dagosc s smlar o Cook s dsace for regresso. However Cook s dsace s calculaed a he observao level whereas B b s a he subjec level. Observaos wh a large value of B b ma be flueal o he parameer esmaes. Baerjee ad Frees 997S showed ha he sasc B b has a approxmae χ ch-square dsrbuo wh K degrees of freedom. Thus we ma use quales of he χ o quaf he adjecve large. Iflueal observaos warra furher vesgao; he ma requre correco for codg errors addoal varable specfcao o accommodae he paers he emphasze or deleo from he daa se. From he defo of B b appears ha he calculao of he fluece sasc s compuaoall esve. Ths s because he defo requres + regresso compuaos oe for b ad oe for each b. However as wh Cook s dsace a he observao level shorcu calculao procedures are avalable. The deals are Appedx A.3. Example Medcare hospal coss Coued Fgure.3 alered us o he uusual value of AVE_DAYS ha occurred he 54 h subjec a he d me po. I urs ou ha hs observao has a subsaal mpac o he fed regresso model. Foruael he graphcal procedure Fgure.3 ad he summar sascs Table. were suffce o deec hs uusual po. Ifluece dagosc sascs provde aoher ool for deecg uusual observaos ad subjecs. Suppose ha he model equao. was f usg he full daa se of N 34 observaos. I urs ou ha Cook s dsace was D 54 7.06 for he 54 po srogl dcag a flueal

Chaper. Fxed Effecs Models / -9 observao. The correspodg subjec-level sasc was B 54 44.3. Compared o a ch-square dsrbuo wh K 4 degrees of freedom hs dcaes ha somehg abou he 54 h subjec was uusual. For comparso he dagosc sascs were calculaed uder a fed regresso model afer removg he 54 po. The larges value of Cook s dsace was 0.0907 ad he larges value of he subjec-level sasc was 0.495. Neher value dcaes subsaal flueal behavor afer he uusual 54 po was removed..4.4. Cross-secoal correlao Our basc model reles o assumpo F4 he depedece amog observaos. I radoal cross-secoal regresso models hs assumpo s uesable whou a paramerc assumpo. However wh repeaed measuremes o a subjec s possble o exame hs assumpo. As s radoal he sascs leraure whe esg for depedece we are reall esg for zero correlao. Tha s we are eresed he ull hpohess H 0 : ρ j Corr j 0 for j. To udersad how volaos of hs assumpo ma arse pracce suppose ha he rue model s λ + x β + ε. Here we use λ for a radom emporal effec ha s commo o all subjecs. Because s commo duces correlao amog subjecs as follows. We frs oe ha he varace of a respose s Var σ λ + σ where Varε σ ad Var λ σ λ. From here basc calculaos show ha he covarace bewee observaos a he same me bu from dffere subjecs s Cov j σ λ for j. Thus he cross-secoal correlao s σ λ Corr j. Hece a posve cross-secoal correlao ma be due o uobserved σ λ + σ emporal effecs ha are commo amog subjecs. Tesg for o-zero cross-secoal correlao To es H 0 : ρ j 0 for all j we use a procedure developed Frees 995E where balaced daa were assumed so ha T T. Procedure for compug cross-secoal correlao sascs. F a regresso model ad calculae he model resduals {e }.. For each subjec calculae he raks of each resdual. Tha s defe {r... r T } o be he raks of {e... e T }. These raks wll var from o T so ha he average rak s T+/. 3. For he h ad jh subjec calculae he rak correlao coeffce Spearma s correlao sr j T r T + / rj T + / T r T + / 4. Calculae he average Spearma s correlao ad he average squared Spearma s correlao R AVE / { < sr j} j ad R AVE / { < sr j} j. Here Σ {<j} meas sum over j... ad... j-.. Calbrao of cross-secoal correlao es sascs Large values of he sascs R AVE ad R AVE dcae he presece of o-zero crosssecoal correlaos. I applcaos where eher posve or egave cross-secoal

-0 / Chaper. Fxed Effecs Models correlaos preval oe should cosder he R AVE sasc. Fredma 937G showed ha FR T--R AVE + follows a ch-square dsrbuo wh T- degrees of freedom asmpocall as becomes large. Fredma devsed he es sasc FR o deerme he equal of reame meas a wo-wa aalss of varace. The sasc FR s also used he problem of -rakgs. I hs coex judges are asked o rak T ems ad he daa are arraged a wo-wa aalss of varace laou. The sasc R AVE s erpreed o be he average agreeme amog judges. The sasc R AVE s useful for deecg a broader rage of aleraves ha he sasc R AVE. For hpohess esg purposes we compare R AVE o a dsrbuo ha s a weghed sum of ch-square radom varables. Specfcall defe Q at χ - T - + bt χ - T T-3/. Here χ ad χ are depede ch-square radom varables wh T- ad T T-3/ degrees of freedom respecvel. The cosas are 4 T + 5T + 6 a T ad b T. 5 T T + 5T T T + Frees 995E showed ha R AVE T - follows a Q dsrbuo asmpocall as becomes large. Thus oe rejecs H 0 f R AVE exceeds T- - + Q q / where Q q s a approprae quale from he Q dsrbuo. Because he Q dsrbuo s a weghed sum of ch-square radom varables compug quales ma be edous. For a approxmao s much faser o compue he varace of Q ad use a ormal approxmao. Exercse.3 llusraes he use of hs approxmao. The sascs R AVE ad R AVE are averages over -/ correlaos whch ma be compuaoall esve for large values of. Appedx A.4 descrbes some shor-cu calculao procedures. Example Medcare hospal coss Coued The ma drawback of he R AVE ad R AVE sascs s ha he asmpoc dsrbuos are ol avalable for balaced daa. To acheve balaced daa for he Medcare hospal coss daa we om he 54 h sae. The model equao. was f o he remag 53 saes ad resduals calculaed. The values of he correlao sascs ured ou o be R AVE 0.8 ad R AVE 0.388. Boh sascs are sascall sgfca wh p-values less ha 0.00. Ths resul dcaes subsaal cross-secoal correlao dcag some pe of co-moveme amog saes over me ha s o capured b our smple model. For comparso he model was re-f usg YEAR as a caegorcal varable leu of a couous oe. Ths s equvale o cludg sx dcaor varables oe for each ear. The values of he correlao sascs ured ou o be R AVE 0.00 ad R AVE 0.49. Thus we have capured some of he posve co-moveme amog sae Medcare hospal coss wh me dcaor varables..4.5. Heeroscedasc Whe fg regresso models o daa a mpora assumpo s ha he varabl s commo amog all observaos. Ths assumpo of commo varabl s called homoscedasc; hs meas same scaer. Ideed he leas squares procedure assumes ha he expeced varabl of each observao s cosa; gves he same wegh o each observao whe mmzg he sum of squared devaos. Whe he scaer vares b observao he daa are sad o be heeroscedasc. Heeroscedasc affecs he effcec of he regresso coeffce esmaors alhough hese esmaors rema ubased eve he presece of heeroscedasc.

Chaper. Fxed Effecs Models / - I he logudal daa coex he varabl Var ma deped o he subjec hrough or he me perod hrough or boh. Several echques are avalable for hadlg heeroscedasc. Frs heeroscedasc ma be mgaed hrough a rasformao of he respose varable. See Carroll ad Rupper 988G for a broad reame of hs approach. Secod heeroscedasc ma be explaed b weghg varables deoed b w. Thrd he heeroscedasc ma be gored he esmao of he regresso coeffces e accoued for he esmao of he sadard errors. Seco.5.3 expads o hs approach. Furher as we wll see Chaper 3 he varabl of ma var over ad hrough a radom effecs specfcao. Oe mehod for deecg heeroscedasc s o perform a prelmar regresso f of he daa ad plo he resduals versus he fed values. The prelmar regresso f removes ma of he major paers he daa ad leaves he ee free o cocerae o oher paers ha ma fluece he f. We plo resduals versus fed values because he fed values are a approxmao of he expeced value of he respose ad ma suaos he varabl grows wh he expeced respose. More formal ess of heeroscedasc are also avalable he regresso leraure. For a overvew see Judge e al. 985E or Greee 00E. To llusrae le us cosder a es due o Breusch ad Paga 980E. Specfcall hs es exames he alerave hpohess H a : Var σ + γ w where w s a kow vecor of weghg varables ad γ s a p-dmesoal vecor of parameers. Thus he ull hpohess s H 0 : Var σ. Procedure for esg for heeroscedasc. F a regresso model ad calculae he model resduals {e }. *. Calculae squared sadardzed resduals e e / Error SS / N. * 3. F a regresso model of e o w. 4. The es sasc s LM Regress SS w / where Regress SS w s he regresso sum of squares from he model f sep 3. 5. Rejec he ull hpohess f LM exceeds a percele from a ch-square dsrbuo wh p degrees of freedom. The percele s oe mus he sgfcace level of he es. Here we use LM o deoe he es sasc because Breusch ad Paga derved as a Lagrage mulpler sasc; see Breusch ad Paga 980E for more deals. Appedx C.7 revews Lagrage mulpler sascs ad relaed hpohess ess. A commo approach for hadlg heeroscedasc volves compug sadard errors ha are robus o he homoscedasc specfcao. Ths s he opc of Seco.5.3..5 Model exesos To roduce exesos of he basc model we frs provde a more compac represeao usg marx oao. A marx form of equao. fuco s E α + X β..3 Here s he T vecor of resposes for he h subjec... T. Furher X s a T K marx of explaaor varables

- / Chaper. Fxed Effecs Models X x x M xt x x x M T L L O L x x x K K M T K x x M x T..4 Ths ca be expressed more compacl as X x x... xt. Fall s he T vecor of oes..5. Seral correlao I logudal daa subjecs are measured repeaedl over me ad repeaed measuremes of a subjec ed o be relaed o oe aoher. As we have see oe wa o capure hs relaoshp s hrough commo subjec-specfc parameers. Aleravel hs relaoshp ca be capured hrough he correlao amog observaos wh a subjec. Because hs correlao s amog observaos ake over me we refer o as a seral correlao. As wh me seres aalss s useful o measure edeces me paers hrough a correlao srucure. Noeheless s also mpora o oe ha me paers ca be hadled hrough he use of me-varg explaaor varables. As a specal case emporal dcaor dumm varables ca be used for me paers he daa. Alhough s dffcul o solae whe examg daa me-varg explaaor varables accou for me paers he mea respose whereas seral correlaos are used o accou for me paers secod mome of he respose. Furher Chapers 6 ad 8 we wll explore oher mehods for modelg me paers such as usg lagged depede varables. Tmg of observaos The acual mes ha observaos are ake are mpora whe examg seral correlaos. Ths seco assumes ha observaos are ake equall spaced me such as quarerl or auall. The ol degree of mbalace ha we explcl allow for s he umber of observaos per subjec deoed b T. Chaper 7 roduces ssues of mssg daa aro ad oher forms of ubalace. Chaper 8 roduces echques for hadlg daa ha are o equall spaced me. Temporal covarace marx For a full se of observaos we use R o deoe he T T emporal varace-covarace marx. Ths s defed b R Var where R rs Cov r s s he eleme he rh row ad sh colum of R. There are a mos TT+/ ukow elemes of R. We deoe hs depedece of R o parameers usg he oao Rτ where τ s a vecor of parameers. For less ha a full se of observaos cosder he h subjec ha has T observaos. Here we defe Var R τ a T T marx. The marx R τ ca be deermed b removg cera rows ad colums of he marx Rτ. We assume ha R τ s posve-defe ad ol depeds o hrough s dmeso. The marx Rτ depeds o τ a vecor of ukow parameers called varace compoes. Le us exame several mpora specal cases of R. Table.4 summarzes hese examples. R σ I where I s a T T de marx. Ths s he case of o seral correlao. R σ -ρ I + ρ J where J s a T T marx of oes. Ths s he compoud smmer model also kow as he uform correlao model. R rs σ ρ r-s. Ths s he auoregressve of order oe model deoed b AR. Make o addoal assumpos o R.

Chaper. Fxed Effecs Models / -3 Table.4. Covarace Srucure Examples Srucure Example Varace Comp τ Srucure Example Varace Comp τ Idepede σ 0 0 0 σ Auoregressve 3 ρ ρ ρ σ ρ of Order 0 σ 0 0 R AR ρ ρ ρ R σ 0 0 σ 0 ρ ρ ρ 3 0 0 0 σ ρ ρ ρ Compoud Smmer ρ R σ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ σ ρ No Srucure σ σ R σ 3 σ 4 σ σ σ σ 3 4 σ σ σ σ 3 3 3 34 σ 4 σ 4 σ 34 σ 4 σ σ σ 3 σ 4 σ σ 3 σ 4 σ 3 σ 4 σ 34 To see how he compoud smmer model ma occur cosder he model α + ε where α s a radom cross-secoal effec. Ths elds R Var σ + σ σ. Smlarl for r s we have R rs σ Cov r s α. To wre hs erms of σ oe ha he correlao σ α s Corr r s ρ. Thus R rs σ ρ ad R σ -ρ I + ρ J. σ α + σ ε The auoregressve model s a sadard represeao used me seres aalss. Suppose ha u are depede errors ad he dsurbace erms are deermed sequeall hrough he relao ε ρ ε - + u. Ths mples he relao R rs σ ρ r-s wh σ Var u /- ρ. Ths model ma also be exeded o he case where me observaos are o equall spaced me; Seco 8. provdes furher deals. For he usrucured model here are TT+/ ukow elemes of R. I urs ou ha hs choce s oesmable for a fxed effecs model wh dvdual-specfc erceps. Chaper 7 provdes addoal deals..5.. Subjec-specfc slopes I he Medcare hospal coss example we foud ha a desrable model of covered clams per dscharge CCPD was of he form E CCPD α + β YEAR + x β. Thus oe could erpre β as he expeced aual chage CCPD; hs ma be due for example o a medcal flao compoe. Suppose ha he aals acpaes ha he medcal flao compoe wll var b sae as suggesed b Fgure.5. To address hs cocer we cosder sead he model E CCPD α + α YEAR + x β. Here subjec-specfc erceps are deoed b {α } ad we allow for subjec-specfc slopes assocaed wh ear hrough he oao {α }. Thus addo o leg erceps var b subjec s also useful o le oe or more slopes var b subjec. We wll cosder regresso fucos of he form E z α + x β..5 α ε

-4 / Chaper. Fxed Effecs Models Here he subjec-specfc parameers are α α... α q ad he q explaaor varables are z z z... z q ; boh colum vecors are of dmeso q. Equao.5 s shor had oao for he fuco E α z + α z +... + α q z q + β x + β x +... + β K x K. To provde a more compac represeao usg marx oao we defe Z z z... z T a T q marx of explaaor varables. Wh hs oao as equao.3 a marx form of equao.5 s E Z α + X β..6 The resposes bewee subjecs are depede e we allow for emporal correlao ad heeroscedasc hrough he assumpo ha Var R τ R. Take ogeher hese assumpos comprse wha we erm he fxed effecs lear logudal daa model. Assumpos of he Fxed Effecs Lear Logudal Daa Model F. E Z α + X β. F. {x... x K } ad {z... z q } are osochasc varables. F3. Var R τ R. F4. { } are depede radom vecors. F5. { } are ormall dsrbued. Noe ha we use he same leers F-F5 o deoe he assumpos of he fxed lear logudal daa model ad he lear regresso model. Ths s because he models dffer ol hrough her mea ad varace fucos. Samplg ad model assumpos We ca use a model ha represes how he sample s draw from he populao o movae he assumpos of he fxed effecs lear logudal daa model. Specfcall assume he daa arse as a srafed sample whch he subjecs are he sraa. For example Seco. example we would def each sae as a sraum. Uder srafed samplg oe assumes depedece amog dffere subjecs assumpo F4. For observaos wh a sraum ulke srafed samplg a surve coex we allow for seral depedece o arse a me seres paer hrough he assumpo F3. I geeral selecg subjecs based o exogeous characerscs suggess srafg he populao ad usg a fxed effecs model. To llusrae ma pael daa sudes have aalzed seleced large coures frms or mpora CEOs chef execuve offcers. Whe a sample s seleced based o exogeous explaaor varables ad hese explaaor varables are reaed as fxed e varable we rea he subjec-specfc erms as fxed e varable. Leas squares esmaors The esmaors are derved Appedx A. assumg ha he emporal correlao marx R s kow. Seco 3.5 wll address he problems of esmag he parameers ha deerme hs marx. Moreover Seco 7. wll emphasze some of he specal problems of esmag hese parameers he presece of fxed effecs heerogee. Wh kow R he regresso coeffce esmaors are gve b / / Q Z R X / / b FE X R X R Q Z R.7

Chaper. Fxed Effecs Models / -5 ad FE FE b X R Z Z R Z a..8 Here / / Z R Z Z R Z Z R I Q..5.3. Robus esmao of sadard errors Equaos.7 ad.8 show ha he regresso esmaors are lear combaos of he resposes ad hus s sraghforward o deerme he varace of hese esmaos. To llusrae we have / / Var Z FE X R Q R X b. Thus sadard errors for he compoes of b FE are readl deermed b usg esmaes of R ad akg square roos of dagoal elemes of Var b FE. I s commo pracce o gore seral correlao ad heeroscedasc all whe esmag β so ha oe ca assume R σ I. Wh hs assumpo he leas squares esmaor of β s X Q X Q X b wh Z Z Z Z I Q. Ths s a ubased ad asmpocall ormal esmaor of β alhough s less effce ha b FE. Basc calculaos show ha has varace Var X X Q Q X X Q R X Q X b. To esmae hs Huber 967G Whe 980E ad Lag ad Zeger 986B suggesed replacg R b e e o ge a esmae ha s robus o ususpeced seral correlao ad heeroscedasc. Here e s he vecor of resduals. Thus a robus sadard error of b j s h j of eleme dagoal j b se X X Q Q X e X Q e Q X X..9 For coras cosder pooled cross-secoal regresso model based o equao. so ha Q I ad assume o seral correlao. The he ordar leas squares esmaor of β has varace Var X X X R X X X b where R σ I for heeroscedasc. Furher usg he esmaor T s / ee for σ elds he usual Whe s robus sadard errors. B wa of comparso he robus sadard error equao.9 accommodaes heerogee hrough he Q marx ad also accous for ususpeced seral correlao b usg he T T marx e e leu of he scalar esmae T s / ee.

-6 / Chaper. Fxed Effecs Models Furher readg Fxed effecs modelg ca be bes udersood based o a sold foudao regresso aalss ad aalss usg he geeral lear model. Draper ad Smh 98G ad Seber 977G are wo classc refereces ha roduce regresso usg marx algebra. Treames ha emphasze caegorcal covaraes he geeral lear model coex clude Searle 987G ad Hockg 985G. Aleravel mos roducor graduae ecoomerc exbooks cover hs maeral; see for example Greee 00E or Haash 000E. Ths book acvel uses marx algebra coceps o develop he sublees ad uaces of logudal daa aalss. Appedx A provdes a bref overvew of he ke resuls. Grabll 969G provdes addoal backgroud. Earl applcaos of basc fxed effecs pael daa models are b Kuh 959E Johso 960E Mudlak 96E ad Hoch 96E. Kefer 980E dscussed he basc fxed effecs model he presece of a usrucured seral covarace marx. He showed how o cosruc wo-sage geeralzed leas squares GLS esmaors of global parameers. Furher he gave a codoal maxmum lkelhood erpreao of he GLS esmaor. Exesos of hs dea ad addoal refereces are Kug 996O; see also Seco 7.. Emprcal work o esmag subjec-specfc slope models has bee lmed a fxed effecs coex. A example s provded b Polachek ad Km 994E; he used subjec-specfc slopes fxed effecs models whe examg gaps eargs bewee males ad females. Mudlak 978bE provded some basc movao ha wll be descrbed Seco 7.3.

Chaper. Fxed Effecs Models / -7 Appedx A - Leas Squares Esmao A. Basc Fxed Effecs Model - Ordar Leas Squares Esmao We frs calculae he ordar leas squares esmaors of he lear parameers he fuco E α + x β T. * * * * To hs ed le a b b...bk be caddae esmaors of he parameers α β β... β K. For hese caddaes defe he sum of squares * * * * * T * * a + x b SS a b * * * where a a... a ad b b... bk. Specfcall a* ad b* are argumes of he sum of squares fuco SS. To mmze hs qua frs exame paral dervaves wh respec o a * o ge T * * * * SS a b a + * a x b. Seg hese paral dervaves o zero elds he leas squares esmaors of α * * * T a b x b where x x / T. The sum of squares evaluaed a hs value of ercep s * * * T * - x x b SS a b b. To mmze hs sum of squares ake a paral dervave wh respec o each compoe of b *. For jh compoe we have T * * * * SS a b b xj xj - x x b. * b j Seg hs equal o zero for each compoe elds he ormal equaos T x. * x x x b x x T These ormal equaos elds he ordar leas squares esmaors ad T b x x x x x x a b. x T A.. Fxed Effecs Models - Geeralzed Leas Squares Esmao We cosder lear logudal daa models wh regresso fucos E Z α + X β where he varace-covarace marx of R s assumed kow. The geeralzed leas squares sum of squares s

-8 / Chaper. Fxed Effecs Models * * * * * * Za + + X b R Z a X b SS a b. Here a * ad b * are caddae esmaors of α α...α ad ββ β K ; he are argumes of he fuco SS. Followg he same sraeg as Appedx A. beg b akg paral dervaves of SS wh respec o each subjec-specfc erm. Tha s * * * * SS a b Z R Za + Xb. * a Seg hs equal o zero elds * * Z R Z Z R X a b b We work wh he projeco / Z R Z Z R / Z R Z P ha s smmerc ad dempoe P Z P Z P Z. Wh hs oao we have / * / * R Za b PZ R Xb ad / * * / * / R Z a + X b R X b P R X b * Z * I P R / X. Z b Now defe he projeco Q Z I - P Z also smmerc ad dempoe. Wh hs oao he sum of squares s * * * / / * SS a b X b R Q R X b. Z To mmze he sum of squares ake a paral dervave wh respec o b*. Seg hs equal o zero elds he geeralzed leas squares esmaors: / / / / b FE XR Q Z R X XR Q Z R ad a Z R Z Z R X b FE FE.. A.3. Dagosc Sascs Observao level dagosc sasc We use Cook s dsace o dagose uusual observaos. For brev we assume R σ I. To defe Cook s dsace frs le a ad b deoe OLS fxed effecs esmaors of α ad β calculae whou he observao from he h subjec a he h me po. Whou hs observao he fed value for he jh subjec a he rh me po s ˆ jr z jra j + x jrb. We defe Cook s dsace o be D T j r ˆ ˆ jr q + K s jr

Chaper. Fxed Effecs Models / -9 where he fed value s calculaed as ˆ jr z jra j FE + x jrbfe. We calbrae D usg a F- dsrbuo wh umeraor df q+k degrees of freedom ad deomaor df N q+k degrees of freedom. The shor-cu calculao form s:. Calculae he leverage for he h subjec ad h me po as h z Z Z z + x X X x.. Resduals are calculaed as e a FE z + b FE x. The mea square error s 3. Cook s dsace s calculaed as s N q + K D T e h q + K s. e h. Subjec level dagosc sasc From Baerjee ad Frees 997S he geeralzao of Seco.4.3 o he fxed effecs logudal daa model defed Seco.5 s: B / b b FE b / FE X R Q Z R X b FE b FE / K Z / where Q I R / Z Z R Z Z R where B. The shor-cu calculao form s: / / b eq Z R I H H I H R Q Z e / / / / / Q Z X XR Q Z R X X Q Z R H R s he leverage marx ad e - X b FE. We calbrae B usg he ch-square dsrbuo wh K degrees of freedom. A.4. Cross-secoal Correlao - Shor-cu calculaos The sascs R AVE ad R AVE are averages over -/ correlaos whch ma be compuaoall esve for large values of. For a shor-cu calculao for R AVE we compue Fredma s sasc drecl T FR 3 + + r T T T ad he use he relao R AVE FR T-/-T-. For a shor-cu calculao for R AVE frs defe he qua Z u r T + / r T + / 3 u T T Wh hs qua a alerave expresso for R AVE s R AVE Z { } u u Z u.. Here Σ {u} meas sum over... T ad u... T. Alhough more complex appearace hs s a much faser compuaoal form for R AVE. K

-30 / Chaper. Fxed Effecs Models. Exercses ad Exesos Seco... Esmae logudal daa models usg regresso roues Cosder a fcous daa se wh x for 3 4 ad. Tha s we have: 3 3 4 4 x 4 3 6 4 8 Cosder he usual regresso model of he form X β + ε where he marx of explaaor varables s x x L xk x x L xk X. M M O M x x L xk You wsh o express our logudal daa model erms of he usual regresso model. a. Provde a expresso for he marx X for he regresso model equao.. Specf he dmeso of he marx as well as each er of he marx erms of he daa provded above. b. Cosder he basc fxed effecs model equao.. Express hs erms of he usual regresso model b usg bar dumm varables. Provde a expresso for he marx X. c. Provde a expresso for he marx X for he fxed effecs model equao.4. d. Provde a expresso for he marx X for he fxed effecs model equao.5. e. Suppose ow ha ou have 400 sead of 4 subjecs ad T 0 observaos per subjec sead of. Wha s he dmeso of our desg marces pars a-d? Wha s he dmeso of he marx X X ha regresso roues eed o ver? Seco.3.. Sadard errors for regresso coeffces Cosder he basc fxed effecs model equao.3 wh {ε } decall ad depedel dsrbued wh mea zero ad varace σ. a. Check equao.0 ha s prove ha Var T b σ W where W x x x x. b. Deerme he varace of he h ercep Var a. c. Deerme he covarace amog erceps ha s deerme Cova a j for j. d. Deerme he covarace bewee a ercep ad he slope esmaor ha s deerme Cova b. e. Deerme Var a + x* b where x* s a kow vecor of explaaor varables. For wha value of x* s hs a mmum?.3. Leas squares a. Suppose ha he regresso fuco s E α. Deerme he ordar leas squares esmaor for α. b. Suppose ha he regresso fuco s E α x where x s a scalar. Deerme he ordar leas squares esmaor for α.

Chaper. Fxed Effecs Models / -3 c. Suppose ha he regresso fuco s E α x + β. Deerme he ordar leas squares esmaor for α..4. Two populao slope erpreaos Cosder he basc fxed effecs model equao.3 ad suppose ha K ad ha x s a T bar varable. Specfcall le x be he umber of oes for he h subjec ad T x le T be he umber of zeroes. Furher defe o be he average T x whe x for he h subjec ad smlarl. a. Show ha we ma wre he fxed effecs slope gve equao.6 as wh weghs w / T. b. Ierpre hs slope coeffce b. Var b σ w. c. Show ha b w w d. Suppose ha ou would lke o mmze Var b ad ha he se of observaos umbers {T T } s fxed. How could ou desg he bar varables x ad hus ad o mmze Var b? e. Suppose ha x 0 for half he subjecs ad x for he oher half. Wha s Var b? Ierpre hs resul. f. Suppose ha he h subjec s desged so ha x 0. Wha s he corbuo of hs subjec o w?.5. Leas squares bas Suppose ha he aals should use he heerogeeous model equao. bu sead decdes o use a smpler homogeeous model of he form E α + x β. a. Call he leas square slope esmaor b H H for homogeeous. Show ha he slope esmaor s T T b H x x x x x x. b. Show ha he devao of b H from he slope β s T T b + H β x x x x x x α α ε ε. c. Assume ha K. Show ha he bas usg b H ca be expressed as E bh β Tα x x N s N where s x x x T x s he sample varace of x.

-3 / Chaper. Fxed Effecs Models.6. Resduals Cosder he basc fxed effecs model equao.3 ad suppose ha K. Defe he resduals of he ordar leas squares f as e - a + x b. a. Show ha he average resdual s zero ha s show e 0. b. Show ha he average resdual for he h subjec s zero ha s show e 0. c. Show ha e x 0. T ` j d. Wh does c mpl ha he esmaed correlao bewee he resduals ad he jh explaaor varable s zero? e. Show ha he esmaed correlao bewee he resduals ad he fed values s zero. f. Show ha he esmaed correlao bewee he resduals ad he observed depede varables s geeral o equal o zero. g. Wha are he mplcaos of pars e ad f for resdual aalss?.7. Group erpreao Cosder he basc fxed effecs model equao.3 ad suppose ha K. Suppose ha we are cosderg 5 groups. Each group was aalzed separael wh sadard devaos ad regresso slope coeffces gve below. For group he sample sadard x. T Group 3 4 5 Observaos per group T 9 9 Sample sadard devao 3 5 8 4 devao of he explaaor varable s gve b s x x s x Slope b 3 4-3 0 a. Use equao.8 o deerme he overall slope esmaor b. b. Dscuss he fluece of he group sample sadard devaos ad sze o b..8. Cossec Cosder he basc fxed effecs model equao.3 ad suppose ha K. A suffce codo for weak cossec of b s he mea square error eds o zero ha s E b - β 0 as ad T remas bouded. a. Show ha we requre a suffce amou of varabl he se of explaaor varables {x } order o esure cossec of b. Explcl wha does he phrase a suffce amou of varabl mea hs coex? b. Suppose ha x - for all ad. Does hs se of explaaor varables mee our suffce codo o esure cossec of b? c. Suppose ha x - for all ad. Does hs se of explaaor varables mee our suffce codo o esure cossec of b? d. Suppose ha x -/ for all ad. Does hs se of explaaor varables mee our suffce codo o esure cossec of b?.9. Leas Squares For he h subjec cosder he regresso fuco E α + x β T. a. Wre hs as a regresso fuco of he form E X * β * b gvg approprae defos for X * ad β *. T

Chaper. Fxed Effecs Models / -33 b. Use a resul o paroed marces equao A. of Appedx A5 o show ha he leas squares esmaor of β s T T b x x x x x x. Seco.4.0. Poolg es a. Assume balaced daa wh T 5 ad K 5. Use a sascal sofware package o show ha he 95 h percele of he F-dsrbuo wh df - ad df N-+K 4 5 behaves as follows. 0 5 0 5 50 00 50 500 000 95 h percele.608.8760.769.635.497.847.739.09.0845 b. For he poolg es sasc defed Seco.4. show ha F-rao as use weak or srog cossec. Ierpre he resuls of par a erms of hs resul... Added varable plo Cosder he basc fxed effecs model equao.3 ad suppose ha K. a. Beg wh he model whou a explaaor varables α + ε. Deerme he resduals for hs model deoed b e. b. Now cosder he model x α + ε. Deerme he resduals for hs represeao deoed b e. c. Expla wh a plo of {e } versus {e } s a specal case of added varable plos. d. Deerme he sample correlao bewee {e } ad {e }. Deoe hs correlao as corre e. e. For he basc fxed effecs model equao.3 wh K show ha T T b e ee. f. For he basc fxed effecs model equao. wh K show ha T T N + s e b ee. g. For he basc fxed effecs model equao. wh K esablsh he relaoshp descrbed Seco.4. bewee he paral correlao coeffce ad he -sasc. Tha s use pars d-f o show b corr e e. b + N +.. Observao level dagoscs We ow esablsh a shor-cu formula for calculag he usual Cook s dsace lear regresso models. To hs ed we cosder he lear regresso fuco E X β ha cosss of N rows ad use he subscrp o for a geerc observao. Thus le x o be he oh row or observao. Furher defe X o o be he marx of explaaor varables whou he oh observao ad smlarl for o. The ordar leas squares esmaor of β wh all observaos s b XX - X. a. Use he equao jus below equao A.3 Appedx A.5 o show ha

-34 / Chaper. Fxed Effecs Models X X X x x X X X X x x X X o o X o o o o + hoo where hoo x o X X x o s he leverage for he oh observao. b. The esmaor of β whou he oh observao s b o X o X o - X o o. Use par a o show ha X X x oeo b o b hoo where e o o x o b s he oh resdual. ˆ ˆ o ˆ ˆ o c. Cook s dsace s defed o be Do where ˆ Xb s he vecor of col X s fed values ad ˆ o Xb o s he vecor of fed values whou he oh observao. Show ha e o hoo Do. s h col hoo oo X d. Use he expresso par c o verf he formula for Cook s Dsace gve Appedx A.3..3. Cross-secoal correlao es sasc a. Calculae he varace of he radom varable Q Seco.4.4. b. The followg able provdes he 95 h percele of he Q radom varable as a fuco of T. T 4 5 6 7 8 9 0 5 95 h percele 0.83 0.683 0.57 0.495 0.43 0.38 0.344 0.86 0.7 Compue he correspodg cu-offs usg he ormal approxmao ad our aswer par a. Dscuss he performace of he approxmao as T creases. Seco.5.4. Seral correlaos Cosder he compoud smmer model where he error varace-covarace marx s gve b R σ -ρ I + ρ J. a. Check ha he verse of R s R - σ - -ρ I -ρ/t-ρ+ J. Do hs b showg ha R R - I he de marx. b. Use hs form of R equao.7 o show ha he fxed effecs esmaor of β b FE equals he ordar leas squares esmaor b gve Seco.5.3..5. Regresso model Cosder he geeral fxed effecs logudal daa model gve equao.6. Wre hs model as a regresso fuco he form E X* β*. Whe dog hs be sure o: a. Descrbe specfcall how o use he marces of explaaor varables {Z } ad {X } o form X*. b. Descrbe specfcall how o use he vecors of parameers {α } ad β o form β*. c. Idef he dmesos of X* ad β*.

Chaper. Fxed Effecs Models / -35.6. Ierpreg he slope as a weghed average of subjec-specfc slopes Cosder he geeral fxed effecs logudal daa model gve equao.6 ad defe / / he wegh marx W X R Q Z R X. a. Based o daa from ol he h subjec show ha he leas squares slope esmaor s / / b FE W X R Q Z R. H: cosder equao.7 wh. b. Show ha he fxed effecs esmaor of β ca be expressed as a marx weghed average of he form b FE W Wb FE..7 Fxed effecs lear logudal daa esmaors Cosder he regresso coeffce esmaors of he fxed effecs lear logudal daa model equaos.7 ad.8. Show ha f we assume o seral correlao q ad z he hese expressos reduce o he esmaors gve equaos.6 ad.7..8 Ordar leas squares based o dffereced daa Cosder he based fxed effecs model equao.3 ad use ordar leas squares based o dfferecg daa. Tha s daa wll be dffereced over me so ha he respose s - - ad he vecor of covaraes s x x - x -. a. Show ha he ordar leas squares esmaor of β based o dffereced daa s T T b x x x. b. Now compue he varace of hs esmaor. To hs ed defe he vecor of dffereced resposes... ad he correspodg marx of dffereced covaraes where X T x... x. Wh hs oao show ha T * X X XR X Var b X X 0 L 0 0 L 0 0 * 0 L 0 0 R Var σ. M M M O M M 0 0 0 L 0 0 0 L c. Now assume balaced daa so ha T T. Furher assume ha {x } are decall ad depedel dsrbued wh mea E x µ x ad varace Var x Σ x. Usg equao.0 show ha lm Var... b X X σ Σ wh probabl oe. T x d. Use he assumpos of par c. Usg par b show ha

-36 / Chaper. Fxed Effecs Models 3T 4 lm Var b X... X σ Σ x wh probabl oe. T e. From he resuls of pars c ad d argue ha he equao.6 fxed effecs esmaor s more effce ha he leas squares esmaor based o dffereced daa. Wha s he lmg as T effcec rao? Emprcal Exercses.9. Charable Corbuos We aalze dvdual come ax reurs daa from he 979-988 Sascs of Icome SOI Pael of Idvdual Reurs. The SOI Pael s a subse of he IRS Idvdual Tax Model Fle ad represes a smple radom sample of dvdual come ax reurs fled each ear. Based o he dvdual reurs daa he goal s o vesgae wheher a axpaer's margal ax rae affecs prvae charable corbuos ad secodl f he ax reveue losses due o charable corbuos deducos s less ha he ga of charable orgazaos. To address hese ssues we cosder a prce ad come model of charable corbuos cosdered b Baerjee ad Frees 997S. The laer defe prce as he compleme of a dvdual's federal margal ax rae usg axable come pror o corbuos. Icome of a dvdual s defed as he adjused gross come. The depede varable s oal charable corbuos whch s measured as he sum of cash ad oher proper corbuos excludg carr overs from prevous ears. Oher covaraes cluded he model are age maral saus ad he umber of depedes of a dvdual axpaer. Age s a dchoomous varable represeg wheher a axpaer s over sx four ears or o. Smlarl maral saus represes f a dvdual s marred or sgle. The populao cosss of all U.S. axpaers who emze her deducos. Specfcall hese are he dvduals who are lkel o have ad o record charable corbuo deducos a gve ear. Amog he 43 axpaers our subse of he SOI Pael approxmael % emzed her deducos each ear durg he perod 979-988. A radom sample of 47 dvduals was seleced from he laer group. These daa are aalzed Baerjee ad Frees 997S. Table E.. Taxpaer Characerscs Varable Descrpo SUBJECT Subjec defer -47. TIME Tme defer -0. CHARITY The sum of cash ad oher proper corbuos excludg carr overs from prevous ears. INCOME Adjused gross come. PRICE Oe mus he margal ax rae. Here he margal ax rae s defed o come pror o corbuos. AGE A bar varable ha equal oe f a axpaer s over sx four ears ad equals zero oherwse. MS A bar varable ha equal oe f a axpaer s marred ad equals zero oherwse. DEPS Number of depedes clamed o he axpaer s form. a Basc summar sascs Summarze each varable. For he bar varables AGE ad MS provde ol averages. For he oher varables CHARITY INCOME PRICE ad DEPS provde he mea meda sadard devao mmum ad maxmum. Furher summarze he average respose varable CHARITY over TIME.

Chaper. Fxed Effecs Models / -37 b c d Creae a mulple me seres plo of CHARITY versus TIME. Summarze he relaoshp amog CHARITY INCOME PRICE DEPS ad TIME. Do hs b calculag correlaos ad scaer plos for each par. Basc fxed effecs model Ru a oe-wa fxed effecs model of CHARITY o INCOME PRICE DEPS AGE ad MS. Sae whch varables are sascall sgfca ad jusf our coclusos. Produce a added varable plo of CHARITY versus INCOME corollg for he effecs of PRICE DEPSAGE ad MS. Ierpre hs plo. Icorporag emporal effecs. Is here a mpora me paer? Re-ru he model b ad clude TIME as a addoal explaaor couous varable. Re-ru he model b ad clude TIME hrough dumm varables oe for each ear. Re-ru he model b ad clude a AR compoe for he error. v Whch of he hree mehods for corporag emporal effecs do ou prefer? Be sure o jusf our cocluso. Uusual observaos Re-ru he model b ad calculae Cook s dsace o def uusual observaos. Re-ru he model b ad calculae he fluece sasc for each subjec. Idef he subjec wh he larges fluece sasc. Re-ru our model b omg he subjec ha ou have defed. Summarze he effecs o he global parameer esmaes..0. Tor Flgs The respose ha we cosder s FILINGS he umber of or acos agas surace compaes per oe hudred housad populao. For each of sx ears 984-989 he daa were obaed from 9 saes. Thus here are 6 9 4 observaos avalable. The ssue s o r o udersad how sae legal ecoomc ad demographc characerscs affec FILINGS. Table E. descrbes hese characerscs. More exesve movao s provded Seco 0.. Table E.. Sae Characerscs Depede Varable FILINGS Number of flgs of or acos agas surace compaes per oe hudred housad populao. Sae Legal Characerscs JSLIAB A dcaor of jo ad several labl reform. COLLRULE A dcaor of collaeral source reform. CAPS A dcaor of caps o o-ecoomc reform. PUNITIVE A dcaor of lms of puve damage Sae Ecoomc ad Demographc Characerscs POPLAWYR The populao per lawer. VEHCMILE Number of auomobles mles per mle of road housads. GSTATEP Perceage of gross sae produc from maufacurg ad cosruco. POPDENSY Number of people per e square mles of lad. WCMPMAX Maxmum workers compesao weekl beef. URBAN Perceage of populao lvg urba areas. UNEMPLOY Sae uemplome rae perceages. Source: Lee H. D. 994O.

-38 / Chaper. Fxed Effecs Models a Basc summar sascs Provde a basc able of uvarae summar sascs cludg he mea meda sadard devao mmum ad maxmum. Calculae he correlao bewee he depede varable FILINGS ad each of he explaaor varables. Noe he correlao bewee WCMPMAX ad FILINGS. Exame he mea of each varable over me. You ma also wsh o explore he daa oher was hrough lookg a dsrbuos ad basc plos. b F a pooled cross-secoal regresso model based o equao. usg VEHCMILE GSTATEP POPDENSY WCMPMAX URBAN UNEMPLOY ad JSLIAB as explaaor varables. c F a oe-wa fxed effecs model usg sae as he subjec defer ad VEHCMILE GSTATEP POPDENSY WCMPMAX URBAN UNEMPLOY ad JSLIAB as explaaor varables. d F a wo-wa fxed effecs model usg sae ad me subjec defers ad VEHCMILE GSTATEP POPDENSY WCMPMAX URBAN UNEMPLOY ad JSLIAB as explaaor varables. e Perform paral F-ess o see whch of he models pars b c ad d ha ou prefer. f The oe-wa fxed effecs model usg sae as he subjec defer produced a sascall sgfca posve coeffce o WCMPMAX. However hs was o cogruece wh he correlao coeffce. To expla hs coeffce produce a added varable plo of FILINGS versus WCMPMAX corollg for sae ad he oher explaaor varables VEHCMILE GSTATEP POPDENSY URBAN UNEMPLOY ad JSLIAB. g For our model sep c calculae he fluece sasc for each subjec. Idef he subjec wh he larges fluece. You ma also wsh o re-ru he par c model whou hs subjec. h Re-ru he par c model bu assumg a auoregressve of order error srucure. Is hs AR coeffce sascall sgfca?.. Housg Prces I hs problem we wll exame models of housg prces US meropola areas. Ma sudes have addressed he housg marke see for example Gree ad Malpezz 003O for a roduco. The prces of houses are flueced b demad-sde facors such as come ad demographc varables. Suppl-sde facors such as he regulaor evrome of a meropola area ma also be mpora. The daa cosss of aual observaos from 36 meropola sascal areas MSAs over he e-ear perod 986-994. The respose varable s NARSP a MSA s average sale prce based o rasacos repored hrough he Mulple Lsg Servce Naoal Assocao of Realors. As par of a prelmar aalss he respose varable has bee rasformed usg a aural logarhm. For hs problem he demad-sde varables are me varg e he suppl-sde varables do o var wh me.

Chaper. Fxed Effecs Models / -39 Table E.3. MSA Characerscs Respose varable NARSP a MSA's average sale prce logarhmc us. I s based o rasacos repored hrough he Mulple Lsg Servce. Demad sde explaaor varables YPC Aual Per Capa come from he Bureau of Ecoomc Aalss POP Populao from he Bureau of Ecoomc Aalss PERYPC Aual perceage growh of per capa come PERPOP Aual perceage growh of populao Suppl sde explaaor varables REGTEST Regulaor dex from Malpezz 996O. RCDUM Re corol dumm varable SREG Sum of Amerca Isue of Plaers sae regulaor quesos regardg use of evromeal plag ad maageme. AJWTR Idcaes wheher he MSA s adjace o a coasle AJPARK Idcaes wheher he MSA s adjace o oe or more large parks mlar bases or reservaos. Addoal Varables MSA Subjec MSA defer - 36. TIME Tme defer -9. a b Basc summar sascs Beg b summarzg he me-cosa suppl sde explaaor varables. Provde meas for he bar varables ad he mea meda sadard devao mmum ad maxmum for he oher varables. Assess he sregh of he relaoshp amog he suppl sde explaaor varables b calculag correlaos. Summarze he addoal varables. Provde mea meda sadard devao mmum ad maxmum for hese varables. v Exame reds over me producg he meas over me for he o-suppl sde varables. v Produce a mulvarae me seres plo of NARSP. v Calculae correlaos amog NARSP PERYPC PERPOP ad YEAR. v Plo PERYPC versus YEAR. Comme o he uusual behavor 993 ad 994. v Produce a added varable plo of NARSP versus PERYPC corollg for he effecs of PERPOP ad YEAR. Ierpre hs plo. Basc fxed effecs model F a homogeeous model as equao. usg PERYPC PERPOP ad YEAR as explaaor varables. Comme o he sascal sgfcace of each varable ad he overall model f. Ru a oe-wa fxed effecs model of NARSP o PERYPC PERPOP ad YEAR. Sae whch varables are sascall sgfca ad jusf our coclusos. Comme o he overall model f. Compare he models pars b ad b usg a paral F-es. Sae whch model ou prefer based o hs es. v Re-ru he sep b b excludg YEAR as a addoal explaaor couous varable e cludg a AR compoe for he error. Sae wheher or o YEAR should be cluded he model.

-40 / Chaper. Fxed Effecs Models c v For he model b calculae he fluece sasc for each MSA. Idef he MSA wh he larges fluece sasc. Re-ru our model b omg he MSA ha ou have defed. Summarze he effecs o he global parameer esmaes. Addoal aalses We have o e red o f a suppl-sde varables. Re-do he model f par b e cludg suppl sde varables REGTEST RCDUM SREG AJPARK ad AJWTR. Comme o he sascal sgfcace of each varable ad he overall model f. Re-ru he model par c ad clude a dumm varable for each MSA resulg a oewa fxed effecs models. Comme o he dffcul of achevg uque parameer esmaes wh hs procedure.. Bod Maur Usrucured problem These daa coss of observaos of 38 o-regulaed frms over he perod 980-989. The goal s o assess he deb maur srucure of a frm. For hs exercse develop a fxed effecs model. For oe approach see Sohs ad Mauer 996O. Table E.4 Bod Maur Varable Descrpo DEBTMAT The book value-weghed average of he frm s deb. Ths s he respose varable. SIC Sadard Idusral Classfcao SIC of he frm. FIRMID Subjec frm defer -38 TIME Tme defer -0 MVBV The marke value of he frm proxed b he sum of he book value of asses ad he marke value of equ less he book value of equ scaled b he book value of asses. SIZE The aural logarhm of he esmae of frm value measured 98 dollars usg he PPI deflaor. CHANGEEPS The dfferece bewee ex ear s eargs per share ad hs ear s eargs per share scaled b hs ear s commo sock prce per share. ASSETMAT The book value-weghed average of he maures of curre asses ad e proper pla ad equpme. VAR Rao of he sadard devao of he frs dfferece eargs before eres deprecao ad axes o he average of asses over he perod 980-989. TERM The dfferece bewee he log-erm ad shor-erm elds o goverme bods. BONDRATE The frm s S&P bod rag. TAXRATE Rao of come axes pad o preax come. LEVERAGE Rao of oal deb o he marke value of he frm.

Chaper 3. Models wh Radom Effecs / 3-003 b Edward W. Frees. All rghs reserved Chaper 3. Models wh Radom Effecs Absrac. Ths chaper cosders he Chaper daa srucure bu here he heerogee s modeled usg radom quaes leu of fxed parameers; hese radom quaes are kow as radom effecs. B roducg radom quaes he aalss of logudal ad pael daa ca ow be cas he mxed lear model framework. Alhough mxed lear models are a esablshed par of sascal mehodolog her use s o as wdespread as regresso. Thus he chaper roduces hs modelg framework begg wh he specal case of a sgle radom ercep kow as he error compoes model ad he focusg o he lear mxed effecs model ha s parcularl mpora for logudal daa. Afer roducg he models hs chaper descrbes esmao of regresso coeffces ad varace compoes as well as hpohess esg for regresso coeffces. 3. Error compoes / radom erceps model Samplg ad ferece Suppose ha ou are eresed sudg he behavor of dvduals ha are radoml seleced from a populao. For example Seco 3. we wll sud he effecs ha a dvdual s ecoomc ad demographc characerscs have o he amou of come ax pad. Here he se of subjecs ha we wll sud s radoml seleced from a larger daabase ha s self a radom sample of he US axpaers. I coras he Chaper Medcare example deal wh a fxed se of subjecs. Tha s s dffcul o hk of he 54 saes as a subse from some super-populao of saes. For boh suaos s aural o use subjec-specfc parameers {α } o represe he heerogee amog subjecs. Ulke Chaper Chaper 3 dscusses suaos whch s more reasoable o represe {α } as radom varables sead of fxed e ukow parameers. B argug ha {α } are draws from a dsrbuo we wll have he abl o make fereces abou subjecs a populao ha are o cluded he sample. Basc model ad assumpos The error compoes model equao s α + x β + ε. 3. Ths poro of he oao s he same as he error represeao of he basc fxed effecs model. However ow he erm α s assumed o be a radom varable o a fxed ukow parameer. The erm α s kow as a radom effec. Mxed effecs models are oes ha clude radom as well as fxed effecs. Because equao 3. cludes radom effecs α ad fxed effecs β he error compoes model s a specal case of he mxed lear model. To complee he specfcao of he error compoes model we assume ha {α } are decall ad depedel dsrbued wh mea zero ad varace σ α. Furher we assume ha {α } are depede of he error radom varables {ε }. For compleeess we sll assume ha x s a vecor of covaraes or explaaor varables ad ha β s a vecor of fxed e ukow populao parameers. Noe ha because E α 0 s cusomar o clude a cosa

3- / Chaper 3. Models wh Radom Effecs wh he vecor x. Ths was o rue of he fxed effecs models Chaper where we dd o ceer he subjec-specfc erms abou 0. Lear combaos of he form x β quaf he effec of kow varables ha ma affec he respose. Addoal varables ha are eher umpora or uobservable comprse he error erm. I he error compoes model we ma hk of a regresso model x β + η where he error erm η s decomposed o wo compoes so ha η α + ε. The erm α represes he me-cosa poro whereas ε represes he remag poro. To def he model parameers we assume ha he wo erms are depede. I he bologcal sceces he error compoes model s kow as he radom erceps model; hs descrpor s used because he ercep α s a radom varable. We wll use he descrpors error compoes ad radom erceps erchageabl alhough for smplc we ofe use ol he former erm. Tradoal ANOVA se-up I he error compoes model he erms {α } accou for he heerogee amog subjecs. To help erpre hs feaure cosder he specal case where K x ad deoe µ β. I hs case equao 3. coas o explaaor varables ad reduces o µ +α + ε he radoal radom effecs oe-wa ANOVA model. Neer ad Wasserma 974G descrbe hs classc model. Ths model ca be erpreed as arsg from a wo-sage samplg scheme: Sage. Draw a radom sample of subjecs from a populao. The subjecspecfc parameer α s assocaed wh he h subjec. Sage. Codoal o α draw realzaos of { } for T for he h subjec. Tha s he frs sage we draw a sample from a populao of subjecs. I he secod sage we observe each subjec over me. Because he frs sage s cosdered a radom draw from a populao of subjecs we represe characerscs ha do o deped o me hrough he radom qua α. Fgure 3. llusraes he wo-sage samplg scheme. α α α 3 α α α 3 Sage Sage Fgure 3.. Two-sage radom effecs samplg. I he lef pael uobserved subjec-specfc compoes are draw from a uobserved populao. I he rgh pael several observaos are draw for each subjec. These observaos are ceered abou he uobserved subjecspecfc compoes from he frs sage. Dffere plog smbols represe draws for dffere subjecs.

Chaper 3. Models wh Radom Effecs / 3-3 Wh hs radoal model eres geerall ceers abou he dsrbuo of he populao of subjecs. For example he parameer Var α σ α summarzes he heerogee amog subjecs. I Chaper o fxed effecs models we examed he heerogee ssue hrough a es of he ull hpohess H 0 : α α α. I coras uder he radom effecs model we exame he ull hpohess H 0 : σ α 0. Furhermore esmaes of σ α are of eres bu requre scalg o erpre. A more useful qua o repor s σ α /σ α + σ he ra-class correlao. As we saw Seco.5. hs qua ca be erpreed as he correlao bewee observaos wh a subjec. The correlao s cosraed o le bewee 0 ad ad does o deped o he us of measureme for he respose. Furher ca also be erpreed as he proporo of varabl of a respose ha s due o heerogee amog subjecs. Samplg ad model assumpos The Seco. basc fxed effecs ad error compoes models are smlar appearace e as wll be dscussed Seco 7. ca lead o dffere subsave coclusos he coex of a specfc applcao. As we have descrbed he choce bewee hese wo models s dcaed prmarl b he mehod whch he sample s draw. O he oe had selecg subjecs based o a wo-sage or cluser sample mples use of he radom effecs model. O he oher had selecg subjecs based o exogeous characerscs suggess a srafed sample ad hus usg a fxed effecs model. The samplg bass allows us o resae he error compoes model as follows. Error Compoes Model Assumpos R. E α α + x β. R. {x... x K } are osochasc varables. R3. Var α σ. R4. { } are depede radom varables codoal o {α α }. R5. s ormall dsrbued codoal o {α α }. R6. E α 0 Var α σ α ad {α α } are muuall depede. R7. {α } s ormall dsrbued. Assumpos R-R5 are smlar o he fxed effecs models assumpos F-F5; he ma dfferece s ha we ow codo o radom subjec-specfc erms {α α }. Assumpos R6 ad R7 summarze he samplg bass of he subjec-specfc erms. Take ogeher hese assumpos comprse our error compoes model. However assumpos R-R7 do o provde a observables represeao of he model because he are based o uobservable quaes {α α }. We summarze he effecs of Assumpos R-R7 o he observable varables {x... x K }. Observables Represeao of he Error Compoes Model RO. E x β. RO. {x... x K } are osochasc varables. RO3. Var σ + σ α ad Cov r s σ α for r s. RO4. { } are depede radom vecors. RO5. { } s ormall dsrbued. To reerae he properes RO-5 are a cosequece of R-R7. As we progress o more complex suaos our sraeg wll coss of usg samplg bases o sugges basc assumpos such as R-R7 ad he cover hem o esable properes such as RO-5. Iferece abou he esable properes he provdes formao abou he more basc assumpos. Whe cosderg olear models begg Chaper 9 hs coverso wll o

3-4 / Chaper 3. Models wh Radom Effecs be as drec. I some saces we wll focus o he observable represeao drecl ad refer o as a margal or populao-averaged model. The margal verso emphaszes he assumpo ha observaos are correlaed wh subjecs Assumpo RO3 o he radom effecs mechasm for ducg he correlao. For more complex suaos wll be useful o descrbe hese assumpos marx oao. As equao.3 he regresso fuco ca be expressed more compacl as E α α + Xβ ad hus E X β. 3. Recall ha s a T vecor of oes ad from equao.4 ha X s a T K marx of explaaor varables X x x L xt. The expresso for E α s a resaeme of Assumpo R marx oao. Equao 3. s a resaeme of Assumpo RO. Aleravel equao 3. s due o he law of eraed expecaos ad assumpos R ad R6 because E E E α E α + X β X β. For Assumpo RO3 we have Var V σ α J +σ I. 3.3 Here recall ha J s a T T marx of oes ad I s a T T de marx. Srucural models Model assumpos are ofe dcaed b samplg procedures. However we also wsh o cosder sochasc models ha represe causal relaoshps suggesed b a subsave feld kow as srucural models. Seco 6. descrbes srucural modelg logudal ad pael daa aalss. To llusrae models of ecoomc applcaos s mpora o cosder carefull wha oe meas b he populao of eres. Specfcall whe cosderg choces of ecoomc ees a sadard defese for a probablsc approach o aalzg ecoomc decsos s ha alhough here ma be a fe umber of ecoomc ees here s a fe rage of ecoomc decsos. For example he Chaper Medcare hospal cos example oe ma argue ha each sae faces a dsrbuo of fel ma ecoomc oucomes ad ha hs s he populao of eres. Ths vewpo argues ha oe should use a error compoes model. Here we erpre {α } o represe hose aspecs of he ecoomc oucome ha are uobservable e cosa over me. I coras Chaper we mplcl used he samplg based model o erpre {α } as fxed effecs. Ths vewpo s he sadard raoale for sudg sochasc ecoomcs. To llusrae a quoe from Haavelmo 944E s relaed o hs po: he class of populaos we are dealg wh does o coss of a f of dffere dvduals cosss of a f of possble decsos whch mgh be ake wh respec o he value of. Ths defese s well summarzed b Nerlove ad Balesra a moograph eded b Máás ad Sevesre 996E Chaper he coex of pael daa modelg. Iferece Whe desgg a logudal sud ad cosderg wheher o use a fxed or radom effecs model keep md he purposes of he sud. If ou would lke o make saemes abou a populao larger ha he sample he use he radom effecs model. Coversel f ou are smpl eresed corollg for subjec-specfc effecs reag hem as usace parameers or makg predcos for a specfc subjec he use he fxed effecs model.

Chaper 3. Models wh Radom Effecs / 3-5 Tme-cosa varables Whe desgg a logudal sud ad cosderg wheher o use a fxed or radom effecs model also keep md he varables of eres. Ofe he prmar eres s esg for he effec of a me-cosa varable. To llusrae our axpaer example we ma be eresed he effecs ha geder ma have o a dvdual s ax labl we assume ha hs varable does o chage for a dvdual over he course of our sud. Aoher mpora example of a me-cosa varable s a varable ha classfes subjecs o groups. Ofe we wsh o compare he performace of dffere groups for example a reame group ad a corol group. I Seco.3 we saw ha me-cosa varables are perfecl collear wh subjecspecfc erceps ad hece are esmable. I coras wll ur ou ha coeffces assocaed wh me-cosa varables are esmable a radom effecs model. Hece f a mecosa varable such as geder or reame group s he prmar varable of eres oe should desg he logudal sud so ha a radom effecs model ca be used. Degrees of freedom Whe desgg a logudal sud ad cosderg wheher o use a fxed or radom effecs model also keep md he sze of he daa se ecessar for ferece. I mos logudal daa sudes ferece abou he populao parameers β s he prmar goal whereas he erms {α } are cluded o corol for he heerogee. I he basc fxed effecs model we have see ha here are +K lear regresso parameers plus varace parameer. Ths s compared o ol +K regresso plus varace parameers he basc radom effecs model. Parcularl sudes where he me dmeso s small such as T or 3 a desg suggesg a radom effecs model ma be preferable because fewer degrees of freedom are ecessar o accou for he subjec-specfc parameers. GLS esmao Equaos 3. ad 3.3 summarze he mea ad varace of he vecor of resposes. To esmae regresso coeffces hs chaper uses geeralzed leas squares GLS equaos of he form: X V X β X V. The soluo of hese equaos elds geeralzed leas square esmaors ha hs coex we call he error compoes esmaor of β. Addoal algebra Exercse 3. shows ha hs esmaor ca be expressed as b ζ EC X I J X T X I ζ J T. 3.4 Tσ α Here he qua ζ s a fuco of he varace compoes σ α ad σ. I Tσ α + σ Chaper 4 we wll refer o hs qua as he credbl facor. Furher he varace of he error compoes esmaor urs ou o be Varb ζ σ X I J X. T EC

3-6 / Chaper 3. Models wh Radom Effecs To erpre b EC we gve a alerave form for he correspodg Chaper fxed effecs esmaor. Tha s from equao.6 ad some algebra we have b X I T J X X I T J. Thus we see ha he radom effecs b EC ad fxed effecs b are approxmael equal whe he credbl facors are close o oe. Ths occurs whe σ α s large relave o σ. Iuvel whe here s subsaal separao amog he ercep erms relave o he ucera he observaos we acpae ha he fxed ad radom effec esmaors wll behave smlarl. Coversel equao 3.3 shows ha b EC s approxmael equal o a ordar leas squares esmaor whe σ s large relave o σ α so ha he credbl facors are close o zero. Seco 7. furher develops he comparso amog hese alerave esmaors. Feasble geeralzed leas squares esmaor The calculao of he GLS esmaor equao 3.4 assumes ha he varace compoes σ α ad σ are kow. Procedure for compug a feasble geeralzed leas squares esmaor. Frs ru a regresso assumg σ α 0 resulg a ordar leas squares esmae of β.. Use he resduals from Sep o deerme esmaes of σ α ad σ. 3. Usg he esmaes of σ α ad σ from Sep deerme b EC usg equao 3.4. For Sep here are ma was of esmag he varace compoes. Seco 3.5 provdes deals. Ths procedure could be eraed. However sudes have show ha eraed versos do o mprove he performace of he oe-sep esmaors. See for example Carroll ad Ruper 988G. To llusrae we cosder some smple mome-based esmaors of σ α ad σ due o Balag ad Chag 994E. Defe he resduals e - a + x b usg a ad b accordg o he Chaper fxed effecs esmaors equaos.6 ad.7. The he esmaor of σ s s as gve equao.. The esmaor of σ α s: a a T w s c α N T / N s where aw N T a ad T c + race x x x x T x x x x. A poeal drawback s ha a parcular realzao of s α udesrable for a varace esmaor. ma be egave; hs feaure s Poolg es As wh he radoal radom effecs ANOVA model he es for heerogee or poolg es s wre as a es of he ull hpohess H 0 : σ α 0. Tha s uder he ull hpohess we do o have o accou for subjec-specfc effecs. Alhough hs s a dffcul ssue for he geeral case he specal case of error compoes desrable es procedures have bee developed. We dscuss here a es ha exeds a Lagrage mulpler es sasc due o

Chaper 3. Models wh Radom Effecs / 3-7 Breusch ad Paga 980E o he ubalaced daa case. See Appedx C.7 for a roduco o Lagrage mulpler sascs. Ths es s a smpler verso of oe developed b Balag ad L 990E for a more complex model specfcall a wo-wa error compoe model ha we wll roduce Chaper 6. Poolg es procedure. Ru he pooled cross-secoal regresso model x β + ε o ge resduals e. T. For each subjec compue a esmaor of σ α s T e e where T T e T T e. s T T 3. Compue he es sasc TS. T N e 4. Rejec H 0 f TS exceeds a percele from a χ ch-square dsrbuo wh oe degree of freedom. The percele s oe mus he sgfcace level of he es. Noe ha he poolg es procedure uses esmaors of σ α s ha ma be egave wh posve probabl. Seco 5.4 dscusses alerave procedures where we resrc varace esmaors o be oegave. 3. Example: Icome ax pames I hs seco we sud he effecs ha a dvdual s ecoomc ad demographc characerscs have o he amou of come ax pad. Specfcall he respose of eres s LNTAX defed as he aural logarhm of he labl o he ax reur. Table 3. descrbes several axpaer characerscs ha ma affec ax labl. The daa for hs sud are from he Sascs of Icome SOI Pael of Idvdual Reurs a par of he Ers ad Youg/Uvers of Mchga Tax Research Daabase. The SOI Pael represes a smple radom sample of uauded dvdual come ax reurs fled for ax ears 979-990. The daa are compled from a srafed probabl sample of uauded dvdual come ax reurs Forms 040 040A ad 040EZ fled b U.S. axpaers. The esmaes ha are obaed from hese daa are eded o represe all reurs fled for he come ax ears uder revew. All reurs processed are subjeced o samplg excep eave ad ameded reurs. We exame a balaced pael from 98-984 ad 986-987 axpaers cluded he SOI pael; a four perce sample of hs comprses our sample of 58 axpaers. These ears are chose because he coa he eresg formao o pad preparer usage. Specfcall hese daa clude le em ax reur daa plus a bar varable og he presece of a pad ax preparer for ears 98-984 ad 986-987. These daa are also aalzed Frschma ad Frees 999O. The prmar goal of hs aalss s o deerme wheher ax preparers sgfcal affec ax labl. To movae hs queso we oe ha preparers have he opporu o mpac vruall ever le em o a ax reur. Our varables are seleced because he appear cossel pror research ad are largel ousde he fluece of ax preparers ha s he are exogeous. Brefl our explaaor varables are as follows: MS HH AGE EMP ad PREP are bar varables coded for marred head-of-household a leas 65 ears of age self-

3-8 / Chaper 3. Models wh Radom Effecs emploed ad pad preparer respecvel. Furher DEPEND s he umber of depedes ad MR s he margal ax rae measure. Fall LNTPI ad LNTAX are he oal posve come ad ax labl as saed o he reur 983 dollars logarhmc us. Table 3. Taxpaer Characerscs Demographc Characerscs MS s a bar varable oe f he axpaer s marred ad zero oherwse. HH s a bar varable oe f he axpaer s he head of household ad zero oherwse. DEPEND s he umber of depedes clamed b he axpaer. AGE s a bar varable oe f he axpaer s age 65 or over ad zero oherwse. Ecoomc Characerscs LNTPI s he aural logarhm of he sum of all posve come le ems o he reur 983 dollars.. MR s he margal ax rae. I s compued o oal persoal come less exempos ad he sadard deduco. EMP s a bar varable oe f Schedule C or F s prese ad zero oherwse. Self-emploed axpaers have greaer eed for professoal asssace o reduce he reporg rsks of dog busess. PREP s a varable dcag he presece of a pad preparer. LNTAX s he aural logarhm of he ax labl 983 dollars. Ths s he respose varable of eres. Tables 3. ad 3.3 descrbe he basc axpaer characerscs used our aalss. The bar varables Table 3. dcae ha over half he sample s marred MS ad approxmael half he sample uses a pad preparer PREP. Preparer use appears hghes 986 ad 987 ears sraddlg sgfca ax law chage. Slghl less ha e perce of he sample s 65 or older AGE 98. The presece of self-emplome come EMP also vares over me. TABLE 3. Averages of Bar Varables 58 YEAR MS HH AGE EMP PREP 98 0.597 0.08 0.085 0.40 0.450 983 0.597 0.093 0.05 0.59 0.44 984 0.64 0.085 0. 0.55 0.484 986 0.647 0.08 0.3 0.47 0.508 987 0.647 0.093 0.47 0.47 0.56 The summar sascs for he oher o-bar varables are Table 3.3. Furher aalses dcae a creasg come red eve afer adjusg for flao as measured b oal posve come LNTPI. Moreover boh he mea ad meda margal ax raes MR are decreasg alhough mea ad meda ax lables LNTAX are sable see Fgure 3.. These resuls are cosse wh cogressoal effors o reduce raes ad expad he ax base hrough broadeg he defo of come ad elmag deducos.

Chaper 3. Models wh Radom Effecs / 3-9 Table 3.3 Summar Sascs for Oher Varables Varable Mea Meda Mmum Maxmum Sadard devao DEPEND.49.000 0.000 6.000.338 LNTPI 9.889 0.05-0.8 3..65 MR 3.53.000 0.000 50.000.454 LNTAX 6.880 7.70 0.000.860.695 LNTAX 0 4 6 8 0 98 983 984 986 987 YEAR Fgure 3.. Boxplo of LNTAX versus YEAR. Logarhmc ax labl real dollars s sable over he ears 98-987. To explore he relaoshp bewee each dcaor varable ad logarhmc ax Table 3.4 preses he average logarhmc ax labl b level of dcaor varable. Ths able shows ha marred flers pa greaer ax head of household flers pa less ax axpaers 65 or over pa less axpaers wh self-emploed come pa less ad axpaers ha use a professoal ax preparer pa more. TABLE 3.4 Averages of Logarhmc Tax b Level of Explaaor Varable Explaaor Varable Level of Explaaor MS HH AGE EMP PREP Varable 0 5.973 7.03 6.939 6.983 6.64 7.430 5.480 6.43 6.97 7.58

3-0 / Chaper 3. Models wh Radom Effecs Table 3.5 summarzes basc relaos amog logarhmc ax ad he oher o-bar explaaor varables. Boh LNTPI ad MR are srogl correlaed wh logarhmc ax whereas he relaoshp bewee DEPEND ad logarhmc ax s posve e weaker. Furher Table 3.5 shows ha LNTPI ad MR are srogl posvel correlaed. TABLE 3.5 Correlao Coeffces DEPEND LNTPI MR LNTPI 0.78 MR 0.8 0.796 LNTAX 0.085 0.78 0.747 Alhough o preseed deal here explorao of he daa revealed several oher eresg relaoshps amog he varables. To llusrae a basc added varable plo Fgure 3.3 shows he srog relao bewee logarhmc ax labl ad oal come eve afer corollg for subjec-specfc me-cosa effecs. Resduals from LNTAX 8 6 4 0 - -4-6 -8-8 -6-4 - 0 4 Resduals from LNTPI Fgure 3.3. Added varable plo of LNTAX versus LNTPI. The error compoes model descrbed Seco 3. was f usg he explaaor varables descrbed Table 3.. The esmaed model appears Dspla 3. from a f usg he sascal package SAS. Dspla 3. shows ha HH EMP LNTPI ad MR are sascall sgfca varables ha affec LNTAX. Somewha surprsgl he PREP varable was o sascall sgfca. To es for he mporace of heerogee he Seco 3. poolg es was performed. A f of he pooled cross-secoal model wh he same explaaor varables produced resduals ad a error sum of squares equal o Error SS 3599.73. Thus wh T 5 ears ad 58 subjecs he es sasc s TS 73.5. Comparg hs es sasc o a ch-square dsrbuo wh oe degree of freedom dcaes ha he ull hpohess of homogee s rejeced. As we

Chaper 3. Models wh Radom Effecs / 3- wll see Chaper 7 here are some uusual feaures of hs daa se ha cause hs es sasc o be large. Dspla 3. Seleced SAS Oupu Ierao Hsor Ierao Evaluaos - Log Lke Crero 0 4984.6806443 479.5465804 0.0000000 Covergece crera me. Covarace Parameer Esmaes Cov Parm Subjec Esmae Iercep SUBJECT 0.97 Resdual.8740 F Sascs - Log Lkelhood 479.3 AIC smaller s beer 483.3 AICC smaller s beer 483.5 BIC smaller s beer 485.3 Soluo for Fxed Effecs Sadard Effec Esmae Error DF Value Pr > Iercep -.9604 0.5686 57-5. <.000 MS 0.03730 0.88 04 0. 0.8375 HH -0.6890 0.3 04 -.98 0.009 AGE 0.0074 0.993 04 0.0 0.97 EMP -0.5048 0.674 04-3.0 0.006 PREP -0.070 0.7 04-0.9 0.8530 LNTPI 0.7604 0.0697 04 0.9 <.000 DEPEND -0.8 0.05907 04 -.9 0.0566 MR 0.54 0.00788 04 5.83 <.000

3- / Chaper 3. Models wh Radom Effecs 3.3 Mxed effecs models Smlar o he exesos for he fxed effecs model descrbed Seco.5 we ow exed he error compoes model o allow for varable slopes seral correlao ad heeroscedasc. 3.3. Lear mxed effecs model We ow cosder codoal regresso fucos of he form E α z α + x β. 3.5 Here he erm z α comprses he radom effecs poro of he model. The erm x β comprses he fxed effecs poro. As wh equao.5 for fxed effecs equao 3.5 s shor-had oao for E α α z + α z +... + α q z q + β x + β x +... + β K x K. As equao.6 a marx form of equao 3.5 s E α Z α + X β. 3.6 We also wsh o allow for seral correlao ad heeroscedasc. Smlar o Seco.5. for fxed effecs we ca corporae hese exesos hrough he oao Var α R. We maa he assumpo ha he resposes bewee subjecs are depede. Furher we assume ha he subjec-specfc effecs {α } are depede wh mea E α 0 ad varace-covarace marx Var α D a q q posve defe marx. B assumpo he radom effecs are mea zero; hus a ozero mea for a radom effec mus be expressed as par of he fxed effecs erms. The colums of Z are usuall a subse of he colums of X. Take ogeher hese assumpos comprse wha we erm he lear mxed effecs model. Lear Mxed Effecs Model Assumpos R. E α Z α + X β. R. {x... x K } ad {z... z q } are osochasc varables. R3. Var α R. R4. { } are depede radom vecors codoal o {α α }. R5. { } s ormall dsrbued codoal o {α α }. R6. E α 0 Var α D ad {α α } are muuall depede. R7. {α } s ormall dsrbued. Wh assumpos R3 ad R6 he varace of each subjec ca be expressed as Var Z D Z + R V τ V. 3.7 The oao V τ meas ha he varace-covarace marx of depeds o varace compoes τ. Seco.5. provded several examples ha llusrae how R ma deped o τ; we wll gve specal cases o show how V ma deped o τ. Wh hs we ma summarze he effecs of Assumpos R-R7 o he observables varables {x... x K z... z q }.

Chaper 3. Models wh Radom Effecs / 3-3 Observables Represeao of he Lear Mxed Effecs Model RO. E X β. RO. {x... x K } ad {z... z q } are osochasc varables. RO3. Var Z D Z + R V τ V. RO4. { } are depede radom vecors. RO5. { } s ormall dsrbued. As Chaper ad Seco 3. he properes RO-5 are a cosequece of R-R7. We focus o hese properes because he are he bass for esg our specfcao of he model. The observable represeao s also kow as a margal or populao-averaged model. Example Trade localzao Feberg Keae ad Bogao 998E suded 70 U.S. based mulaoal corporaos over he perod 983-99. Usg frm-level daa avalable from he Bureau of Ecoomc Aalss of he U.S. Deparme of Commerce he documeed how large corporao s allocao of emplome ad durable asses proper pla ad equpme of Caada afflaes chaged respose o chages Caada ad U.S. arffs. Specfcall her model was l β CT + β UT + β 3 Tred + x * β* + ε β +α CT + β +α UT + β 3 +α 3 Tred + x * β* + ε α CT + α UT + α 3 Tred + x β + ε. Here CT s he sum over all dusr Caada arffs whch frm belogs ad smlarl for UT. The vecor x cludes CT UT ad Tred for he mea effecs as well as real U.S. ad Caada wages gross domesc produc prce eargs rao real U.S. eres raes ad a measure of rasporao coss. For he respose he used boh Caada emplome ad durable asses. The frs equao emphaszes ha respose o chages Caada ad U.S. arffs as well as me reds s frm-specfc. The secod equao provdes he lk o he hrd expresso ha s erms of he lear mxed effecs model form. Here we have cluded CT UT ad Tred x * o ge x. Wh hs reformulao he mea of each radom slope s zero ha s E α E α E α 3 0. I he frs specfcao he meas are E β β E β β ad E β 3 β 3. Feberg Keae ad Bogao foud ha a sgfca poro of he varao was due o frmspecfc slopes; he arbue hs varao o doscrac frm dffereces such as echolog ad orgazao. The also allowed for heerogee he me red. Ths allows for uobserved me-varg facors such as echolog ad demad ha affec dvdual frms dfferel. A major fdg of hs paper s ha Caada arff levels were egavel relaed o asses ad emplome Caada; hs fdg coradcs he hpohess ha lower arffs would uderme Caada maufacurg.

3-4 / Chaper 3. Models wh Radom Effecs Specal cases To help erpre lear mxed effecs models we cosder several mpora specal cases. We beg b emphaszg he case where q ad z. I hs case he lear mxed effecs model reduces o he error compoes model roduced Seco 3.. For hs model we have ol subjec-specfc erceps o subjec-specfc slopes ad o seral correlao. Repeaed measures desg Aoher classc model s he so-called repeaed measures desg. Here several measuremes are colleced o a subjec over a relavel shor perod of me uder corolled expermeal codos. Each measureme s subjec o a dffere reame bu he order of reames s radomzed so ha o seral correlao s assumed. Specfcall we cosder... subjecs. A respose for each subjec s measured based o each of T reames where he order of reames s radomzed. The mahemacal model s: respose radom subjec effec + fxed reame effec + error 443 444444 3 444444 3 3 ε α β The ma research queso of eres s H 0 : β β... β T ha s he ull hpohess s o reame dffereces. The repeaed measures desg s a specal case of equao 3.5 akg q z T T K T ad usg he h explaaor varable x o dcae wheher he h reame has bee appled o he respose. Radom coeffces model We ow reur o he lear mxed effecs model ad suppose ha q K ad z x. I hs case he lear mxed effecs model reduces o a radom coeffces model of he form E α x α + β x β. 3.8 Here {β } are radom vecors wh mea β. The radom coeffces model ca be easl erpreed as a wo-sage samplg model. I he frs sage oe draws he h subjec from a populao ha elds a vecor of parameers β. From he populao hs vecor has mea E β β ad varace Var β D. A he secod sage oe draws T observaos for he h observao codoal o havg observed β. The mea ad varace of he observaos are E β X β ad Var β R. Pug hese wo sages ogeher elds E X E β X β ad Var E Var β + Var E β R + Var X β R + X D X V. Example Taxpaer sud Coued The radom coeffces model was f usg he Taxpaer daa wh K 8 varables. The model fg was doe usg he sascal package SAS wh he MIVQUE0 varace compoes esmao echques descrbed Seco 3.5. The resulg fg D marx appears Table 3.6. Dspla 3. provdes addoal deals of he model f.

Chaper 3. Models wh Radom Effecs / 3-5 Table 3.6 Values of he Esmaed D Marx INTERCEPT MS HH AGE EMP PREP LNTPI MR DEPEND INTERCEPT 47.86 MS -0.40 0.64 HH -.6.5 3.46 AGE 8.48.6-0.79.33 EMP -8.53 0.9 0. 0. 0.60 PREP 4. -0.50 -.85-0. -0.50.35 LNTPI -4.54-0.7 0.5 -.38.8-0.38.44 MR 0.48 0.06-0.03 0.4-0.09 0.04-0.09 0.68 DEPEND 3.07 0.4 0.9-0.60-0.40-0.35-0.34 0.0 0.68 Dspla 3. Seleced SAS Oupu for he Radom Coeffces Model F Sascs - Res Log Lkelhood 7876.0 AIC smaller s beer 7968.0 AICC smaller s beer 797.5 BIC smaller s beer 83.4 Soluo for Fxed Effecs Sadard Effec Esmae Error DF Value Pr > Iercep -9.5456.475 53-4.44 <.000 MS -0.383.0664 4-0.30 0.7668 HH -.054.448 6-0.73 0.4764 AGE -0.407.533 0-0.35 0.7306 EMP -0.498 0.909 3-0.7 0.869 PREP -0.56 0.68 67-0.35 0.757 LNTPI.68 0.37 57 4.34 <.000 DEPEND -0.84 0.48 70-0.58 0.563 MR 0.09303 0.853 50 0.33 0.7446 Varaos of he radom coeffces model Cera varaos of he wo-sage erpreao of he radom coeffces models lead o oher forms of he radom effecs model equao 3.6. To llusrae equao 3.6 we ma ake he colums of Z o be a src subse of he colums of X. Ths s equvale o assumg ha cera compoes of β assocaed wh Z are sochasc whereas oher compoes ha are assocaed wh X bu o Z are osochasc. Noe ha he coveo equao 3.6 s o assume ha he mea of he radom effecs α are kow ad equal o zero. Aleravel we could assume ha he are ukow wh mea sa α ha s E α α. However hs s equvale o specfg addoal fxed effecs erms Z α equao 3.6. B coveo we absorb hese addoal erms o he X β poo of he model. Thus s cusomar o clude hose explaaor varables he Z desg marx as par of he X desg marx. Aoher varao of he wo-sage erpreao uses kow varables B such ha E β B β. The we have E X B β ad Var R + X D X. Ths s equvale o our equao 3.6 model replacg X wh X B ad Z wh X. Hsao 986E Seco 6.5 refers o hs as a

3-6 / Chaper 3. Models wh Radom Effecs varable-coeffces model wh coeffces ha are fucos of oher exogeous varables. Chaper 5 descrbes hs approach greaer deal. Example Loer sales Seco 4.5 wll descrbe a case whch we wsh o predc loer sales. The respose varable s logarhmc loer sales week for a geographc u. No me-varg varables are avalable for hese daa so a basc explaao of loer sales s hrough he oe-wa radom effecs ANOVA model of he form α * + ε. We ca erpre α * o be he codoal mea loer sales for he h geographc u. I addo we wll have avalable several me-cosa varables ha descrbe he geographc u cludg populao meda household come meda home value ad so forh. Deoe hs se of varables ha descrbe he h geographc u as B. Wh he wo-sage erpreao we could use hese varables o expla he mea loer sales wh he represeao α * B β + α. Noe ha he varable α * s uobservable so hs model s o esmable b self. However whe combed wh he ANOVA model we have α +B β + ε our error compoes model. Ths combed model s esmable. Group effecs I ma applcaos of logudal daa aalss s of eres o assess dffereces of resposes from dffere groups. I hs coex he erm group refers o a caegor of he populao. For example he Seco 3. Taxpaer example we ma be eresed sudg he dffereces ax labl due o geder male/female or due o polcal par afflao democra/republca/lberara/ ad so o. A pcal model ha cludes group effecs ca be expressed as a specal case of he lear mxed effecs model usg q z ad he expresso E g α g α g + δ g + x g β. Here he subscrps rage over g... G groups... g subjecs each group ad... T g observaos of each subjec. The erms {α g } represe radom subjec-specfc effecs ad {δ g } represe fxed dffereces amog groups. A eresg aspec of radom effecs poro s ha subjecs eed o chage groups over me for he model o be esmable. To llusrae f we were eresed geder dffereces ax labl we would o expec dvduals o chage geder over such a small sample. Ths s coras o he fxed effecs model where group effecs are o esmable due o her collear wh subjec-specfc effecs. Tme-cosa varables The sud of me-cosa varables provdes srog movao for desgg a pael or logudal sud ha ca be aalzed as a lear mxed effecs model. Wh a lear mxed effecs model boh he heerogee erms {α } ad parameers assocaed wh me-cosa varables ca be aalzed smulaeousl. Ths was o he case for he fxed effecs models where he heerogee erms ad me-cosa varables are perfecl collear. The group effec dscussed above s a specfc pe of me-cosa varable. Of course s also possble o aalze group effecs where dvduals swch groups over me such as wh polcal par afflao. Ths pe of problem ca be hadled drecl usg bar varables o dcae he presece or absece of a group pe ad represes o parcular dffcules.

Chaper 3. Models wh Radom Effecs / 3-7 We ma spl he explaaor varables assocaed wh he populao parameers o hose ha var b me ad hose ha do o me-cosa. Thus we ca wre our lear mxed effecs codoal regresso fuco as E α α z + x β + x β. Ths model s a geeralzao of he group effecs model. 3.3. Mxed lear models I he Seco 3.3. lear mxed effecs models we assumed depedece amog subjecs Assumpo RO4. Ths assumpo s o eable for all models of repeaed observaos o a subjec over me so s of eres o roduce a geeralzao kow as he mxed lear model. Ths model equao s gve b Z α + X β + ε. 3.9 Here s a N vecor of resposes ε s a N vecor of errors Z ad X are N q ad N K marces of explaaor varables respecvel ad α ad β are q ad K vecors of parameers. For he mea srucure we assume E α Z α + X β ad E α 0 so ha E X β. For he covarace srucure we assume Var α R Var α D ad Cov α ε 0. Ths elds Var Z D Z+ R V. Ulke he lear mxed effecs model Seco 3.3. he mxed lear model does o requre depedece bewee subjecs. Furher he model s suffcel flexble so ha several complex herarchcal srucures ca be expressed as specal cases of. To see how he lear mxed effecs model s a specal case of he mxed lear model ake K ε ε ε K α α α α ε K X Z 0 0 L 0 X 0 Z 0 L 0 X X 3 ad Z 0 0 Z 3 L 0. M M M M O M X 0 0 0 L Z Wh hese choces he mxed lear model reduces o he lear mxed effecs model. The wo-wa error compoes model s a mpora pael daa model ha s o a specfc pe of lear mxed effecs model alhough s a specal case of he mxed lear model. Ths model ca be expressed as α + λ + x β + ε. 3.0 Ths s smlar o he error compoes model bu we have added a radom me compoe λ. We assume ha {λ } {α } ad {ε } are muuall depede. See Chaper 8 for addoal deals regardg hs model. To summarze he mxed lear model geeralzes he lear mxed effecs model ad cludes oher models ha are of eres logudal daa aalss. Much of he esmao ca be accomplshed drecl erms of he mxed lear model. To llusrae hs book ma of he examples are aalzed usg PROC MIXED a procedure wh he sascal package SAS specfcall desged o aalze mxed lear models. The prmar advaage of he lear mxed effecs model s ha provdes a more uve plaform for examg logudal daa.

3-8 / Chaper 3. Models wh Radom Effecs Example Icome equal Zhou 000 examed chages come deermas for a sample of 4730 urba Chese resdes over a perod from 955-994. Subjecs were seleced as a srafed radom sample from 0 ces wh he sraa sze beg proporoal o he c sze. The come formao was colleced rerospecvel as a sgle po me for sx me pos before 985 955 960 965 975 978 ad 984 ad fve me pos afer 984 987 99 99 993 ad 994; he ear 985 marks he offcal begg of a urba reform. Specfcall he model was l c - z α c + z α c + λ + x c β + z x c β + ε c. Here c represes come for he h subjec he ch c a me. The vecor x c represes several corol varables ha clude geder age age squared educao occupao ad work orgazao goverme oher publc ad prvae frms. The varable z s a bar varable defed o be oe f 985 ad zero oherwse. Thus he vecor β represes parameer esmaes for he explaaor varables before 985 ad β represes he dffereces afer urba reform. The prmar eres s he chage of he explaaor varable effecs β. For he oher varables he radom effec λ s mea o corol for udeeced me effecs. There are wo c effecs: - z α c s for ces before 985 ad z α c s for afer 984. Noe ha hese radom effecs are a he c level ad o a he subjec level. Zhou used a combao of error compoes ad auoregressve srucure o model he seral relaoshps of he dsurbace erms. Icludg hese radom effecs accoued for cluserg of resposes wh boh ces ad me perods hus provdg more accurae assessme of he regresso coeffces β ad β. Zhou foud sgfca reurs o educao ad hese reurs creased he pos-reform era. Lle chage was foud amog orgazao effecs wh he excepo of sgfcal creased effecs for prvae frms. 3.4 Iferece for regresso coeffces Esmao of he lear mxed effecs model proceeds wo sages. I he frs sage we esmae he regresso coeffces β assumg kowledge of he varace compoes τ. The he secod sage he varace compoes τ are esmaed. Seco 3.5 dscusses varace compoe esmao whereas hs seco dscusses regresso coeffce ferece assumg ha he varace compoes are kow. GLS esmao From Seco 3.3 we have ha he vecor has mea X β ad varace Z D Z + R V τ V. Thus drec calculaos show ha he geeralzed leas squares GLS esmaor of β s X V X bgls X V. 3. The GLS esmaor of β akes he same form as he error compoes model esmaor equao 3.4 e wh a more geeral varace covarace marx V. Furhermore drec calculao show ha he varace s Var b GLS X V X. 3.

Chaper 3. Models wh Radom Effecs / 3-9 As wh fxed effecs esmaors s possble o express b GLS as a weghed average of subjec-specfc esmaors. To hs ed for he h subjec defe he GLS esmaor b GLS X V - X - X V - ad he wegh W GLS X V - X. The we ca wre GLS W GLS b GLS b GLS W. Marx verso formula To smplf he calculaos ad o provde beer uo for our expressos we ce a formula for verg V. Noe ha he marx V has dmeso T T. From Appedx A.5 we have V - R + Z D Z - R - - R - Z D - + Z R - Z - Z R -. 3.3 The expresso o he rgh-had sde of equao 3.3 s easer o compue ha he lef-had sde whe he emporal covarace marx R has a easl compuable verse ad he dmeso q s smaller ha T. Moreover because he marx D - + Z R - Z s ol a q q marx s easer o ver ha V a T T marx. Some specal cases are of eres. Frs oe ha he case of o seral correlao we have R σ I ad equao 3.3 reduces o I + Z DZ I Z σ D + Z Z Z V σ. 3.4 σ Furher he error compoes model cosdered Seco 3. we have q D σ α Z so ha equao 3.3 reduces o σ α ζ + V σ I σ α ZZ I J I J 3.5 σ Tσ α + σ σ T Tσ α where ζ as Seco 3.. Ths demosraes ha equao 3.4 s a specal case Tσ α + σ of equao 3.. For aoher specal case cosder he radom coeffces model z x wh o seral correlao so ha R σ I. Here he wegh W GLS akes o a smple form: D + X X W GLS σ see Exercse 3.8. From hs form we see ha subjecs wh large values of X X have a greaer effec o b GLS ha subjecs wh smaller values. Maxmum lkelhood esmao Wh assumpo RO5 he log-lkelhood of a sgle subjec s l β τ T + V τ + Xβ V τ lπ l de X β. 3.6 Wh equao 3.6 he log-lkelhood for he ere daa se s l L β τ β τ. The values of β ad τ ha maxmze Lβ τ are he maxmum lkelhood esmaors MLEs whch we deoe as b MLE ad τ MLE. Noe o Reader: We ow beg o use lkelhood ferece exesvel. You ma wsh o revew Appedx B for addoal backgroud o jo ormal ad he relaed lkelhood fuco. Appedx C revews lkelhood esmao a geeral coex.

3-0 / Chaper 3. Models wh Radom Effecs The score vecor s he vecor of dervaves of he log-lkelhood ake wh respec o he parameers. We deoe he vecor of parameers b θ β τ. Wh hs oao he score vecor s Lθ/ θ. Tpcall f hs score has a roo he he roo s a maxmum lkelhood esmaor. To compue he score vecor frs ake dervaves wh respec o β ad fd he roo. Tha s L β τ l β τ β β Seg he score vecor equal o zero elds β X β V τ X β X X V τ β. b MLE XV τ X X V τ b GLS. 3.7 Tha s for fxed covarace parameers τ he maxmum lkelhood esmaor ad he geeral leas squares esmaor are he same. Robus esmao of sadard errors Eve whou he assumpo of ormal he maxmum lkelhood esmaor b MLE has desrable properes. I s ubased effce ad asmpocall ormal wh covarace marx gve equao 3.. However he esmaor does deped o kowledge of varace compoes. As a alerave ca be useful o cosder a alerave weghed leas squares esmaor b W X W RE X X W RE 3.8 where he weghg marx W RE depeds o he applcao a had. To llusrae oe could use he de marx so ha b W reduces o he ordar leas squares esmaor. Aoher choce s Q from Seco.5.3 ha elds fxed effecs esmaors of β. We explore hs choce furher Seco 7.. The weghed leas squares esmaor s a ubased esmaor of β ad s asmpocall ormal alhough o effce uless W RE V -. Basc calculaos show ha has varace XW RE X XW REV W RE X Var b W X W RE X. As Seco.5.3 we ma cosder esmaors ha are robus o ususpeced seral correlao ad heeroscedasc. Specfcall followg a suggeso made depedel b Huber 967G Whe 980E ad Lag ad Zeger 986B we ca replace V b e e where e - X b W s he vecor of resduals. Thus a robus sadard error of b Wj he jh eleme of b W s se b h j W j dagoal eleme of W RE X X W REee W RE X X X W RE X. Tesg hpoheses For ma sascal aalses esg he ull hpohess ha a regresso coeffce equals a specfed value ma be he ma goal. Tha s he eres ma be esg H 0 : β j β j0

Chaper 3. Models wh Radom Effecs / 3- where he specfed value β j0 s ofe alhough o alwas equal o 0. The cusomar procedure s o compue he releva b j GLS β j0 -sasc. se b j GLS Here b jgls s he jh compoe b GLS from equao 3.7 ad seb jgls s he square roo of he jh dagoal eleme of X V ˆ τ X where τˆ s he esmaor of he varace compoe ha wll be descrbed Seco 3.5. The oe assesses H 0 b comparg he -sasc o a sadard ormal dsrbuo. There are wo wdel used varas of hs sadard procedure. Frs oe ca replace seb jgls b seb jw o ge so-called robus -sascs. Secod oe ca replace he sadard ormal dsrbuo wh a -dsrbuo wh he approprae umber of degrees of freedom. There are several mehods for calculag he degrees of freedom ha deped o he daa ad he purpose of he aalss. To llusrae Dspla 3. ou wll see ha he approxmae degrees of freedom uder he DF colum s dffere for each varable. Ths s produced b he SAS defaul coame mehod. For he applcaos hs ex we pcall wll have large umber of observaos ad wll be more cocered wh poeal heeroscedasc ad seral correlao; hus we wll use robus -sascs. For readers wh smaller daa ses eresed he secod alerave Lell e al. 996S descrbes he -dsrbuo approxmao deal. For esg hpoheses cocerg several regresso coeffces smulaeousl he cusomar procedure s he lkelhood rao es. Oe ma express he ull hpohess as H 0 : C β d where C s a p K marx wh rak p d s a p vecor pcall 0 ad recall ha β s he K vecor of regresso coeffces. Boh C ad d are user specfed ad deped o he applcao a had. Ths ull hpohess s esed agas he alerave H 0 : C β d. Lkelhood rao es procedure. Usg he ucosraed model calculae maxmum lkelhood esmaes ad he correspodg lkelhood deoed as L MLE.. For he model cosraed usg H 0 : C β d calculae maxmum lkelhood esmaes ad he correspodg lkelhood deoed as L Reduced. 3. Compue he lkelhood rao es sasc LRT L MLE - L Reduced. 4. Rejec H 0 f LRT exceeds a percele from a χ ch-square dsrbuo wh p degrees of freedom. The percele s oe mus he sgfcace level of he es. Of course oe ma also use p-values o calbrae he sgfcace of he es. See Appedx C.7 for more deals o he lkelhood rao es. The lkelhood rao es s he dusr sadard for assessg hpoheses cocerg several regresso coeffces. However we oe ha beer procedures ma exs parcularl for small daa ses. To llusrae Phero ad Baes 000S recommed he use of codoal F-ess whe p s large relave o he sample sze. As wh esg dvdual regresso coeffces we shall be more cocered wh poeal heeroscedasc for large daa ses. I hs case a modfcao of he Wald es procedure s avalable. For he case of o heeroscedasc ad/or seral correlao he Wald procedure for esg H 0 : C β d s o compue he es sasc

3- / Chaper 3. Models wh Radom Effecs Cb d C X V τ X C Cb d MLE ad compare hs sasc o a ch-square dsrbuo wh p degrees of freedom. Compared o he lkelhood rao es he advaage of he Wald procedure s ha he sasc ca be compued wh jus oe evaluao of he lkelhood o wo. However he dsadvaage s ha for geeral cosras such as C β d specalzed sofware s requred. A advaage of he Wald procedure s ha s sraghforward o compue robus aleraves. For a robus alerave we use he regresso coeffce esmaor defed equao 3.8 ad compue MLE Cb d C X W X X W e e W X X W X C Cb d W RE We compare hs sasc o a ch-square dsrbuo wh p degrees of freedom. RE RE MLE RE W. 3.5 Varace compoes esmao I hs seco we descrbe several mehods for esmag he varace compoes. The wo prmar mehods eal maxmzg a lkelhood fuco coras o mome esmaors. I sascal esmao heor Lehma 99G here are well-kow rade-offs whe cosderg mome compared o lkelhood esmao. Tpcall lkelhood fucos are maxmzed b usg erave procedures ha requre sarg values. A he ed of hs seco we descrbe how o oba reasoable sarg values for he erao usg mome esmaors. 3.5. Maxmum lkelhood esmao The log-lkelhood was preseed Seco 3.4. Subsug he expresso for he geeralzed leas squares esmaor equao 3. o he log-lkelhood equao 3.6 elds he coceraed or profle log-lkelhood L bgls τ T lπ + l de V τ + Error SS τ 3.9 a fuco of τ. Here he error sum of squares for he h subjec s Error SS τ X b V τ X b. 3.0 Thus we ow maxmze he log-lkelhood as a fuco of τ ol. I ol a few specal cases ca oe oba closed form expressos for he maxmzg varace compoes. Exercse 3.0 llusraes oe such specal case. GLS GLS Specal case Error compoes model For hs specal case he varace compoes are τ σ σ α. Usg equao A.5 Appedx A.5 we have ha l de V l de σ α J + σ I T l σ + l + T σ α / σ. From hs ad equao 3.9 we have ha he coceraed lkelhood s

Chaper 3. Models wh Radom Effecs / 3-3 + σ + GLS + α L b σ α σ T lπ T lσ l T σ σ α + XbGLS I J XbGLS σ Tσ α + σ where b GLS s gve equao 3.4. Ths lkelhood ca be maxmzed over σ σ α usg erave mehods. Ierave esmao I geeral he varace compoes are esmaed recursvel. Ths ca be doe usg eher he Newo-Raphso or Fsher scorg mehod see for example Harvlle 977S ad Wolfger e al. 994S. Newo-Raphso. Le L Lb GLS τ τ ad use he erave mehod: Here he marx τ NEW τ OLD L τ τ L τ τ τold L s called he sample formao marx. τ τ Fsher scorg. Defe he expeced formao marx τ NEW τ OLD. L I τ E ad use τ τ L + I τ OLD. τ τ τold 3.5. Resrced maxmum lkelhood As he ame suggess resrced maxmum lkelhood REML s a lkelhood-based esmao procedure. Thus shares ma of he desrable properes of maxmum lkelhood esmaors MLEs. Because s based o lkelhoods s o specfc o a parcular desg marx as are aalss of varace esmaors Harvlle 977S. Thus ca be readl appled o a wde vare of models. Lke MLEs REML esmaors are raslao vara. Maxmum lkelhood ofe produces based esmaors of he varace compoes τ. I coras esmao based o REML resuls ubased esmaors of τ a leas for ma balaced desgs. Because maxmum lkelhood esmaors are egavel based he ofe ur ou o be egave a uvel udesrable suao for ma users. Because of he ubasedess of ma REML esmaors here s less of a edec o produce egave esmaors Corbel ad Searle 976aS. As wh MLEs REML esmaors ca be defed o be parameer values for whch he resrced lkelhood acheves a maxmum value over a cosraed parameer space. Thus as wh maxmum lkelhood s sraghforward o modf he mehod o produce oegave varace esmaors. The dea behd REML esmao s o cosder he lkelhood of lear combaos of he resposes ha do o deped o he mea parameers. To llusrae cosder he mxed lear model. We assume ha he resposes deoed b he vecor are ormall dsrbued have mea E Xβ ad varace-covarace marx Var V Vτ. The dmeso of s N ad he dmeso of X s N p. Wh hs oao defe he projeco marx Q I -X X X - X ad cosder he lear combao of resposes Q. Sraghforward calculaos show ha

3-4 / Chaper 3. Models wh Radom Effecs Q has mea 0 ad varace-covarace marx VarQ QVQ. Because Q has a mulvarae ormal dsrbuo ad he mea ad varace-covarace marx do o deped o β he dsrbuo of Q does o deped o β. Furher Appedx 3A. shows ha Q s depede of he geeralzed leas squares esmaor b GLS X V X X V. The vecor Q s he resdual vecor from a ordar leas squares f of he daa. Hece REML s also referred o as resdual maxmum lkelhood esmao. Because he rak of Q s N - p we lose some formao b cosderg hs rasformao of he daa; hs movaes he use of he descrpor resrced maxmum lkelhood. There s some formao abou τ he vecor b GLS ha we are o usg for esmao. Furher oe ha we could also use a lear rasform of Q such as AQ ha AQY also has a mulvarae ormal dsrbuo wh a mea ad varace-covarace marx ha do o deped o β. Paerso ad Thompso 97S ad Harvlle 974S 977S showed ha he lkelhood does o deped o he choce of A. The roduced he resrced log-lkelhood: L b τ τ l de V τ + l de X V τ X Error SS τ 3. [ ] REML GLS + up o a addve cosa. See Appedx 3A. for a dervao of hs lkelhood. REML esmaors τ REML are defed o be maxmzers of he fuco L REML b GLS τ τ. Here he error sum of squares s Error SS τ X b GLS τ V τ Xb GLS τ. 3. Aalogous o equao 3.9 he usual log-lkelhood s L b GLS τ τ [ l de V τ + Error SS τ ] up o a addve cosa. The ol dfferece bewee he wo lkelhoods s he erm l dex Vτ - X. Thus erave mehods of maxmzao are he same ha s usg eher Newo-Raphso or Fsher scorg. For lear mxed effecs models hs addoal erm s l de V τ X X. For balaced aalss of varace daa T T Corbel ad Searle 976aS esablshed ha he REML esmao reduces o sadard aalss of varace esmaors. Thus REML esmaors are ubased for hese desgs. However REML esmaors ad aalss of varace esmaors dffer for ubalaced daa. REML esmaors acheve her ubasedess b accoug for he degrees of freedom los esmag he fxed effecs β; MLEs do o accou for hs loss of degrees of freedom. Whe p s large he dfferece bewee REML esmaors ad MLEs s sgfca. Corbel ad Searle 976bS showed ha erms of mea square errors MLEs ouperform REML esmaors for small p < 5 alhough he suao s reversed for large p wh a suffcel large sample. Harvlle 974S gave a Baesa erpreao of REML esmaors. He poed ou ha usg ol Q o make fereces abou τ s equvale o gorg pror formao abou β ad usg all he daa. Some sascal packages prese maxmzed values of resrced lkelhoods suggesg o users ha hese values ca be used for fereal echques such as lkelhood rao ess. For lkelhood rao ess oe should use ordar lkelhoods eve whe evaluaed a REML esmaors o he resrced lkelhoods ha are used o deerme REML esmaors. Appedx 3A.3 llusraes he poeall dsasrous cosequeces of usg REML lkelhoods for lkelhood rao ess.

Chaper 3. Models wh Radom Effecs / 3-5 Sarg values Boh he Newo-Raphso ad Fsher scorg algorhms ad he ML ad REML esmao mehods volve recursve calculaos ha requre sarg values. We ow descrbe wo o-recursve mehods due o Swam 970E ad Rao 970S respecvel. Oe ca use he resuls of hese o-recursve mehods as sarg values he Newo-Raphso ad Fsher scorg algorhms. Swam s mome-based procedure appeared he ecoomercs pael daa leraure. We cosder a radom coeffces model; ha s equao 3.8 wh x z ad R σ I. Procedure for compug mome-based varace compoe esmaors. Compue a ordar leas squares esmaor of σ s I X X X X T K. Ths s a ordar leas squares procedure ha gores D.. Nex calculae b OLS X X X a predcor of β + α. 3. Fall esmae D usg D SWAMY b OLS OLS s b b b X X where b b. OLS The esmaor of D ca be movaed b examg he varace of b OLS Var b OLS X X X Var X β + α + ε Var β + α + X X X ε D + σ XX Usg b b b b OLS OLS ad s as esmaors of Varb OLS ad σ respecvel elds D SWAMY as a esmaor of D. Varous modfcaos of hs esmaor are possble. Oe ca erae he procedure b usg D SWAMY o mprove he esmaors s ad so o. Homoscedasc of he ε erms could also be assumed. Hsao 986E recommeds droppg he secod erm s X X esure ha D SWAMY s o-egave defe. o 3.5.3 MIVQUE esmaors Aoher o-recursve mehod s Rao s 970S mmum varace quadrac ubased esmaor MIVQUE. To descrbe hs mehod we reur o he mxed lear model X β + ε whch Var V Vτ. We wsh o esmae he lear combao of varace compoes r c k k τ k where he c k are specfed cosas ad τ τ τ r. We assume ha V s lear he sese ha r V τ k V. k τ k Thus wh hs assumpo we have ha he marx of secod dervaves he Hessa of V s zero Grabll 969G. Alhough hs assumpo s geerall vable s o sasfed b for

3-6 / Chaper 3. Models wh Radom Effecs example auoregressve models. I s o resrcve o assume ha V s kow eve hough τ k he varace compoe τ k s ukow. To llusrae cosder a error compoes srucure so ha V σ α J + σ I. The V J ad V I are boh kow. σ α σ r Quadrac esmaors of c k k τ k are based o A where A s a smmerc marx o be specfed. The varace of A assumg ormal ca easl be show o be racevava. We would lke he esmaor o be vara o raslao of β. Tha s we requre A X b 0 A X b 0 for each b 0. Thus we resrc our choce of A o hose ha sasf A X 0. r For ubasedess we would lke c τ E A. Usg A X 0 we have k k E A E ε Aε racee εε A race VA r race τ. k V A k τ k Because hs equal should be vald for all varace compoes τ k we requre ha A sasf race c k V A for k r. 3.3 τ k Rao showed ha he mmum value of racevava sasfg A X 0 ad he cosras equao 3.3 s aaed a r A V λ k V Q V V Q * k τ k where Q QV I - X X V - X - X V - ad λ λ r s he soluo of S λ λ r c c r. Here he jh eleme of S s gve b race V Q. V V Q V τ τ j Thus he MIVQUE esmaor of τ s he soluo of S τ MIVQUE G 3.4 where he k h eleme of G s gve b V Q V V Q. τ k Whe comparg Rao s o Swam s mehod we oe ha he MIVQUE esmaors are avalable for a larger class of models. To llusrae he logudal daa coex s possble o hadle seral correlao wh he MIVQUE esmaors. A drawback of he MIVQUE esmaor s ha ormal s assumed; hs ca be weakeed o zero kuross for cera forms of V Swallow ad Searle 978S. Furher MIVQUE esmaors requre a pre-specfed esmae of V. A wdel used specfcao s o use he de marx for V equao 3.4. Ths specfcao produces so-called MIVQUE0 esmaors a opo wdel avalable sascal packages. I s he defaul opo PROC MIXED of he sascal package SAS. k

Chaper 3. Models wh Radom Effecs / 3-7 Furher readg Whe compared o regresso ad lear models here are fewer exbook roducos o mxed lear models alhough more are becomg avalable. Searle Casella ad McCulloch 99S gve a earl echcal reame. A slghl less echcal s Logford 993EP. McCulloch ad Searle 00S gve a excelle rece echcal reame. Oher rece corbuos egrae sascal sofware o her exposo. Lle e al. 996D ad Verbeke ad Moleberghs 000D roduce mxed lear models usg he SAS sascal package. Phero ad Baes 000D provde a roduco usg he S ad S-Plus sascal packages. Radom effecs ANOVA ad regresso models have bee par of he sadard sascal leraure for que some me; see for example Scheffé 959G Searle 97G or Neer ad Wasserma 974G. Balesra ad Nerlove 966E roduced he error compoes model o he ecoomerc leraure. The radom coeffces model was descrbed earl o b Hldreh ad Houck 968S. As descrbed Seco 3.5 mos of he developme of varace compoe esmaors occurred he 970 s. More recel Balag ad Chag 994E compared he relave performace of several varace compoes esmaors for he error compoes model.

3-8 / Chaper 3. Models wh Radom Effecs Appedx 3A. REML Calculaos Appedx 3A. Idepedece of Resduals ad Leas Squares Esmaors Assume ha has a mulvarae ormal dsrbuo wh mea Xβ ad varacecovarace marx V where X has dmeso N p wh rak p. Recall ha he marx V depeds o he parameers τ. We use he marx Q I X X X X. Because Q s dempoe ad has rak N - p we ca fd a N N - p marx A such ha A A Q ad A A I N. We also eed G V - X X V - X - a N p marx. Noe ha G b GLS he geeralzed leas squares esmaor of β. Wh hese wo marces defe he rasformao marx H A G a N N marx. Cosder he rasformed varables A A H. G b GLS Basc calculaos show ha A ~ N 0 A V A ad G b GLS ~ N β X V X whch z ~ Nµ V deoes ha a radom vecor z has a mulvarae ormal dsrbuo wh mea µ ad varace V. Furher we have ha A ad b GLS are depede. Ths s due o ormal ad zero covarace marx: Cov A b GLS E A G A VG A X X V X 0. We have A X 0 because A X A AA X A Q X ad Q X 0. Zero covarace ogeher wh ormal mpl depedece. Appedx 3A. Resrced Lkelhoods To develop he resrced lkelhood we frs check he rak of he rasformao marx H. Thus wh H as Appedx 3A. ad equao A. of Appedx A.5 we have A A A A G de H de H H de [ A G] de G G A G G de A A de G G G A A A A G de G G G QG de G X X X X G de X X usg G X I. Thus he rasformao H s o-sgular f ad ol f X X s o-sgular. I hs case o formao s los b cosderg he rasformao H. We ow develop he resrced lkelhood based o he probabl des fuco of A. We frs oe a relaoshp used b Harvlle 974S cocerg he probabl des fuco of G. We wre f G z β o deoe he probabl des fuco of he radom vecor G evaluaed a he vecor po z wh mea vecor parameer β. Because probabl des fucos egrae o we have he relao

Chaper 3. Models wh Radom Effecs / 3-9 f G π f G z β dz p / de X V π X p / / z β dβ for each z de X V X / exp z β X V exp z β X V X z β dβ X z β dz wh a chage of varables. Because of he depedece of A ad G b GLS we have f f f. Here f f ad f are he des fucos of he radom vecors H A ad G H A G respecvel. For oao le * be a poeal realzao of he radom vecor. Thus he probabl des fuco of A s f A * f A * f G * β dβ A f A H G H * β dβ de H f H * β dβ usg a chage of varables. Now le b * GLS be he realzao of b GLS usg *. The from a sadard equal from aalss of varace * * * * * Xβ V * Xβ * Xb V * Xb + b β X V X b β GLS GLS GLS GLS. Wh hs equal he probabl des fuco f ca be expressed as f * β exp * Xβ V * Xβ N / / π de V π N * * * * exp * * exp. / / XbGLS V XbGLS bgls β X V X bgls β de V p / π de X V X N / / π de V / * * * GLS V * XbGLS b G GLS Thus p / π de X V X f A * A N / / π de V π N p / de V / exp * Xb / de X X / de H de X V * exp * Xb X / exp GLS V f * Xb * Xb * GLS V A * GLS β. G f G * Xb Ths elds he REML lkelhood Seco 3.5 afer akg logarhms ad droppg cosas ha do o volve τ. b * GLS * GLS β dβ.

3-30 / Chaper 3. Models wh Radom Effecs Appedx 3A.3 Lkelhood Rao Tess ad REML Recall he lkelhood rao sasc LRT Lθ MLE - Lθ Reduced. Ths s evaluaed usg he so-called coceraed or profle log-lkelhood gve equaos 3.9 ad 3.0. For comparso from equao 3. he resrced log-lkelhood s REML GLS + + Error L b τ T lπ l de V τ SS τ l de X V τ X. 3A. To see wh a REML lkelhood does o work for lkelhood rao ess cosder he followg example. Specal Case. Tesg he Imporace of a Subse of Regresso Coeffces. For smplc we assume ha V σ I so ha here s o seral correlao. For hs specal case we have he fe ad asmpoc dsrbuo of he paral F-es Chow. Because he asmpoc dsrbuo s well kow we ca easl judge wheher or o REML lkelhoods are approprae. Wre β β β ad suppose ha we wsh o use he ull hpohess H 0 : β 0. Assumg o seral correlao he geeralzed leas square esmaor of β reduces o he ordar leas squares esmaor ha s b GLS b OLS X X X. Thus from equao 3.9 he coceraed lkelhood s: L b OLS σ T lπ + T lσ + Xb OLS Xb OLS σ N lπ + N lσ + Error SS Full σ where Error SS X b X b MLE Full σ s σ Error SS N Full OLS OLS so he maxmum lkelhood s. The maxmum lkelhood esmaor of L b OLS σ MLE N lπ + N l Error SS Full N l N + N. Now wre X X X where X has dmeso T K-r ad X has dmeso T r. Uder H 0 he esmaor of β s b OLS Reduced X X X. Thus uder H 0 he log-lkelhood s: L b OLSReduce d σ MLE Reduced N lπ + N l Error SS Reduced N l N + N where Error SS Reduced X b OLS Reduced Xb OLS Reduced. Thus he lkelhood rao es sasc s: LRTMLE L bols σ MLE L bols Reduced σ MLE Reduced N l Error SS Reduced l Error SS Full. x x From a Talor seres approxmao we have l l x + +.... Thus we x x have Error SS Reduced Error SS Full LRT MLE N +... Error SS Full

Chaper 3. Models wh Radom Effecs / 3-3 whch has a approxmae ch-square dsrbuo wh r degrees of freedom. For comparso from equao 3A. he resrced log-lkelhood s L REML b OLS σ N lπ + N K lσ + Error SS Full l de X X σ The resrced maxmum lkelhood esmaor of σ s σ Error SS N K. REML Thus he resrced maxmum lkelhood s L REML b OLS σ REML N lπ + N K l Error SS + N K l N K N K. Uder H 0 he resrced log-lkelhood s: L b σ REML OLS Reduced REML Reduced Full Full l de X X + N K q l N K q N K q. Thus he lkelhood rao es sasc usg a resrced lkelhood s: LRT L b σ L b σ + N lπ + N K q l Error SS Reduced l de X X REML REML OLS REML REML OLSReduced REML Reduced l Error SS Reduced l Error SS Full q l Error SS Reduced N K + l de X X l de X X + N K l q l N K q q N K LRT MLE + N Error SS + q l N K q N K N K q l de X X l de X X Reduced q + N K l. N K q The frs erm s asmpocall equvale o he lkelhood rao es usg ordar maxmzed lkelhoods. The hrd ad fourh erms ed o cosas. The secod erm l de X X l de X X ma ed o plus or mus f depedg o he values of he explaaor varables. For example he specal case ha X X 0 we have l de r X l de X X l de X X X. Thus hs erm wll ed o plus or mus f for mos explaaor varable desgs..

3-3 / Chaper 3. Models wh Radom Effecs 3. Exercses ad Exesos Seco 3. 3.. Geeralzed leas squares GLS esmaors For he error compoes model he varace of he vecor of resposes s gve as V σ α J + σ ε I. a. B mulplg V b V - check ha V I ζ σ T J. ε - b. Use hs form of V ad he expresso for a GLS esmaor X V X b EC X V o esablsh he formula equao 3.3. c. Use equao 3.4 o show ha he basc radom effecs esmaor ca be expressed as: T T b EC xx ζ T xx x ζ T x. d. Show ha X I T J X X I T J b s a alerave expresso for he basc fxed effecs esmaor gve equao.6. e. Suppose ha σ α s large relave o σ so ha we assume ha σ /σ α. Gve a expresso ad erpreao for b EC. f. Suppose ha σ α s small relave o σ so ha we assume ha σ α / σ 0. Gve a expresso ad erpreao for b EC. 3.. GLS esmaor as a weghed average Cosder he basc radom effecs model ad suppose ha K ad ha x. Show ha ζ bec. ζ 3.3. Error compoes wh oe explaaor varable Cosder he error compoes model α + β 0 + β x + ε. Tha s cosder he model equao 3. wh K ad x x. σ a. Show ha ζ T ζ. σ α b. Show ha we ma wre he geeralzed leas squares esmaors of β 0 ad β as ad b EC x x T ζ x T ζ x ζ T ζ T x b 0 EC w xwb EC x w w w

Chaper 3. Models wh Radom Effecs / 3-33 where ζ x ζ xw ad w. ζ ζ H: use he expresso of b EC Exercse 3.c. c. Suppose ha σ α s large relave o σ so ha we assume ha σ /σ α. Gve a expresso ad erpreao for b EC. d. Suppose ha σ α s small relave o σ so ha we assume ha σ α / σ 0. Gve a expresso ad erpreao for b EC. 3.4. Two-populao slope erpreao Cosder he basc radom effecs model ad suppose ha K ad ha x s bar varable. Suppose furher ha x akes o he value of for hose from populao ad for hose from populao. Aalogous o Exercse.4 le ad be he umber of oes ad mus oes for he h subjec respecvel. Furher le ad be he average respose whe x s oe ad mus oe for he h subjec respecvel. Show ha we ma wre he error compoes esmaor as b w w EC w + w wh weghs w +ζ / T ad w +ζ / T ζ H: use he expresso of b EC Exercse 3.c. ζ. 3.5. Ubased varace esmaors Perform he followg seps o check ha he varace esmaors gve b Balag ad Chag 994E are ubased varace esmaors for he ubalaced error compoes model roduced Seco 3.. For oaoal smplc assume he model follows he form µ α + α + x β + ε where µ α s a fxed parameer represe he model ercep. As descrbed Seco 3. we wll use he resduals e - a + x b usg a ad b accordg o he Chaper fxed effecs esmaors equaos.6 ad.7. a. Show ha respose devaos ca be expressed as x x β + ε ε. b. Show ha he fxed effecs slope esmaor ca be expressed as T T b β + x x x x x x ε. c. Show ha he resdual ca be expressed as e x x β b + ε ε. d. Show ha he mea square error defed equao. s a ubased esmaor for hs model. Tha s show ha E E T s e σ N + K. e. Show ha a a w α α w + x x β b + ε ε where α w N T α. E α α + N T T N. f. Show ha g. Show ha w σ α / α σ α E s.

3-34 / Chaper 3. Models wh Radom Effecs 3.6. Ordar leas squares esmaor Perform he followg seps o check ha he ordar leas square esmaor of he slope coeffce sll performs well whe he error compoes model s rue. To hs ed: a. Show ha he ordar leas squares esmaor for he model x β + ε ca be expressed T T as b OLS xx x. b. Assumg he error compoes model α + x β + ε show ha he dfferece bewee he par a esmaor ad he vecor of parameers s T T b + OLS β xx x α ε. c. Use par b o argue ha he esmaor gve par a s ubased. d. Calculae he varace of b OLS. e. For K show ha he varace calculaed par d s larger ha he varace of he radom effecs esmaor Var b EC gve Seco 3.. 3.7. Poolg es Perform he followg seps o check ha he es sasc for he poolg es gve Seco 3. has a approxmae ch-square dsrbuo uder he ull hpohess of a homogeeous model of he form x β + ε. a. Check ha he resduals ca be expressed as e ε + x β bols where b OLS s he ordar leas squares esmaor of β Exercse 3.5a. T b. Check ha E σ e. N K T c. Check ha T T s T e e er es where he laer sum s over {r s such ha s r ad r T s T }. r s T r e s h r s σ where hr s x r x x x s d. Check for s r ha Ee s a eleme of he ha marx. e. Esablsh codos so ha he bas s eglgble. Tha s check ha / h / T T 0. r s r s f. Deerme he approxmae varace of s b showg ha 4 E σ ε rε s. T T r s T T g. Oule a argume o show ha s T T s approxmael sadard σ ormal hus compleg he argume for he behavor of he poolg es sasc uder he ull hpohess.

Chaper 3. Models wh Radom Effecs / 3-35 Seco 3.3 3.8. Nesed models Le j be he oupu of he jh frm he h dusr for he h me perod. Assume ha he error srucure s gve b j E j + δ j where δ j α + ν j + ε j. Here assume ha each of {α } {ν j } ad {ε j } are depedel ad decall dsrbued ad depede of oe aoher. a. Le be he vecor of resposes for he h dusr. Wre as a fuco of { j }. b. Use σ α σ ν ad σ ε o deoe he varace of each error compoe respecvel. Gve a expresso for Var erms of hese varace compoes. c. Cosder he lear mxed effecs model Z α + X β + ε. Show how o wre he qua Z α erms of he error compoes α ad ν j ad he approprae explaaor varables. Seco 3.4 3.9. GLS esmaor as a weghed average of subjec-specfc GLS esmaors Cosder he radom coeffces models ad cosder he weghed average expresso for he GLS esmaor GLS W GLSb GLS b GLS W. a. Show ha he weghs ca be expressed as GLS D + XX b. Show ha Varb D + σ X X. GLS W σ. 3.0. Mached pars desg Cosder a par of observaos ha have bee mached some wa. The par ma coss of sblgs frms wh smlar characerscs bu from dffere dusres or he same e observed before ad afer some eve of eres. Assume ha here s reaso o beleve ha he par of observaos are depede some fasho. Le be he se of resposes ad x x be he correspodg se of covaraes. Because of he samplg desg he assumpo of depedece bewee ad s o accepable. a. Oe alerave s o aalze he dfferece bewee he resposes. Thus le -. Assumg perfec machg we mgh assume ha x x x sa ad use he model x γ + ε. Whou perfec machg oe could use he model x β - x β + η. Calculae he ordar leas squares esmaor ββ β ad call hs esmaor b PD because he resposes are pared dffereces. b. As aoher alerave form he vecors ad ε ε ε as well as he marx x 0 X. 0 x Now cosder he lear mxed effecs model X β + ε where he depedece bewee resposes s duced b he varace R Var ε. Uder hs model specfcao show ha b PD s ubased. Compue he varace of b PD. Calculae he geeralzed leas squares esmaor of β sa b GLS. c. For e aoher alerave assume ha he depedece s duced b a commo lae radom varable α. Specfcall cosder he error compoes model j α + x j β j + ε j. Uder hs model specfcao show ha b PD s ubased.

3-36 / Chaper 3. Models wh Radom Effecs Calculae he varace of b PD. Le b EC be he geeralzed leas squares esmaor uder hs model. Calculae he varace. v Show ha f σ α he he varace of b EC eds o he varace of b PD. Thus for hghl correlaed daa he wo esmaors have he same effcec. d. Coue wh he model par c. We kow ha Var b EC s smaller ha Var b PD because b EC s he geeralzed leas squares esmaor of β. To quaf hs a specal case assume asmpocall equvale machg so ha x x Σ j. Moreover le x Σ x ad assume ha Σ s smmerc. Suppose ha we are eresed dffereces of he respodes so ha he vecor of parameers of eres are β - β. Le b EC b EC b EC ad b PD b PD b PD. Show ha Var b EC b EC Σ x + zσ σ ζ / where ζ / σ α z ad ζ. ζ / σ + σ α Show ha Var b PD b PD 4 Σ x + zσ σ Use pars d ad d o quaf he relave varaces. For example f Σ 0 he he relave varaces effcec s / ζ whch s bewee 0.5 ad.0. j j x 3.. Robus sadard errors To esmae he lear mxed effecs model cosder he weghed leas squares esmaor gve equao 3.8. The varace of hs esmaor Var b W s also gve Seco 3.4 alog wh he correspodg robus sadard error of he jh compoe of b W deoed as seb Wj. Le us re-wre hs as: se b j W ˆ h j dagoal eleme of X W REX XW REV W REX X W RE X where V ˆ ee s a esmaor of V. I parcular expla: a. How oe goes from a varace-covarace marx o a sadard error? b. Wha abou hs sadard error makes robus? c. Le s derve a ew robus sadard error. For smplc drop he subscrps ad defe he ha marx / / H W WRE X X WRE X X WRE. Show ha he weghed resduals ca be expressed as a lear combao of weghed errors. Specfcall show / / WRE e I HW WRE ε Show ha E W / ee W / I H W / VW / I H. RE RE W Show ha e e s a ubased esmaor of a lear rasform of V. Specfcall show ha RE RE W

Chaper 3. Models wh Radom Effecs / 3-37 * * ee I H V I E W H W * / / where HW WRE HW WRE. d Expla how he resul c suggess defg a alerave esmaor of V ˆ * * V I H e e I H. W W Use hs alerave esmaor o sugges a ew robus esmaor of he sadard error of b Wj. See Frees ad J 004 f ou would lke more deals abou he properes of hs esmaor. Seco 3.5 3.. Bas of MLE ad REML varace compoe esmaors Cosder he basc radom effecs model ad suppose ha T T K ad ha x. Furher do o mpose boudar codos so ha he esmaors ma be egave. a. Show ha he maxmum lkelhood esmaor of σ ma be expressed as: T ˆ σ ML. T b. Show ha ˆ σ ML s a ubased esmaor of σ. c. Show ha he maxmum lkelhood esmaor of σ α ma be expressed as: ˆ σ α ˆ ML σ ML. T d. Show ha ˆ σ α ML s a based esmaor of σ α ad deerme he bas. e. Show ha he resrced maxmum lkelhood esmaor of σ equals he correspodg maxmum lkelhood esmaor ha s show ˆ σ ˆ REML σ ML. f. Show ha he resrced maxmum lkelhood esmaor of σ α ma be expressed as: ˆ σ α ˆ REML σ ML. T g. Show ha σ s a ubased esmaor of σ α. ˆα REML. Emprcal Exercses 3.3. Charable Corbuos refer o Exercse.9 for he problem descrpo. a Error compoes model Ru a error compoes model of CHARITY o INCOME PRICE DEPS AGE ad MS. Sae whch varables are sascall sgfca ad jusf our coclusos. b Re-ru he sep par a b cludg he suppl-sde measures as addoal explaaor varables. Sae wheher or o hese varables should be cluded he model. Expla our reasog. c Icorporag emporal effecs. Is here a mpora me paer? For he model par a: re-ru excludg YEAR as a explaaor varable e cludg a AR seral compoe for he error. re-ru cludg YEAR as a explaaor varable ad cludg AR seral compoe for he error. re-ru cludg YEAR as a explaaor varable ad cludg a usrucured seral compoe for he error. Ths sep ma be dffcul o acheve covergece of he algorhm!

3-38 / Chaper 3. Models wh Radom Effecs d v Whch model do ou prefer or? Jusf our choce. I our jusfcao dscuss he osaoar of errors. Varable slope models Re-ru he model par a cludg a varable slope for INCOME. Sae whch of he wo models s preferred ad sae our reaso. Re-ru he model par a cludg a varable slope for PRICE. Sae whch of he wo models s preferred ad sae our reaso. Fal Par. Whch model do ou hk s bes? Do o cofe ourself o he opos ha ou esed he precedg pars. Jusf our choce. 3.4. Tor Flgs refer o Exercse.0 for he problem descrpo. a Ru a error compoes model usg sae as he subjec defer ad VEHCMILE GSTATEP POPDENSY WCMPMAX URBAN UNEMPLOY ad JSLIAB as explaaor varables. b Re-ru he error compoes model par a ad clude he addoal explaaor varables COLLRULE CAPS ad PUNITIVE. Tes wheher hese addoal varables are sascall sgfca usg he lkelhood rao es. Sae our ull ad alerave hpoheses our es sasc ad decso-makg rule. c Nowhsadg our aswer par b re-ru he model par a bu also clude varable radom coeffces assocaed wh WCMPMAX. Whch model do ou prefer he model par a or hs oe? d Jus for fu re-ru he model par b ad cludg varable radom coeffces assocaed wh WCMPMAX e Re-ru he error compoes model par a bu clude a auoregressve error of order. Tes for he sgfcace of hs erm. f Ru he model par a bu wh fxed effecs. Compare hs model o he radom effecs verso. 3.5. Housg Prces refer o Exercse. for he problem descrpo. a Basc summar sascs Produce a mulple me seres plo of NARSP. Produce a mulple me seres plo of YPC. Produce a scaer plo of NARSP versus YPC. v Produce a added varable plo of NARSP versus YPC corollg for he effecs of MSA. v Produce a scaer plo of NARSP versus YPC. b Error compoes model Ru a oe-wa error compoes model of NARSP o YPC ad YEAR. Sae whch varables are sascall sgfca. Re-ru he sep b b cludg he suppl-sde measures as addoal explaaor varables. Sae wheher or o hese varables should be cluded he model. Expla our reasog. c Icorporag emporal effecs. Is here a mpora me paer? Ru a oe-wa error compoes model of NARSP o YPC. Calculae resduals from hs model. Produce a mulple me seres plo of resduals. Re-ru he model par c ad clude a AR seral compoe for he error. Dscuss he saoar of errors based o he oupu of hs model f ad our aalss par c.

Chaper 3. Models wh Radom Effecs / 3-39 d Varable slope models Re-ru he model par c cludg a varable slope for YPC. Assume ha he wo radom effecs erceps ad slopes are depede. Re-ru he model par d bu allow for depedece bewee he wo radom effecs. Sae whch of he wo models ou prefer ad wh. Re-ru he model par d bu corporae he me-cosa suppl-sde varables. Esmae he sadard errors usg robus sadard errors. Sae whch varables are sascall sgfca; jusf our saeme. v Gve he dscusso of o-saoar par c descrbe wh robus varace esmaors are preferred whe compared o he model-based sadard errors. 3.6 Capal Srucure Usrucured problem Durg he 980s Japa s real ecoom was exhbg a healh rae of growh. The ose of he crash he sock ad real esae markes bega a he ed of December 989 ad he facal crss soo spread o Japa s bakg ssem. Afer more ha e ears he bakg ssem remaed weak ad he ecoom sruggles. These daa provde formao o 36 dusral Japaese frms before ad afer he crash. 355 of he 36 frms he sample are from he Frs Seco of he Toko Sock Exchage; he remag sx are from he Secod Seco. Togeher he cosue abou 33 perce of he marke capalzao of he Toko Sock Exchage. The sample frms are classfed as keresu or o-keresu. Tha s he ma bak ssem s ofe par of a broader cross share-holdg srucure ha cludes corporae groups called keresu. A useful fuco of such corporae groupgs ad he ma bak ssem s o mgae some of he formaoal ad ceve problems Japa s facal markes. A mpora feaure of he sud s o def chages facal srucure before ad afer he crash ad o see how hese chages are affeced b wheher or o a frm s classfed as keresu. For hs exercse develop a model wh radom effecs. Assess wheher or o he keresu srucure s a mpora deerma of leverage. For oe approach see Paker 000O. Varable TANG MTB LS PROF STD LVB Table 3E. Corporae Srucure Descrpo Tagbl. Ne oal fxed asses as a proporo of he book value of oal asses. Marke-o-Book. Rao of oal asses a marke value o oal asses a book value. Logarhmc Sales. The aural logarhm of he amou of sales of goods ad servces o hrd pares relag o he ormal acves of he compa. Profabl. Eargs before eres ad axes plus deprecao all dvded b he book value of oal asses. Volal. The sadard devao of weekl ulevered sock reurs durg he ear. Ths varable proxes for he busess rsk facg he frm. Toal Leverage Book. Toal deb as a proporo of oal deb plus book equ. Toal deb s he sum of shor-erm ad log-erm deb. Ths s he depede varable. Source: Paker 000O.

Chaper 4. Predco ad Baesa Iferece / 4-003 b Edward W. Frees. All rghs reserved Chaper 4. Predco ad Baesa Iferece Absrac. Chapers ad 3 roduced models wh fxed ad radom effecs focusg o esmao ad hpohess esg. Ths chaper dscusses he hrd basc pe of sascal ferece predco. Predco s parcularl mpora mxed effecs models where oe ofe eeds o summarze a radom effecs compoe. I ma cases he predcor ca be erpreed as a shrkage esmaor ha s a lear combao of local ad global effecs. I addo o predcg he radom effecs compoe hs chaper also shows how o predc dsurbace erms mpora for dagosc work as well as fuure resposes ha are he ma focus forecasg applcaos. The predcors are opmal he sese ha he are derved as mmum mea square bes lear ubased predcors kow as BLUPs. As a alerave o hs classc freques seg Baesa predcors are also defed. Moreover Baesa predcors wh a dffuse pror are equvale o he mmum mea square lear ubased predcors hus provdg addoal movao for hese predcors. Esmaors versus predcors I he lear mxed effecs model z α + x β + ε he radom varables {α } descrbe effecs ha are specfc o a subjec. Gve he daa { z x } some problems s of eres o summarze subjec effecs. Chapers ad 3 dscussed how o esmae fxed ukow parameers. Ths chaper dscusses applcaos where s of eres o summarze radom subjec-specfc effecs. I geeral we use he erm predcor for a esmaor of a radom varable. Lke esmaors a predcor s sad o be lear f s formed from a lear combao of he resposes. Whe would a aals be eresed esmag a radom varable? I some applcaos he eres s predcg he fuure values of a respose kow as forecasg. To llusrae Seco 4.4 demosraes forecasg he coex of Wscos loer sales. Ths chaper also roduces ools ad echques for predco coexs oher ha forecasg. For example amal ad pla breedg oe wshes o predc he produco of mlk for cows based o her leage radom ad herds fxed. I surace oe wshes o predc expeced clams for a polcholder gve exposure o several rsk facors kow as credbl heor. I sample surves oe wshes o predc he sze of a specfc age-sex-race cohor wh a small geographcal area kow as small area esmao. I a surve arcle Robso 99S also ces ore reserve esmao geologcal surves measurg qual of a produco pla ad 3 rakg baseball plaers ables.

4- / Chaper 4. Predco ad Baesa Iferece 4. Predcos for oe-wa ANOVA models To beg recall a specal case of lear mxed effecs models he radoal oe-wa radom effecs ANOVA aalss of varace model µ + α + ε. 4. As descrbed Seco 3. we assume ha boh α ad ε are mea zero depede quaes. Suppose ow ha we are eresed summarzg he codoal mea effec of he h subjec µ + α. For coras recall he correspodg fxed effecs model. I hs case we dd o explcl express he overall mea bu used he oao α + ε. Wh hs oao α represes he mea of he h subjec. We saw ha s he bes Gauss-Markov esmaor of α. Ths esmaor s ubased ha s E α. Furher s mmum varace bes amog all lear ubased esmaors kow b he acrom BLUE. Shrkage esmaor For he model equao 4. seems uvel plausble ha s a desrable esmaor of µ ad ha - s a desrable esmaor of α. Thus s a desrable predcor of µ + α. More geerall cosder predcors of µ + α ha are lear combaos of ad ha s c + c for cosas c ad c. To rea he ubasedess we use c c. Some basc calculaos see Exercse 4. show ha he bes value of c ha mmzes E c + c µ + α s where T c * T σ σ + T σ T α * α + T j N N * j. T N Here we use he oao σ α ad σ for he varace of α ad ε respecvel. For erpreao s helpful o cosder he case where he umber of subjecs eds o f. Ths elds he shrkage esmaor or predcor of µ + α defed as ζ + 4. α / α s ζ Tσ α T where ζ s he h credbl facor. Tσ + σ T + σ σ Example Cosder he followg llusrave daa: 4 0 9 6 5 ad 3 8 0 7 7. Tha s we have 3 subjecs each of whch has T4 observaos. The sample mea s ; he subjec-specfc sample meas are 3 ad 3 8. We ow f he oe-wa radom effecs ANOVA model equao 4.. From he varace esmao procedures descrbed Seco 3.5 we have ha he REML esmaes of σ ad σ α are 4.889 ad 5.778 respecvel. I follows ha he esmaed ζ wegh s 0.85 ad he correspodg predcos for he subjecs are.85.650 ad 8.55 respecvel.

Chaper 4. Predco ad Baesa Iferece / 4-3 Fgure 4. compares subjec-specfc meas o he correspodg predcos. Here we see less spread he predcos compared o he subjec-specfc meas; each subjec s esmae s shruk o he overall mea. These are he bes predcors assumg α are radom. I coras he subjec-specfc meas are he bes predcors assumg α are deermsc. Thus hs shrkage effec s a cosequece of he radom effecs specfcao. 3 8 3 8.55.85.650 3 s s s Fgure 4. Comparso of Subjec-Specfc Meas o Shrkage Esmaes. For a llusrave daa se subjec-specfc ad overall meas are graphed o he upper scale. The correspodg shrkage esmaes are graphed o he lower scale. Ths fgure shows he shrkage aspec of models wh radom effecs. Uder he radom effecs ANOVA model we have ha s a ubased predcor of µ +α he sese ha E - µ +α 0. However s effce he sese ha he shrkage esmaor s has a smaller mea square error ha. Iuvel because s s a lear combao of ad we sa ha has bee shruk owards he esmaor. Furher because of he addoal formao s cusomar o erpre a shrkage esmaor as borrowg sregh from he esmaor of he overall mea. Noe ha he shrkage esmaor reduces o he fxed effecs esmaor whe he credbl facor ζ becomes. I s eas o see ha ζ as eher T or σ α /σ. Tha s he bes predcor approaches he subjec mea as eher he umber of observaos per subjec becomes large or he varabl amog subjecs becomes large relave o he respose varabl. I acuaral laguage eher case suppors he dea ha he formao from he h subjec s becomg more credble. Bes predcors Whe he umber of observaos per subjec vares he shrkage esmaor defed equao 4. ca be mproved. Ths s due o he fac ha s o he opmal esmaor of µ. Usg echques descrbed Seco 3. s eas o check ha see Exercse 3. he geeralzed leas squares GLS esmaor of µ s ζ mα GLS 4.3 ζ I Seco 4. we wll see ha he lear predcor of µ + α ha has mmum varace s

4-4 / Chaper 4. Predco ad Baesa Iferece ζ + - ζ m αgls. 4.4 BLUP The acrom BLUP sads for bes lear ubased predcor. Tpes of predcors Ths chaper focuses o predcors for hree pes of radom varables.. Lear combaos of regresso parameers ad subjec-specfc effecs. The sasc BLUP provdes a opmal predcor of µ + α. Thus BLUP s a example of a predcor of a lear combao of a global parameer ad a subjec-specfc effec.. Resduals. Here we wsh o predc ε. The BLUP urs ou o be:. 4.5 e BLUP BLUP These quaes called BLUP resduals are useful dagosc sascs for developg a model. Furher uusual resduals help us udersad f a observao s le wh ohers he daa se. To llusrae f he respose s a salar or a sock reur uusual resduals ma help us deec uusual salares or sock reurs. 3. Forecass. Here we wsh o predc for L lead me us o he fuure T + L µ + α + ε. 4.6 T + L Forecasg s smlar o predcg a lear combao of global parameers ad subjecspecfc effecs wh a addoal fuure error erm. I he absece of seral correlao we wll see ha he predcor s he same as he predcor of µ + α alhough he mea square error urs ou o be larger. Seral correlao wll lead o a dffere forecasg formula. I hs seco we have movaed BLUP s usg mmum varace ubased predco. Oe ca also movae BLUP s usg ormal dsrbuo heor. Tha s cosder he case where α ad K } have a jo mulvarae ormal dsrbuo. The ca be show ha { T E µ +α K ζ + - ζ µ. T Ths calculao s of eres because f oe were eresed esmag he uobservable α based o he observed resposes K } he ormal heor suggess ha he expecao { T s a opmal esmaor. Tha s cosder askg he queso: wha realzao of µ +α could be assocaed wh K }? The expecao! The BLUP s he bes lear ubased esmaor { T BLUE of E µ +α K } specfcall we eed ol replace µ b m αgls. Seco 4.5 { T wll dscuss hese deas more formall a Baesa coex. 4. Bes lear ubased predcors Ths seco develops bes lear ubased predcors he coex of mxed lear models. Seco 4.3 he specalzes he cosderao o lear mxed effecs models. Seco 8.3 wll cosder aoher specalzao o me-varg coeffce models. As descrbed Seco 4. we develop BLUP s b examg he mmum mea square error predcor of a radom varable w. Ths developme s due o Harvlle 976S whch also appears hs dscusso of Robso 99S. However he argume s orgall due o Goldberger 96E who coed

Chaper 4. Predco ad Baesa Iferece / 4-5 he phrase bes lear ubased predcor. The acrom BLUP was frs used b Hederso 973B. Recall he mxed lear model preseed Seco 3.3.. Tha s suppose ha we observe a N radom vecor wh mea E X β ad varace Var V. The geerc goal s o predc a radom varable w such ha E w λ β ad Var w σ w. Deoe he covarace bewee w ad as he N vecor Covw E { w E w E }. The choce of w ad hus λ ad σ w wll deped o he applcao a had; several examples wll be gve Seco 4.3. Beg b assumg ha he global regresso parameers β are kow. The Appedx 4A. shows ha he bes lear predcor of w s w * E w + Covw V - - E λ β + Covw V - - X β. As we wll see he Baesa coex Seco 4.4 f w have a mulvarae jo ormal dsrbuo he w * equals E w so ha w * s a mmum mea square predcor of w. Appedx 4A. shows ha he predcor w * s also a mmum mea square predcor of w whou he assumpo of ormal. BLUP s as predcors Nex we assume ha he global regresso parameers β are o kow. As Seco 3.5. we use b GLS X V - X - X V o be he geeralzed leas squares GLS esmaor of β. Ths s he bes lear ubased esmaor BLUE of β. Replacg β b b GLS he defo of w * we arrve a a expresso for he BLUP of w w BLUP λ b GLS + Covw V - - X b GLS λ - Covw V - X b GLS + Covw V -. 4.7 Appedx 4A. esablshes ha w BLUP s he bes lear ubased predcor of w he sese ha s he bes lear combao of resposes ha s ubased ad has he smalles mea square error over all lear ubased predcors. From Appedx 4A.3 we also have he form for he mea square error ad varace: Var ad Var w BLUP w BLUP w Cov w V λ Cov w V X X V X λ Cov w V X Cov w Cov w V Cov w + σ w 4.8 Cov λ w V X X V X λ Cov w V X. 4.9 From equaos 4.8 ad 4.9 we see ha σ w Var w Varw - w BLUP + Var w BLUP. Hece he predco error w BLUP - w ad he predcor w BLUP are ucorrelaed. Ths fac wll smplf calculaos subseque examples. The BLUP predcors are opmal assumg he varace compoes mplc V ad Covw are kow. Applcaos of BLUP pcall requre ha he varace compoes be esmaed as descrbed Seco 3.5. BLUP wh esmaed varace compoes are kow as emprcal BLUPs or EBLUPs. The formulas equaos 4.8 ad 4.9 do o accou for he ucera varace compoe esmao. Iflao facors ha accou for hs addoal ucera have bee proposed Kackar ad Harvlle 984S bu he ed o be small a leas

4-6 / Chaper 4. Predco ad Baesa Iferece for daa ses commol ecouered pracce. McCulloch ad Searle 00G ad Keward ad Roger 997B provde furher dscussos. Specal case - Oe-wa radom effecs ANOVA model We ow esablsh he oe-wa radom effecs model BLUPs ha were descrbed equaos 4.4-4.6 of Seco 4.. To do hs we frs wre he oe-wa radom effecs ANOVA model as a specal case of he mxed lear model. We he esablsh he predcos as specal cases of w BLUP gve equao 4.7. To express equao 4. erms of a mxed lear model recall he error compoes formulao Seco 3.. Thus we wre 4. vecor form as µ + α + ε where Var V σ α J + σ I ad V I ζ σ T J. Thus we have X ad V block dagoal V V. To develop expressos for he bes lear ubased predcors we beg wh GLS esmaor gve equao 4.3. Thus from equao 4.7 ad he block dagoal aure of V we have w BLUP λ m αgls + Covw V - - X m αgls λ m α GLS + Cov w - m V. α GLS Now we have he relao ζ V - mα GLS I J - mα GLS σ T - mα GLS ζ mα GLS. σ Ths elds w BLUP λ mα GLS + Cov w - m GLS m GLS α ζ α. σ Now suppose ha he eres s predcg w µ + α. The we have λ ad Covw σ α for he h subjec ad Covw 0 for all oher subjecs. Thus we have σ α wblup mα GLS + - mα GLS ζ mα GLS σ σ α mα GLS + T - mα GLS ζ mα GLS σ σ α mα GLS + T ζ - mα GLS mα GLS + ζ - mα GLS σ whch cofrms equao 4.4. For predcg resduals we assume ha w ε. Thus we have λ 0 ad Covε σ ε for he h subjec ad Covw 0 for all oher subjecs. Here deoes a T vecor wh a oe he h row ad zeroes oherwse. Thus we have wblup whch cofrms equao 4.5. σ V - X bgls - BLUP

Chaper 4. Predco ad Baesa Iferece / 4-7 For forecasg we use equao 4.6 ad choose w µ + α + ε T + L. Thus we have λ ad Covw σ α for he h subjec ad Covw 0 for all oher subjecs. Wh hs our expresso for w BLUP s he same as he case predcg w µ + α. 4.3 Mxed model predcors Bes lear ubased predcors for mxed lear models were preseed equao 4.7 wh correspodg mea square errors ad varaces equaos 4.8 ad 4.9 respecvel. Ths seco uses hese resuls b preseg hree broad classes of predcors ha are useful for lear mxed effecs models ogeher wh a hos of specal cases ha provde addoal erpreao. I some of he specal cases we po ou ha hese resuls also pera o: cross-secoal models b choosg D o be a zero marx ad fxed effecs models b choosg D o be a zero marx ad corporag Z α as fxed effecs o he expeced value of. The hree broad classes of predcors are lear combaos of global parameers β ad subjec-specfc effecs α resduals ad 3 forecass. 4.3. Lear mxed effecs model To see how he geeral Seco 4. resuls appl o he lear mxed effecs model recall from equao 3.5 our marx oao of he model: Z α + X β + e. As descrbed equao 3.8 we sack hese equaos o ge Z α + X β + e. Thus E X β wh X X X X. Furher we have Var V where he varace-covarace marx s block dagoal of he form V block dagoal V V... V where V Z D Z + R. Thus from equao 4.7 we have w BLUP λ b GLS + Covw V - - X b GLS λ b + Cov w V - X b. 4.0 Exercse 4.9 provdes expressos for he BLUP mea square error ad varace. 4.3. Lear combaos of global parameers ad subjec-specfc effecs Cosder predcg lear combaos of he form w c α + c β. Here c ad c are kow vecors of cosas ha are user-specfed. The wh hs choce of w sragh-forward calculaos show ha E w c β so ha λ c. Furher we have c DZ for j Cov w j. 0 for j Pug hs equao 4.0 elds w BLUP c D Z V - - X b GLS + c b GLS. To smplf hs expresso we ake c 0 ad use Wald s devce. Ths elds he BLUP of α a BLUP D Z V - - X b GLS. 4. Wh hs oao our BLUP predcor of w c α + c β s w BLUP c a BLUP + c b GLS. 4. Some addoal specal cases are of eres. For he radom coeffces model roduced Seco 3.3. wh equao 4. s eas o check ha he BLUP of β + α s GLS GLS

4-8 / Chaper 4. Predco ad Baesa Iferece wblup ζ b + ζ b. GLS Here b X V X - X V s he subjec-specfc GLS esmaor ad ζ D X V - X s a wegh marx. Ths resul geeralzes he oe-wa radom effecs predcors preseed Seco 4.. I he case of he error compoes model descrbed Seco 3. we have q ad z. Usg equao 4. he BLUP of α reduces o a ζ - x b. BLUP GLS For comparso recall from Chaper ha he fxed effecs parameer esmae s The oher poro - ζ s borrowg sregh from zero he mea of α. Seco 4.6 descrbes furher examples from surace credbl. a b. x Example Trade localzao Coued Feberg Keae ad Bogao 998E used frm-level daa o vesgae U.S. based mulaoal corporaos emplome ad capal allocao decsos. From Chaper 3 her model ca be wre as l β CT + β UT + β 3 Tred + x * β* + ε β +α CT + β +α UT + β 3 +α 3 Tred + x * β* + ε α CT + α UT + α 3 Tred + x β + ε. where CT UT s a measure of Caada U.S. arffs for frm ad he respose s eher emplome or durable asses for he Caada afflae. Feberg Keae ad Bogao preseed predcors of β ad β usg Baesa mehods see he Seco 4.5 dscusso. I our oao he predced he lear combaos β +α ad β +α. The red erm was o of prmar scefc eres ad was cluded as a corol varable. Oe major fdg was ha predcors for β were egave for each frm dcag ha emplome ad asses Caada afflaes creased as Caada arffs decreased. 4.3.3 BLUP resduals For he secod broad class cosder predcg a lear combao of resduals w c ε ε where c ε s a vecor of cosas. Wh hs choce we have E w 0; follows ha λ 0. Sraghforward calculaos show ha c ε R for j Cov w j. 0 for j Thus from equao 4.0 ad Wald s devce we have he vecor of BLUP resduals e BLUP R V - - X b GLS 4.3a ha ca also be expressed as e BLUP Z a BLUP + X b GLS. 4.3b Equao 4.3a s appealg because allows for drec compuao of BLUP resduals; equao 4.3b s appealg because s he radoal observed mus expeced form for

Chaper 4. Predco ad Baesa Iferece / 4-9 resduals. We remark ha he BLUP resdual equals he GLS resdual he case ha D 0; hs case e BLUP - X b GLS e GLS. Furher recall he smbol ha deoes a T vecor ha has a oe he h poso ad s zero oherwse. Thus we ma defe he BLUP resdual as e BLUP e BLUP R V - - X b GLS. Equaos 4.3a ad 4.3.b provde a geeralzao of he BLUP resdual for he oewa radom effecs model descrbed equao 4.5. Furher usg equao 4.9 oe ca show ha he BLUP resdual has varace Var e BLUP R V V X XV X X V R. Takg he square roo of Var e BLUP wh a esmaed varaces elds a sadard error; hs cojuco wh he BLUP resdual s useful for dagosc checkg of he fed model. 4.3.4 Predcg fuure observaos For he hrd broad class suppose ha he h subjec s cluded he daa se ad we wsh o predc w T + L z T + Lα + x T + Lβ + ε T L for L lead me us he fuure. Assume ha z T + L ad x T + L are kow. Wh hs choce of w follows ha λ x +. Furher we have T Cov w j z 0 T + L DZ + Cov ε T + L Thus usg equaos 4.0 4. ad 4.3 we have ˆ w DZ + Cov ε ε V ε for j. for j z T + L T + L XbGLS + x T + L bgls T + L BLUP x T + L b GLS + z T + L a BLUP + Cov T + L ε R e BLUP ε. 4.4 Thus he forecas s he esmae of he codoal mea plus he seral correlao correco facor Cov ε T + L ε R e BLUP. Usg equao 4.8 oe ca show ha he varace of he forecas error as Var ˆ + T + L T + L T + L T + L T + L x z DZ Cov ε ε V X XV X x z DZ + Covε ε V X T + L T + L T + L T + L DZ V Z Dz T + L + z T + L Dz T + L + Var ε T + L z. 4.5 Specal case Auoregressve seral correlao To llusrae cosder he specal case where we have auoregressve of order AR serall correlaed errors. For saoar AR errors he lag j auocorrelao coeffce ρ j ca be expressed as ρ j. Thus wh a AR specfcao we have

/ Chaper 4. Predco ad Baesa Iferece 4-0 3 3 L M O M M M L L L T T T T T T ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ σ R where we have omed he subscrp. Sraghforward marx algebra resuls show ha + + + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ σ L L M M O M M M L L L R. Furher omg he subscrp o T we have + + + + + L L L T L T L T L T ρ ρ ρ ρ ρ σ ε 3 Cov L ε. Thus + Cov L T R ε ε + + + + + + + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ L L M M O M M M L L L L L L L T L T L T L L L ρ ρ ρ ρ 0 0 0 0 0 0 0 0 L L + +. To summarze he L sep forecas s BLUP T L BLUP L T GLS L T L T e ˆ ρ + + + + + a z b x. Tha s he L sep forecas equals he esmae of he codoal mea plus a correco facor of ρ L mes he mos rece BLUP resdual BLUP T e. Ths resul was orgall gve b Goldberger 96E he coex of ordar regresso whou radom effecs ha s assumg D 0. 4.4 Example: Forecasg Wscos loer sales I hs seco we forecas he sale of sae loer ckes from 50 posal ZIP codes Wscos. Loer sales are a mpora compoe of sae reveues. Accurae forecasg helps he budge plag process. Furher a model s useful assessg he mpora deermas of loer sales. Udersadg he deermas of loer sales s useful for mprovg he desg of he loer sales ssem. Addoal deals of hs sud are Frees ad Mller 003O.

Chaper 4. Predco ad Baesa Iferece / 4-4.4. Sources ad characerscs of daa Sae of Wscos loer admsraors provded weekl loer sales daa. We cosder ole loer ckes ha are sold b seleced real esablshmes Wscos. These ckes are geerall prced a $.00 so he umber of ckes sold equals he loer reveue. We aalze loer sales OLSALES over a for-week perod Aprl 998 hrough Jauar 999 from ff radoml seleced ZIP codes wh he sae of Wscos. We also cosder he umber of realers wh a ZIP code for each me NRETAIL. A buddg leraure such as Ashle Lu ad Chag 999O sugges varables ha fluece loer sales. Table 4. lss ecoomc ad demographc characerscs ha we cosder hs aalss. Much of he emprcal leraure o loeres s based o aual daa ha exames he sae as he u of aalss. I coras we exame much fer ecoomc us he ZIP code level ad exame weekl loer sales. The ecoomc ad demographc characerscs were absraced from he Ued Saes cesus. These varables summarze characerscs of dvduals wh ZIP codes a a sgle po me ad hus are o me varg. Table 4.. Loer ecoomc ad demographc characerscs of ff Wscos ZIP codes Loer characerscs OLSALES Ole loer sales o dvdual cosumers NRETAIL Number of lsed realers Ecoomc ad demographc characerscs PERPERHH Persos per household MEDSCHYR Meda ears of schoolg MEDHVL Meda home value $000s for ower-occuped homes PRCRENT Perce of housg ha s reer occuped PRC55P Perce of populao ha s 55 or older HHMEDAGE Household meda age MEDINC Esmaed meda household come $000s POPULATN Populao housads Source: Frees ad Mller 003O Table 4. summarzes he ecoomc ad demographc characerscs of ff Wscos ZIP codes. To llusrae for he populao varable POPULATN we see ha he smalles ZIP code coaed 80 people whereas he larges coaed 39098. The average over ff ZIP codes was 93.04. Table 4. also summarzes average ole sales ad average umber of realers. Here hese are averages over for weeks. To llusrae we see ha he for-week average of ole sales was as low as $89 ad as hgh as $338. Table 4.. Summar sascs of loer ecoomc ad demographc characerscs of ff Wscos ZIP codes Varable Mea Meda Sadard Mmum Maxmum Devao Average OLSALES 6494.83 46.4 803.0 89 338 Average NRETAIL.94 6.36 3.9 68.65 PERPERHH.7.7 0.. 3. MEDSCHYR.70.6 0.55. 5.9 MEDHVL 57.09 53.90 8.37 34.50 0 PRCRENT 4.68 4 9.34 6 6 PRC55P 39.70 40 7.5 5 56 HHMEDAGE 48.76 48 4.4 4 59 MEDINC 45. 43.0 9.78 7.90 70.70 POPULATN 9.3 4.405.098 0.80 39.098

4- / Chaper 4. Predco ad Baesa Iferece I s possble o exame cross-secoal relaoshps bewee sales ad ecoomc demographc characerscs. For example Fgure 4. shows a posve relaoshp bewee average ole sales ad populao. Furher he ZIP code correspodg o c of Keosha Wscos has uusuall large average sales for s populao sze. Average Loer Sales 30000 Keosha 0000 0000 0 0 0000 0000 Populao 30000 40000 Fgure 4.. Scaer plo of average loer sales versus populao sze. Sales for Keosha are uusuall large for s populao sze. However cross-secoal relaoshps such as correlaos ad plos smlar o Fgure 4. hde damc paers of sales. Fgure 4.3 s a mulple me seres plo of weekl sales over me. Here each le races he sales paers for a ZIP code. Ths fgure shows he dramac crease sales for mos ZIP codes a approxmael weeks egh ad eghee. For boh me pos he jackpo prze of oe ole game PowerBall grew o a amou excess of $00 mllo. Ieres loeres ad sales creases dramacall whe jackpo przes reach large amous. Loer Sales 300000 00000 00000 0 0 0 0 30 40 Week Number Fgure 4.3. Mulple me seres plo of loer sales. Sales a ad aroud weeks 8 ad 8 are uusuall large due o large PowerBall jackpos.

Chaper 4. Predco ad Baesa Iferece / 4-3 Fgure 4.4 shows he same formao as Fgure 4.3 bu o a commo base e logarhmc scale. Here we sll see he effecs of he PowerBall jackpos o sales. However Fgure 4.4 suggess a damc paer ha s commo o all ZIP codes. Specfcall logarhmc sales for each ZIP code are relavel sable wh he same approxmae level of varabl. Furher logarhmc sales for each ZIP code peaks a he same me correspodg o large PowerBall jackpos. Logarhmc Loer Sales 6 5 4 3 0 0 0 30 40 Week Number Fgure 4.4. Mulple me seres plo of logarhmc base 0 loer sales. Aoher form of he respose varable o cosder s he proporoal or perceage chage. Specfcall defe he perceage chage o be sales pchage 00. 4.9 sales A mulple mes seres plo of he perceage chages o dsplaed here shows auocorrelaed seral paers. We cosder models of hs rasformed seres he followg subseco o model seleco. 4.4. I-sample model specfcao Ths subseco cosders he specfcao of a model a ecessar compoe pror o forecasg. We decompose model specfcao crera o wo compoes -sample ad ouof-sample crera. To hs ed we paro our daa o wo subsamples; we use he frs 35 weeks o develop alerave fed models ad use he las fve weeks o predc our held-ou sample. The choce of fve weeks for he ou-of-sample valdao s somewha arbrar; was made wh he raoale ha loer offcals cosder reasoable o r o predc fve weeks of sales based o hr-fve weeks of hsorcal sales daa. Our frs forecasg model s he pooled cross-secoal model. The model fs he daa well; he coeffce of deermao urs ou o be R 69.6%. The esmaed regresso

4-4 / Chaper 4. Predco ad Baesa Iferece coeffces appear Table 4.3. From he correspodg -sascs we see ha each varable s sascall sgfca. Our secod forecasg model s a error compoes model. Table 4.3 provdes parameer esmaes ad he correspodg -sascs as well as esmaes of he varace compoes σ α ad σ. As we have see oher examples allowg erceps o var b subjec ca resul regresso coeffces for oher varables becomg sascall sgfca. Whe comparg hs model o he pooled cross-secoal model we ma use he Lagrage mulpler es descrbed Seco 3.. The es sasc urs ou o be TS 395.5 dcag ha error compoes model s srogl preferred o he pooled cross-secoal model. Aoher pece of evdece s Akake s Iformao Crero AIC. Ths crero s defed as AIC - lmaxmzed lkelhood + umber of model parameers. The smaller hs crero he more preferred s he model. Appedx C.9 descrbes hs crero furher deal. Table 4.3 shows aga ha he error compoes model s preferred compared o he pooled cross-secoal model based o he smaller value of he AIC sasc. Table 4.3 Loer model coeffce esmaes Based o -sample daa of 50 ZIP codes ad T 35 weeks. The respose s aural logarhmc sales. Pooled cross-secoal model Error compoes model Error compoes model wh AR erm Varable Parameer -sasc Parameer -sasc Parameer -sasc esmae esmae esmae Iercep 3.8 0.3 8.096.47 5.55.8 PERPERHH -.085-6.77 -.87 -.45 -.49 -.36 MEDSCHYR -0.8 -.90 -.078 -.87-0.9 -.53 MEDHVL 0.04 5.9 0.007 0.50 0.0 0.8 PRCRENT 0.03 8.5 0.06.7 0.030.53 PRC55P -0.070-5.9-0.073-0.98-0.07 -.0 HHMEDAGE 0.8 5.64 0.9.0 0.0.09 MEDINC 0.043 8.8 0.046.55 0.044.58 POPULATN 0.057 9.4 0. 4.43 0.080.73 NRETAIL 0.0 5. -0.07 -.56 0.004 0.0 Var α σ α 0.607 0.58 Var ε σ 0.700 0.63 0.79 AR corr ρ 0.555 5.88 AIC 4353.5 86.74 70.97 To assess furher he adequac of he error compoes model resduals from he fed model were calculaed. Several dagosc ess ad graphs were made usg hese resduals o mprove he model f. Fgure 4.5 represes oe such dagosc graph a plo of resduals versus lagged resduals. Ths fgure shows a srog relaoshp bewee resduals ad lagged resduals whch we ca represe usg a auocorrelao srucure for he error erms. To accommodae hs paer we also cosder a error compoe model wh a AR erm; he fed model appears Table 4.3. Fgure 4.5 also shows a srog paer of cluserg correspodg o weeks wh large PowerBall jackpos. A varable ha capures formao abou he sze of PowerBall jackpos would help developg a model of loer sales. However for forecasg purposes we requre oe or more varables ha acpaes large PowerBall jackpos. Tha s because he sze of

Chaper 4. Predco ad Baesa Iferece / 4-5 PowerBall jackpos s o kow advace varables ha prox he eve of large jackpos are o suable for forecasg models. These varables could be developed hrough a separae forecasg model of PowerBall jackpos. Oher pes of radom effecs models for forecasg loer sales could also be cosdered. To llusrae we also f a more parsmoous verso of he AR verso of he error compoes model; specfcall we re-f hs model deleg hose varables wh sgfca -sascs. I ured ou ha hs fed model dd o perform subsaall beer erms of overall model f sascs such as AIC. We explore alerave rasforms of he respose whe examg a held-ou sample he followg subseco. Resduals 3 Tme7 or 7 Tme8 or 8 Tme9 or 9 Oher Tmes 0 - - 0 3 Lag Resduals Fgure 4.5. Scaer plo of resduals versus lagged resduals from a error compoe model. The plo shows he srog auocorrelao edec amog resduals. Dffere plog smbols dcae he cluserg accordg o me. The four me smbols correspod o mmedae pror o a jackpo me7 or 7 he week durg a jackpo me8 or 8 he week followg a jackpo me9 or 9 ad oher weeks me-6 0-6 or 0-35. 4.4.3 Ou-of-sample model specfcao Ths subseco compares he abl of several compeg models o forecas values ousde of he sample used for model parameer esmao. As Seco 4.4. we use he frs 35 weeks of daa o esmae model parameers. The remag fve weeks are used o assess he vald of model forecass. For each model we compue forecass of loer sales for weeks 36 hrough 40 b ZIP code level based o he frs 35 weeks. Deoe hese forecas values as ZOL ŜALES 35 +L for L o 5. We summarze he accurac of he forecass hrough wo sascs he mea absolue error 5 MAE ZOLŜALES 35+ L ZOLSALES 35+ L 4.0 5 L ad he mea absolue perceage error

4-6 / Chaper 4. Predco ad Baesa Iferece 00 MAPE 5 5 ZOLŜALES 35+ L ZOLSALES ZOLSALES L 35+ L 35+ L. 4. The several compeg models clude he hree models of logarhmc sales summarzed Table 4.3. Because he auocorrelao erm appears o be hghl sascall sgfca Table 4.3 we also f a pooled cross-secoal model wh a AR erm. Furher we f wo modfcaos of he error compoes model wh he AR erm. I he frs case we use loer sales as he respose o he logarhmc verso ad he secod case we use perceage chage of loer sales defed equao 4.9 as he respose. Fall he seveh model ha we cosder s a basc fxed effecs model α + ε wh a AR error srucure. Recall ha for he fxed effecs models he erm α s reaed as a fxed o radom parameer. Because hs parameer s me-vara s o possble o clude our me-vara demographc ad ecoomc characerscs as par of he fxed effecs model. Table 4.4 preses he model forecas crera equaos 4.0 ad 4. for each of hese seve models. We frs oe ha Table 4.4 re-cofrms he po ha he AR erm mproves each model. Specfcall for boh he pooled cross-secoal ad he error compoes model he verso wh a AR erm ouperforms he aalogous model whou hs erm. Table 4.4 also shows ha he error compoes model domaes he pooled cross-secoal model. Ths was also acpaed b our poolg es a -sample es procedure. Table 4.4 cofrms ha he error compoes model wh a AR erm wh logarhmc sales as he respose s he preferred model based o eher he MAE or MAPE crero. The ex bes model was he correspodg fxed effecs model. I s eresg o oe ha he models wh sales as he respose ouperformed he model wh perceage chage as he respose based o he MAE crero alhough he reverse s rue based o he MAPE crero. Table 4.4. Ou-of-sample forecas comparso of sx alerave models Model forecas crera Model Model respose MAE MAPE Pooled cross-secoal model logarhmc sales 30.68 83.4 Pooled cross-secoal model wh AR erm logarhmc sales 680.64.9 Error compoes model logarhmc sales 38.05 33.85 Error compoes model wh AR erm logarhmc sales 57.4 8.79 Error compoes model wh AR erm sales 409.6 40.5 Error compoes model wh AR erm perceage chage 557.8 48.70 Fxed effecs model wh AR erm logarhmc sales 584.55 9.07 4.4.4 Forecass We ow forecas usg he model ha provdes he bes f o he daa he error compoes model wh a AR erm. The forecass ad varace of forecas errors for hs model are specal cases of he resuls for he lear mxed effecs model gve equaos 4.4 ad 4.5 respecvel. Forecas ervals are calculaed usg a ormal curve approxmao as he po forecas plus or mus.96 mes he square roo of he esmaed varace of he forecas error.

Chaper 4. Predco ad Baesa Iferece / 4-7 Fgure 4.6 dsplas he forecass ad forecas ervals. Here we use T 40 weeks of daa o esmae parameers ad provde forecass for L 5 weeks. Calculao of he parameer esmaes po forecass ad forecas ervals were doe usg logarhmc sales as he respose. The po forecass ad forecas ervals were covered o dollars o dspla he ulmae mpac of he model forecasg sraeg. Fgure 4.6 shows he forecass ad forecas ervals for wo seleced posal codes. The lower forecas represes a posal code from Dae Cou whereas he upper represes a posal code from Mlwaukee. For each posal code he mddle le represes he po forecas ad he upper ad lower les represe he bouds o a 95% forecas erval. Compared o he Dae Cou code he Mlwaukee posal code has hgher forecas sales. Thus alhough sadard errors o a logarhmc scale are abou he same as Dae Cou hs hgher po forecas leads o a larger erval whe rescaled o dollars. Sales 50000 45000 40000 35000 30000 5000 0000 5000 0000 5000 0 Aug- 98.. Sep- 98... Oc- 98.... Nov- 98... Dec- 98... Ja- 99 Tme Perod Fgure 4.6 Forecas Iervals for Two Seleced Posal Codes. For each posal code he mddle le correspods o po forecass for fve weeks. The upper ad lower les correspod o edpos of 95% predco ervals. 4.5 Baesa ferece Wh Baesa sascal models oe vews boh he model parameers ad he daa as radom varables. I hs seco we use a specfc pe of Baesa model he ormal lear herarchcal model dscussed b for example Gelma e al. 004S. As wh he wo-sage

4-8 / Chaper 4. Predco ad Baesa Iferece samplg scheme descrbed Seco 3.3. he herarchcal lear model s oe ha s specfed sages. Specfcall we cosder he followg wo-level herarch:. Gve he parameers β ad α he respose model s Z α + X β + ε. Ths level s a ordar fxed lear model ha was roduced Seco 3.3.. Specfcall we assume ha he resposes codoal o α ad β are ormall dsrbued ad ha E α β Z α + X β ad Var α β R.. Assume ha α s dsrbued ormall wh mea µ α ad varace D ad ha β s dsrbued ormall wh mea µ β ad varace Σ β each depede of he oher. The echcal dffereces bewee he mxed lear model ad he ormal herarchcal lear model are: he mxed lear model β s a ukow fxed parameer whereas he ormal herarchcal lear model β s a radom vecor ad he mxed lear model s dsrbuo-free whereas dsrbuoal assumpos are made each sage of he ormal herarchcal lear model. Moreover here are mpora dffereces erpreao. To llusrae suppose ha β 0 wh probabl oe. I he classc o-baesa also kow as he freques erpreao we hk of he dsrbuo of {α} as represeg he lkelhood of drawg a realzao of α. The lkelhood erpreao s mos suable whe we have a populao of frms or people ad each realzao s a draw from ha populao. I coras he Baesa case oe erpres he dsrbuo of {α} as represeg he kowledge ha oe has of hs parameer. Ths dsrbuo ma be subjecve ad allows he aals a form mechasm o jec hs or her assessmes o he model. I hs sese he freques erpreao ma be regarded as a specal case of he Baesa framework. The jo dsrbuo of α β s kow as he pror dsrbuo. To summarze he jo dsrbuo of α β s α µ α D 0 DZ β N µ β 0 Σβ ΣβX 4.6 Zµ α + Xµ β ZD XΣ β V + XΣ βx where V R + Z D Z. The dsrbuo of parameers gve he daa s kow as he poseror dsrbuo see Appedx 9A. To calculae hs codoal dsrbuo we use sadard resuls from mulvarae aalss see Appedx B. Specfcall he poseror dsrbuo of α β gve s ormal. The codoal momes are α µ α + DZ V + XΣ βx Zµ α Xµ β E 4.7 β µ β + ΣβX V + XΣ βx Zµ α Xµ β ad D 0 DZ Var α 0 Σ V XΣ βx ZD XΣ β β β ΣβX +. 4.8 Up o hs po he reame of parameers α ad β has bee smmerc. I logudal daa applcaos oe pcall has more formao abou he global parameers β ha subjecspecfc parameers α. To see how he poseror dsrbuo chages depedg o he amou of formao avalable we cosder wo exreme cases. Frs cosder he case Σ β 0 so ha β

Chaper 4. Predco ad Baesa Iferece / 4-9 µ β wh probabl oe. Iuvel hs meas ha β s precsel kow geerall from collaeral formao. The from equaos 4.7 ad 4.8 we have E α µ α + D Z V - Z µ α X β ad Var α D - D Z V - Z D. Assumg ha µ α 0 he bes lear ubased esmaor BLUE of E α s a BLUP D Z V - X b GLS Recall from equao 4. ha a BLUP s also he bes lear ubased predcor BLUP he freques o-baesa model framework. - Secod cosder he case where Σ β 0. I hs case pror formao abou he parameer β s vague; hs s kow as usg a dffuse pror. To aalze he mpac of hs assumpo use equao A.4 of Appedx A.5 o ge V + X Σ β X - V - - V - X X V - X + Σ - β - X V - V - - V - X X V - X - X V - Q V as Σ β - 0. Noe ha Q V X 0. Thus wh Σ β - 0 ad µ α 0 we have α N wh mea E α D Z Q V ad varace Var α D - D Z Q V Z D. Ths summarzes he poseror dsrbuo of α gve. Ieresgl from he expresso for Q V we have E α D Z V - - V - X X V - X - X V - D Z V - - D Z V - X b GLS a BLUP. Smlarl oe ca check ha E β b GLS as Σ β - 0. Thus s eresg ha boh exreme cases we arrve a he sasc a BLUP as a predcor of α. Ths aalss assumes D ad R are marces of fxed parameers. I s also possble o assume dsrbuos for hese parameers; pcall depede Wshar dsrbuos are used for D - ad R - as hese are cojugae prors. Appedx 9A roduces cojugae prors. Aleravel oe ca esmae D ad R usg mehods descrbed Seco 3.5. The geeral sraeg of subsug po esmaes for cera parameers a poseror dsrbuo s called emprcal Baes esmao. To exame ermedae cases we look o he followg specal case. Geeralzaos ma be foud Luo Youg ad Frees 00O. Specal case Oe-wa radom effecs ANOVA model We reur o he model cosdered Seco 4. ad for smplc assume balaced daa so ha T T. The goal s o deerme he poseror dsrbuos of he parameers. For llusrave purposes we derve he poseror meas ad leave he dervao of poseror varaces as a exercse for he reader. Thus wh equao 4. he model s β + α + ε where we use he radom β ~N µ β σ β leu of he fxed mea µ. The pror dsrbuo of α s depede wh α ~N 0σ α. Usg equao 4.7 he poseror mea of β s

4-0 / Chaper 4. Predco ad Baesa Iferece βˆ E β + Σ X V + XΣ X X µ β β β µ β T + σ β σ ε + Tσ α T σ ε + Tσ α µ + σ β β afer some algebra. Thus βˆ s a weghed average of he sample mea ad he pror mea µ β. I s eas o see ha βˆ approaches he sample mea as σ β ha s as pror formao abou β becomes vague. Coversel βˆ approaches he pror mea µ β as σ β 0 ha s as formao abou β becomes precse. Smlarl usg equao 4.7 he poseror mea of α s ˆ α E α ζ µ ζ where we recall ha ε α α Tσ ζ ad defeζ σ + Tσ β β β µ β ε α β Tσ. Noe ha ζ β measures σ + Tσ + Tσ he precso of kowledge abou β. Specfcall we see ha ζ β approaches oe as σ β ad approaches zero as σ β 0. Combg hese wo resuls we have ha ˆ α + ˆ β ζ ζ µ + ζ + ζ ζ ζ. β β β β + Thus f our kowledge of he dsrbuo of β s vague he ζ β ad he predcor reduces o he expresso equao 4.4 for balaced daa. Coversel f our kowledge of he dsrbuo of β s precse he ζ β 0 ad he predcor reduces o he expresso gve a he ed of Seco 4.. Wh he Baesa formulao we ma eera suaos where kowledge s avalable alhough mprecse. To summarze here are several advaages of he Baesa approach. Frs oe ca descrbe he ere dsrbuo of parameers codoal o he daa such as hrough equaos 4.7 ad 4.8. Ths allows oe for example o provde probabl saemes regardg he lkelhood of parameers. Secod hs approach allows aalss o bled formao kow from oher sources wh he daa a cohere maer. I our developme we assumed ha formao ma be kow hrough he vecor of β parameers wh her relabl corol hrough he dsperso marx Σ β. Values of Σ β 0 dcae complee fah values of µ β whereas values of Σ β - 0 dcae complee relace o he daa leu of pror kowledge. Thrd he Baesa approach provdes for a ufed approach for esmag α β. Chaper 3 o o-baesa mehods requred a separae seco o varace compoes esmao. I coras Baesa mehods all parameers ca be reaed a smlar fasho. Ths s covee for explag resuls o cosumers of he daa aalss. Fourh Baesa aalss s parcularl useful for forecasg fuure resposes; we develop hs aspec Chaper 0. Seco 4.6 Credbl heor Credbl s a echque for prcg surace coverages ha s wdel used b healh group erm lfe ad proper ad casual acuares. I he Ued Saes he praccal sadards of applcao are descrbed uder he Acuaral Sadard of Pracce Number 5 publshed b he Acuaral Sadards Board of he Amerca Academ of Acuares web se: hp://www.acuar.org/. Furher several surace laws ad regulao requre he use of credbl.

Chaper 4. Predco ad Baesa Iferece / 4- The heor of credbl has bee called a corersoe of he feld of acuaral scece Hckma ad Heacox 999O. The basc dea s o use clams experece ad addoal formao o develop a prcg formula hrough he relao New Premum ζ Clams Experece + ζ Old Premum. 4. Here ζ s kow as he credbl facor; values geerall le bewee zero ad oe. The case ζ s kow as full credbl where clams experece s used solel o deerme he premum. The case ζ 0 ca be hough of as o credbl where clams experece s gored ad exeral formao s used as he sole bass for prcg. Credbl has log foud use pracce wh applcaos dag back o Mowbra 94. See Hckma ad Heacox 999O ad Veer 996O for hsorcal accous. The moder heor of credbl bega wh he work of Bühlma 967O who showed how o express equao 4. wha we ow call a radom effecs framework hus removg he seemgl ad hoc aure of hs procedure. Bühlma expressed radoal credbl surace prces as codoal expecaos where he codog s based o a uobserved rsk pe ha he called a srucure varable. Applcaos of credbl heor are cosderabl ehaced b accoug for kow rsk facors such as reds hrough couous explaaor varables dffere rsk classes hrough caegorcal explaaor varables ad damc behavor hrough evolvg dsrbuos. These pes of applcaos ca be hadled uder he framework of mxed lear models; see Norberg 986O ad Frees Youg ad Luo 999O. Ths seco shows ha hs class of models coas he sadard credbl models as a specal case. B demosrag ha ma mpora credbl models ca be vewed a logudal daa framework we resrc our cosderao o cera pes of credbl models. Specfcall he logudal daa models accommodae ol uobserved rsks ha are addve. Thus we do o address models of olear radom effecs ha have bee vesgaed he acuaral leraure; see for example Talor 977O ad Norberg 980. Talor 977O allowed surace clams o be possbl fe dmesoal usg Hlber space heor ad esablshed credbl formulas hs geeral coex. Norberg 980O cosdered he more cocree coex e sll geeral of mulvarae clams ad esablshed he relaoshp bewee credbl ad sascal emprcal Baes esmao. B expressg credbl raemakg applcaos he framework of logudal daa models acuares ca realze several beefs: Logudal daa models provde a wde vare of models from whch o choose. Sadard sascal sofware makes aalzg daa relavel eas. Acuares have aoher mehod for explag he raemakg process. Acuares ca use graphcal ad dagosc ools o selec a model ad assess s usefuless. 4.6. Credbl heor models I hs subseco we demosrae ha commol used credbl echques are specal cases of bes lear ubased predco appled o he logudal/pael daa model. For addoal examples see Frees Youg ad Luo 00O. Specal Case - Basc credbl model of Bühlma 967O Bühlma 967O cosdered he oe-wa radom effecs ANOVA model ha we ow wre as β + α + ε see Seco 4.. If represes he clams of he h subjec perod he β s he grad mea of he clams over he colleco of subjecs polcholders geographcal regos occupaoal classes ec. ad α s he devao of he h subjec s hpohecal mea from he overall mea β. Here he hpohecal mea s he codoal expeced value E α

4- / Chaper 4. Predco ad Baesa Iferece β + α. The dsurbace erm ε s he devao of from s hpohecal mea. Oe calls σ α he varace of he hpohecal meas ad σ he process varace. Specal case - Heeroscedasc model of Bühlma-Sraub 970O Coue wh he basc Bühlma model ad chage ol he varace-covarace marx of he errors o R Var dagoal... ε σ. B hs chage we allow each w wt observao o have a dffere exposure wegh Bühlma ad Sraub 970O. For example f a subjec s a polcholder he w measures he sze of he h polcholder s exposure durg he h perod possbl va paroll as for workers compesao surace. Specal case - Regresso model of Hachemeser 975O Now assume a radom coeffces model so ha x z. The wh R as he Bühlma-Sraub model we have he regresso model of Hachemeser 975O. Hachemeser focused o he lear red model for whch K q x z. Specal case - Nesed classfcao model of Jewell 975O Suppose j β + µ + γ j + ε j a sum of ucorrelaed compoes whch β s he overall expeced clams µ s he devao of he codoal expeced clams of he h secor from β γ j s he devao of he codoal expeced clams of he jh subjec he h secor from he secor expecao of β + µ j ad ε j s he devao of he observao j from β + µ + γ j T j Jewell 975O. The codoal expeced clams from he h secor s E j µ β + µ ad he codoal expeced clams of subjec j wh he h secor s E j µ γ j β + µ + γ j. If oe were o appl hs model o prvae passeger auomoble surace for example he he secor mgh be age of he sureds whle he subjec s geographcal rego of he sureds. Noe ha oe assumes wh hs model ha he clams rego j for dffere ages are ucorrelaed. If oe beleves hs o be a ureasoable assumpo he a cross classfcao model mgh be approprae; see Daeburg Kaas ad Goovaers 996O. As a example for whch hs esed model mgh be more approprae oe could le he secor be geographcal rego whle he subjec s he polcholder. 4.6. Credbl raemakg I credbl raemakg oe s eresed predcg he expeced clams codoal o he radom rsk parameers of he h subjec for me perod T +. I our oao he credbl raemakg problem s o esmae E T T T. + α x β z α + + + From Seco 4.3. he BLUP predcor of clams s x T bgls z T a BLUP. + + + I Seco 4. we saw how o predc clams for he Bühlma model. We ow llusrae hs predco for oher famlar credbl models. A summar s Table 4.5. Specal case - Heeroscedasc model of Bühlma-Sraub 970O - Coued I he Bühlma-Sraub case we have x z so ha he predcor of clams s b GLS + a BLUP. Sraghforward calculaos smlar o he Bühlma case show ha he predcor of clams s ζ α + ζ m GLS w

Chaper 4. Predco ad Baesa Iferece / 4-3 where ow he credbl facor s ζ T w T w + σ σ α see Table 4.5. Specal case - Regresso model of Hachemeser 975O - Coued I he Hachemeser case we have x z. Defe b X V X XV o be he GLS esmaor of α + β based ol o he h subjec. I Exercse 4.3 we ask he reader o show ha he BLUP credbl esmaor of α + β s a + b I ζ b + ζ b BLUP GLS GLS whch ζ DX V X s he credbl facor. As he Bühlma case aga we see ha he credbl esmaor s a weghed average of a subjec-specfc sasc ad a sasc ha summarzes formao from all subjecs. Ths example s prome credbl heor because oe ca furher express he GLS esmaor of β as a weghed average of he b usg he credbl facors as weghs b ζ ζ b. GLS I Table 4.5 we show how o predc expeced clams he oher examples ha we cosdered Seco 4.6.. I each case he predced clams for he h subjec s a weghed average of ha subjec s experece wh he b GLS usg he h credbl facor as a wegh.

4-4 / Chaper 4. Predco ad Baesa Iferece Bühlma m Table 4.5 Credbl Facors ad Predco of Clams Noao Credbl Facors Predco of Clams T T α GLS Bühlma-Sraub m w T T ζ ζ w w w α GLS ζ ζ Hachemeser Lear red W T T w w T T w w ζ ζ ζ T T T For subjec + σ ζ mα GLS + ζ w T σ α w + σ σ α de DW I + σ DW 4 de DW + σ race DW + σ For subjec ζ α + ζ m GLS w For perod T + T + b GLS + ablup where b GLS + a BLUP ζ bgls T w + ζ W T w Jewell m A α GLS Tj σ γ w Tj σ γ w j Tj wj Tj w j j ζ r j ζ j w j j + σ j ζ j j w ζ j σµ A ζ σ A + σ ζ j σ ad µ γ γ γ Tj σ Tj w j w j + σ For secor ζ α + ζ m GLS w For subjec j secor ζ ζ j m α GLS + ζ ζ + ζ j w j j w

Chaper 4. Predco ad Baesa Iferece / 4-5 Furher Readg For readers who would lke more backgroud small area esmao please refer o Ghosh ad Rao 994S. For readers who would lke more backgroud credbl heor please refer o Daeburg Kaas ad Goovaers 996O Klugma Pajer ad Wllmo 998O ad Veer 996O. The Seco 4.6 roduco o credbl heor does o clude he mpora coeco o Baesa ferece ha was frs poed ou b Bale 950O. See for example Klugma 99O ad Pque 997O. For coecos wh credbl ad he Chaper 8 Kalma fler see Klugma 99O ad Ledoler Klugma ad Lee 99O. Baesa ferece s furher descrbed Chaper 0.

4-6 / Chaper 4. Predco ad Baesa Iferece Appedx 4A. Lear Ubased Predco Appedx 4A. Mmum Mea Square Predcor Le c be a arbrar cosa ad c be a vecor of cosas. For hs choce of c ad c he mea square error usg c + c o predc w s MSEc c Ec + c - w Varc + c - w + Ec + c E E w. Usg E X β ad E w λ β we have MSE c c c + c X λ β c + c X λ β. c c Equag hs o zero elds c * c c λ - c X β. For hs choce of c we have MSE c c c E c E w E w w c Vc + w Cov w c Var c σ. To fd he bes choce of c we have MSE c c c Vc Cov w. c Seg hs equal o zero elds c * V - Covw. Thus he mmum mea square predcor s c * + c * λ - Covw V - X β + Covw V - as requred. Appedx 4A. Bes Lear Ubased Predcor To check ha w BLUP equao 4.7 s he bes lear ubased predcor cosder all oher ubased lear esmaors of he form w BLUP + c where c s a vecor of cosas. B he ubasedess we have ha E c E w - E w BLUP 0. Thus c s a ubased esmaor of 0. Followg Harvlle 976S we requre hs of all possble dsrbuos so ha a ecessar ad suffce codo for E c 0 s c X 0. We wsh o mmze he mea square predco error over all choces of c so cosder Ew BLUP + c w. Now Cov w w c Cov w c Cov w c BLUP BLUP Cov c + λ Cov w V X Cov b c Cov w c λ Cov w V X X V X X V Cov c Cov w c Cov V X X V X X c 0. Cov w V GLS Cov w c + λ w 4A. The las equal follows from c X 0. Thus we have Ew BLUP + c w Varw BLUP - w + Varc ha ca be mmzed b choosg c 0.

Chaper 4. Predco ad Baesa Iferece / 4-7 Appedx 4A.3 BLUP Varace Frs oe ha Cov Covw V - w Cov V - Covw - Cov w 0. The we have Var w w Var λ Cov w V X b + Cov w V w BLUP GLS λ Cov w V X bgls + Var Cov w V w Var. Also we have Var Cov w V w Var Cov w V + Var w Cov Cov w V w w Cov w V Cov w Cov w V Cov w + Var σ Cov w V Cov w. w Thus Var w BLUP w λ Cov w V X X V X λ Cov w V Cov w V Cov w + σ w X as equao 4.8. From equao 4A. we have Covw BLUP - w w BLUP 0. Thus σ w Varw - w BLUP + w BLUP Varw - w BLUP + Var w BLUP. Wh equao 4A. we have Var w BLUP σ w - Varw - w BLUP Cov w V Cov w λ Cov w V whch s equao 4.9. X X V X λ Cov w V X

4-8 / Chaper 4. Predco ad Baesa Iferece 4. Exercses ad Exesos Seco 4. 4.. Shrkage esmaor Cosder he Seco 4. oe-wa radom effecs model wh K so ha µ +α + ε. a. Show ha E c + α s mmzed over choces of c a Cov α c. Var σ T b. Show ha Var σ α + Cov α σ α Cov α σ α T N σ + Tσ Cov α σ σ α ad Var + T j. N N N c. Use par b o show ha Cov α σ. N T d. Use par b o show ha + T Var + σ σ α T j. T N N N j σ e. Use pars a c ad d o show ha he opmal choce of c s c ad * σ + T * + T T σ j α * N N j c c where * T. σ + T σ α T N f. Use par e o show ha for balaced daa wh T T we have T *T. g. Use par e o show ha as N we have T * T. j T σ α Seco 4.3 4.. BLUP predcor of radom effecs error compoes model Cosder he Seco 4. oe-wa radom ANOVA model. Use equao 4. o show ha he BLUP predcor of α s a ζ - x b. BLUP GLS 4.3. BLUP predco radom coeffces model Cosder he radom coeffces model wh Kq ad z x. Use equao 4. o show ha he BLUP predcor of β + α s wblup ζ b + ζ b GLS where b X V X - X V s he subjec-specfc GLS esmaor ad ζ D X V - X. 4.4. BLUP resduals Use equaos 4. ad 4.3 o show a BLUP D Z R - e BLUP.

Chaper 4. Predco ad Baesa Iferece / 4-9 4.5. BLUP subjec-specfc effecs Use equaos 4. ad A.4 of Appedx A.5 o show a D + Z R Z Z R X b BLUP GLS. For hs alerave expresso oe eeds o ol ver R ad q q marces o a T T marx. 4.6.BLUP resduals Use equao 4. o show ha he BLUP resdual equao 4.3a ca be expressed as equao 4.3b ha s as e Z a + X b 4.7. Covarace BLUP BLUP GLS. Use equao 4. o show ha a b 0 Cov BLUP GLS. 4.8. BLUP forecass radom walk Assume a radom walk seral error correlao srucure Seco 8.. wll provde addoal movao. Specfcall suppose ha he subjec-level damcs are specfed hrough ε ε - + η. Here {η } s a..d. sequece wh Var η σ η ad assume ha {ε 0 } are uobserved cosas. The we have Var ε σ η ad Cov ε r ε s Var ε r r σ η for r < s. Ths elds R σ η R RW where L L R RW 3 L 3. M M M O M 3 L T a. Show ha 0 L 0 0 L 0 0 0 L 0 0 R. RW M M M O M M 0 0 0 L 0 0 0 L b. Show ha Cov ε T L L T + ε σ η. c. Deerme he sep forecas ha s deerme he BLUP predcor of T +. d. Deerme he L sep forecas ha s deerme he BLUP predcor of T + L. 4.9. BLUP mea square errors lear mxed effecs model Cosder he lear mxed effecs model roduced Seco 4.3. a. Use he geeral expresso for he BLUP mea square error o show ha he mea square error for he lear mxed effecs model ca be expressed as:

/ Chaper 4. Predco ad Baesa Iferece 4-30 BLUP w w w w Cov Cov Var X V λ X V X X V λ + w w w Cov Cov σ V. b. Use he geeral expresso for he BLUP varace o show ha he varace for he lear mxed effecs model ca be expressed as: Var w BLUP w w Cov Cov V w w Cov Cov X V λ X V X X V λ. c. Now suppose ha he BLUP of eres s a lear combao of global parameers ad subjec-specfc effecs of he form w c α + c β. Use par a o show ha he mea square error s BLUP w w X DZ V c c X X V X DZ V c c Var Dc c Z Dc DZ V c +. d. Use drec calculaos o show ha he varace of he BLUP of α s D Z V X X V X X I V DZ a BLUP Var. e. Use par b o esablsh he form of he varace of he BLUP resdual Seco 4.3.3. f. Use par a o esablsh he varace of he forecas error equao 4.5. 4.0. Hederso s mxed lear model jusfcao of BLUPs Cosder he model equao 3.8 Seco 3.3. Z α + X β + ε. I addo assume ha α ε are jol mulvarae ormall dsrbued such ha R Xβ Zα α ~ + N ad D 0 α ~ N. a. Show ha he jo logarhmc probabl des fuco of α s + + + + l de l Xβ Zα R Xβ Zα R α π N l α D α D l de l + + π q. b. Trea hs as a fuco of α ad β. Take paral dervaves wh respec o α β o eld Hederso s 984B mxed model equaos Z R α D Z Z R Xβ Z R X R Zα X R Xβ X R + + +. c. Show ha solvg Hederso s mxed model equaos for ukows α β elds

Chaper 4. Predco ad Baesa Iferece / 4-3 GLS X V X a BLUP DZ V H: Use equao A.4 of Appedx A.5. b X V Xb Emprcal Exercses 4.. Housg Prces refer o Exercse.9 for he problem descrpo. Here we wll calculae 95% predco ervals for Chcago he h meropola area. Below are he 9 aual values of NARSP PERPYC ad PERPOP for Chcago. GLS. a Assume ha ou have f a oe-wa fxed effecs model: NARSP α + β PERPYC + β PERPOP + β 3 YEAR + ε You have f he model usg leas squares ad arrved a he esmaes b -0.008565 b - 0.004347 b 3 0.036750 α 0.85 ad σ e 0.0738. Assume ha ex ear s 995 values for he explaaor varables are PERPYC 0 3.0 ad PERPOP 0 0.0. Calculae a 95% predco erval for Chcago s 995 average sale prce. Whe expressg our fal aswer cover o dollars leu of logarhmc dollars. b. Assume ha ou have f a error compoes model ha ou have esmaed usg geeralzed leas squares. You have f he model usg geeralzed leas squares ad arrved a he esmaes b -0.0 b -0.004 b 3 0.0367 σ α 0.0 ad σ e 0.005. Assume ha ex ear s 995 values for he explaaor varables are PERPYC 0 3.0 ad PERPOP 0 0.0. Calculae a 95% predco erval for Chcago s 995 average sale prce. Whe expressg our fal aswer cover o dollars leu of logarhmc dollars. c Assume ha ou have f a error compoes model wh a AR auocorrelao srucure ha ou have esmaed usg geeralzed leas squares. You have f he model usg geeralzed leas squares ad arrved a he esmaes b -0.0 b -0.004 b 3 0.0367 ρ 0. σ α 0.0 ad σ e 0.005. Assume ha ex ear s 995 values for he explaaor varables PERPYC 0 3.0 ad PERPOP 0 0.0. Calculae a 95% predco erval for Chcago s 995 average sale prce. Whe expressg our fal aswer cover o dollars leu of logarhmc dollars. YEAR NARSP PERYPC PERPOP 3 4 5 6 7 4.4555 4.50866 4.48864 4.6783 4.76046 4.87596 4.985 5.8387 5.5969 7.8083 7.7689 5.90655.074-0.735 0.983 0.347 0.3056 0.33683 0.47377 0.99697-0.77503 8 4.95583 3.8004 0.976 9 4.96564 3.667 0.973

4-3 / Chaper 4. Predco ad Baesa Iferece 4.. Workers Compesao Usrucured problem Cosder a example from workers compesao surace examg losses due o permae paral dsabl clams. The daa are from Klugma 99O who cosders Baesa model represeaos ad are orgall from he Naoal Coucl o Compesao Isurace. We cosder occupao or rsk classes over T 7 ears. To proec he daa sources furher formao o he occupao classes ad ears are o avalable. The respose varable of eres s he pure premum PP defed o be losses due o permae paral dsabl per dollar of PAYROLL. The varable PP s of eres o acuares because worker compesao raes are deermed ad quoed per u of paroll. The exposure measure PAYROLL s oe of he poeal explaaor varables. Oher explaaor varables are YEAR... 7 ad occupao class. For hs exercse develop a radom effecs model. Use hs model o provde forecass of he codoal mea of pure premum kow as credbl esmaes he acuaral leraure. For oe approach see Frees e al. 00O. 4.3. Group Term Lfe Isurace Usrucured problem We ow cosder clams daa provded b a Wscos-based cred surer. The daa coas clams ad exposure formao for 88 Florda cred uos. These are lfe savgs clams from a corac bewee he cred uo ad her members ha provdes a deah beef based o he member s savgs deposed he cred uo. The depede varable s LN_LSTC l + LSTC/000 where LSTC s he aual oal clams from he lfe savgs corac. The exposure measure s LN_LSCV l + LSCV/000000 where LSCV s he aual coverage for he lfe savgs corac. Also avalable s he corac YEAR. For hs exercse develop a radom effecs model. Use hs model o provde forecass of he codoal mea of pure premum kow as credbl esmaes he acuaral leraure. For oe approach see Frees e al. 00O.

Chaper 5. Mullevel Models / 5- Chaper 5. Mullevel Models 003 b Edward W. Frees. All rghs reserved Absrac. Ths chaper descrbes a codoal modelg framework ha akes o accou herarchcal ad clusered daa srucures. The daa ad models kow as mullevel are used exesvel educaoal scece ad relaed dscples he socal ad behavoral sceces. We show ha a mullevel model ca be vewed as a lear mxed effecs model ad hece he sascal ferece echques roduced Chaper 3 are readl applcable. B cosderg mullevel daa ad models as a separae u we expad he breadh of applcaos ha lear mxed effecs models ejo. 5. Cross-secoal mullevel models Educaoal ssems are ofe descrbed b srucures whch he us of observao a oe level are grouped wh us a a hgher level of srucure. To llusrae suppose ha we are eresed assessg sude performace based o a acheveme es. Sudes are grouped classes classes are grouped schools ad schools are grouped o dsrcs. A each level here are varables ha ma affec resposes from a sude. For example a he class level educao of he eacher ma be mpora a he school level he school sze ma be mpora ad a he dsrc level fudg ma be mpora. Furher each level of groupg ma be of scefc eres. Fall here ma be o ol relaoshps amog varables wh each group bu also across groups ha should be cosdered. The erm mullevel s used for hs esed daa srucure. I he above suao we cosder sudes o be he basc u of observao; he are kow as he level- us of observao. The ex level up s called level- classes hs example ad so forh. We ca mage mullevel daa beg colleced b a cluser samplg scheme. A radom sample of dsrcs s defed. For each dsrc seleced a radom sample of schools s chose. From each school a radom sample of classes s ake ad from each class seleced a radom sample of sudes. Mechasms oher ha radom samplg ma be used ad hs wll fluece he model seleced o represe he daa. Mullevel models are specfed hrough codoal relaoshps where he relaoshps descrbed a oe level are codoal o geerall uobserved radom coeffces of upper levels. Because of hs codoal modelg framework mullevel daa ad models are also kow as herarchcal. 5.. Two-level models To llusrae he mpora feaures of he model all cosder ol wo levels. Suppose ha we have a sample of schools ad for he h school we radoml selec sudes omg class for he mome. For he jh sude he h school we assess he sude s performace o a acheveme es j ad formao o he sude s socoecoomc saus z j for example he oal faml come. To assess acheveme erms of soco-ecoomc saus we could beg wh a smple model of he form j β 0 + β z j + ε j. 5. Equao 5. descrbes a lear relao bewee soco-ecoomc saus ad expeced performace alhough we allow he lear relaoshp o var b school hrough he oao β 0

5- / Chaper 5. Mullevel Models ad β for school-specfc erceps ad slopes. Equao 5. summarzes he level- model ha cocers sude performace as he u of observao. If we have defed a se of schools ha are of eres he we ma smpl hk of he quaes {β 0 β } as fxed parameers of eres. However educaoal research s cusomar o cosder hese schools o be a sample from a larger populao; he eres s makg saemes abou hs larger populao. Thkg of he schools as a radom sample we model {β 0 β } as radom quaes. A smple represeao for hese quaes s: β 0 β 0 + α 0 ad β β + α 5. where α 0 α are mea zero radom varables. Dspla 5. represes a relaoshp abou he schools ad summarzes he level- model. Dsplas 5. ad 5. descrbe models a wo levels. For esmao we combe 5. ad 5. o eld j β 0 + α 0 + β + α z j + ε j α 0 + α z j + β 0 + β z j + ε j. 5.3 Equao 5.3 shows ha he wo-level model ma be wre as a sgle lear mxed effecs model. Specfcall we defe α α 0 α z j z j β β 0 β ad x j z j o wre j z j α + x j β + ε j smlar o equao 3.5. Because we ca wre he combed mullevel model as a lear mxed effecs model we ca use he Chaper 3 echques o esmae he model parameers. Noe ha we are ow usg he subscrp j o deoe replcaos wh a sraum such as a school. Ths s because we erpre he replcao o have o me orderg; geerall we wll assume o correlao amog replcaos codoal o he subjec. Seco 5. wll re-roduce he subscrp whe we cosder me-ordered repeaed measuremes. Oe desrable aspec of he mullevel model formulao s ha we ma modf codoal relaoshps a each level of he model depedg o he research eress of he sud. To llusrae we ma wsh o udersad how characerscs of he school affec sude performace. For example Raudebush ad Brk 00EP dscussed a example where x dcaes wheher he school was a Caholc based or a publc school. A smple wa o roduce hs formao s o modf he level- model dspla 5. o β 0 β 0 + β 0 x + α 0 ad β β + β x + α. 5.a There are wo level- regresso models dspla 5.a; aalss fd uvel appealg o specf regresso relaoshps ha capure addoal model varabl. Noe however ha for each model he lef-had sde quaes are o observed. To emphasze hs Raudebush ad Brk 00EP call hese models erceps-as-oucomes ad slopes-as-oucomes. I Seco 5.3 we wll lear how o predc hese quaes. Combg dspla 5.a wh he level- model equao 5. we have j β 0 + β 0 x + α 0 + β + β x + α z j + ε j α 0 + α z j + β 0 + β 0 x + β z j + β x z j + ε j. 5.4 B defg α α 0 α z j z j β β 0 β 0 β β ad x j x z j x z j we ma aga express hs mullevel model as a sgle lear mxed effecs model. The erm β x z j eracg bewee he level- varable z j ad he level- varable x s kow as a cross-level eraco. For hs example suppose ha we use x for Caholc schools ad x 0 for publc schools. The β represes he dfferece bewee he margal chage acheveme scores per u of faml come bewee Caholc ad publc schools.

Chaper 5. Mullevel Models / 5-3 Ma researchers see for example Raudebush ad Brk 00EP argue ha udersadg cross-level eracos s a major movao for aalzg mullevel daa. Ceerg of varables I s cusomar educaoal scece o ceer explaaor varables order o ehace he erpreabl of model coeffces. To llusrae cosder he herarchcal models 5. 5.a ad 5.4. Usg he aural merc for z j we erpre β 0 o be he mea codoal o he h subjec respose whe z 0. I ma applcaos such as where z represes oal come or es scores a value of zero falls ousde a meagful rage of values. Oe possbl s o ceer level- explaaor varables abou her overall mea ad use z j z as a explaaor varable equao 5.. I hs case we ma erpre he ercep β 0 o be he expeced respose for a dvdual wh a score equal o he grad mea. Ths ca be erpreed as a adjused mea for he h group. Aoher possbl s o ceer each level- explaaor varable abou s level- mea ad use zj z as a explaaor varable equao 5.. I hs case we ma erpre he ercep β 0 o be he expeced respose for a dvdual wh a score equal o he mea of he h group. For logudal applcaos ou ma wsh o ceer he level- explaaor varables so ha he ercep equals he expeced radom coeffce a a specfc po me for example a he sar of a rag program see for example Kref ad deleeuw 998. Exeded wo-level models To cosder ma explaaor varables we exed equaos 5. ad 5.. Cosder a level- model of he form j z j β + x j β + ε j. 5.5 Here z j ad x j represe he se of level- varables assocaed wh varg over level- ad fxed coeffces respecvel. The level- model s of he form: β X β + α 5.6 where E α 0.Wh hs oao he erm X β forms aoher se of effecs wh parameers o be esmaed. Aleravel we could wre equao 5.5 whou explcl recogzg he fxed coeffces β b cludg hem he radom coeffces equao 5.6 bu wh zero varace. However we prefer o recogze her presece explcl because hs helps raslag equaos 5.5 ad 5.6 o compuer sascal roues for mplemeao. Combg equaos 5.5 ad 5.6 elds j z j X β + α + x j β + ε j z j α +x j β + ε j 5.7 wh he oao x j x j z j X z j z j ad β β β. Aga equao 5.7 expresses hs mullevel model our usual lear mxed effecs model form. I wll be helpful o cosder a umber of specal cases of equaos 5.5-5.7. To beg suppose ha β s a scalar ad ha z j. The he model equao 5.7 reduces o he error compoes model roduced Seco 3.. Raudebush ad Brk 00EP furher dscuss he specal case where equao 5.5 does o coa he fxed effecs x j β poro. I hs case equao 5.7 reduces o j α + X β + ε j

5-4 / Chaper 5. Mullevel Models ha Raudebush ad Brk refer o as he meas-as-oucomes model. Ths model wh ol level- explaaor varables avalable ca be used o predc he meas or expeced values of each group. We wll sud hs predco problem formall Seco 5.3. Aoher specal case of equaos 5.5-5.7 s he radom coeffces model. Here we om he level- fxed effecs poro x j β ad use he de marx for X. The equao 5.7 reduces o j z j β + α + ε j. Example As repored Lee 000EP Lee ad Smh 997EP suded 98 Grade sudes 99 who aeded 789 publc Caholc ad ele prvae hgh schools draw from a aoall represeave sample from he Naoal Educao Logudal Sud. The resposes were acheveme gas readg ad mahemacs over four ears of hgh school. The ma varable of eres was a school level varable sze of he hgh school. Educaoal research had emphaszed ha larger schools ejo ecoomes of scale ad are able o offer a broader currculum whereas smaller schools offer more posve socal evromes as well as a more homogeous currculum. Lee ad Smh sough o vesgae he opmal school sze. To corol for addoal sude level effecs level- explaaor varables cluded geder mor saus abl ad soco-ecoomc saus. To corol for addoal school level characerscs level- explaaor varables cluded school average mor cocerao school average socoecoomc saus ad pe of school Caholc publc ad ele prvae. Lee ad Smh foud ha a mddle school sze of approxmael 600-900 sudes produced he bes acheveme resuls. Movao for mullevel models As we have see mullevel models allow aalss o assess he mporace of cross-level effecs. Specfcall he mullevel approach allows ad/or forces researchers o hpohesze relaoshps a each level of aalss. Ma dffere us of aalss wh he same problem are possble hus permg modelg of complex ssems. The abl o esmae cross-level effecs s oe advaage of mullevel modelg whe compared o a alerae research sraeg callg for he aalss of each level solao of he ohers. As descrbed he roducor Chaper mullevel models allow aalss o address problems of heerogee wh samples of repeaed measuremes. Wh he educaoal research leraure o accoug for heerogee from dvduals s kow as aggregao bas; see for example Raudebush ad Brk 00EP. Eve f he eres s udersadg level- relaoshps we wll ge a beer pcure b corporag a level- model of dvdual effecs. Moreover mullevel modelg allows us o predc quaes a boh level- ad level-; Seco 5.3 descrbes hs predco problem. Secod ad hgher levels of mullevel models also provde us wh a opporu o esmae he varace srucure usg a parsmoous paramerc srucure. Improved esmao of he varace srucure provdes a beer udersadg of he ere model ad wll ofe resul mproved precso of our usual regresso coeffce esmaors. Moreover as dscussed above ofe hese relaoshps a he secod ad hgher levels are of heorecal eres ad ma represe he ma focus of he sud. However echcal dffcules arse whe esg cera hpoheses abou varace compoes. These dffcules ad soluos are preseed Seco 5.4.

Chaper 5. Mullevel Models / 5-5 5.. Mulple level models Exesos o more ha wo levels follow he same paer as wo level models. To be explc we gve a hree-level model based o a example from Raudebush ad Brk 00EP. Cosder modelg a sude s acheveme as he respose. The level- model s jk z jk β j + x jk β + ε jk 5.8 where here are schools j J classrooms he h school ad k K j sudes he jh classroom wh he h school. The explaaor varables z jk ad x jk ma deped o he sude geder faml come ad so o classroom eacher characerscs classroom facles ad so o or school orgazao srucure locao ad so o. The parameers ha deped o eher school or classroom j appear as par of he β j vecor whereas parameers ha are cosa appear he β vecor. Ths depedece s made explc he hgher-level model formulao. Codoal o he classroom ad school he dsurbace erm ε jk s mea zero ad has a varace ha s cosa over all sudes classrooms ad schools. The level- model descrbes he varabl a he classroom level. The level- model s of he form β j Z j γ + X j β + ε j. 5.9 Aalogous o level- he explaaor varables Z j ad X j ma deped o he classroom or school bu o he sude. The parameers assocaed wh he Z j explaaor varables γ ma deped o school whereas he parameers assocaed wh he X j explaaor varables are cosa. Codoal o he school he dsurbace erm ε j s mea zero ad has a varace ha s cosa over classrooms ad schools. The level- parameers β j ma be varg bu osochasc or sochasc. Wh hs oao we use a zero varace o model parameers ha are varg bu osochasc. The level-3 model descrbes he varabl a he school level. Aga he level- parameers γ ma be varg bu osochasc or sochasc. The level-3 model s of he form γ X 3 β 3 + ε 3. 5.0 Aga he explaaor varables X 3 ma deped o he school. Codoal o he school he dsurbace erm ε 3 s mea zero ad has a varace ha s cosa over schools. Pug equaos 5.8-5.0 ogeher we have jk z jk Z j X 3 β 3 + ε 3 + X j β + ε j + x jk β + ε jk x jk β + z jk X j β + z jk Z j X 3 β 3 + z jk Z j ε 3 + z jk ε j + ε jk x jk β + z jk α j + ε jk 5. where x jk x jk z jk X j z jk Z j X 3 β β β β 3 z jk z jk z jk Z j ad α j ε j ε 3. We have alread specfed he usual assumpo of homoscedasc for each radom qua ε jk ε j ad ε 3. Moreover s cusomar o assume ha hese quaes are ucorrelaed wh oe aoher. Our ma po s ha as wh he wo-level model equao 5. expresses he hree-level model as a lear mxed effecs model. Coverg he model equao 5. o he lear mxed effecs model equao 3.5 s a maer of defg vecor expressos carefull. Seco 5.3 provdes furher deals. Thus parameer esmao s a drec cosequece of our Chaper 3 resuls. Ma varaos of he basc assumpos ha we have descrbed are possble. I Seco 5. o logudal mullevel models we wll gve a more dealed descrpo of a example of a hree-level model. Appedx 5A exeds he dscusso o hgher order mullevel models.

5-6 / Chaper 5. Mullevel Models For applcaos several sascal sofware packages exs such as HLM MLwN ad MIXREG ha allow aalss o f mullevel models whou combg he several equaos o a sgle expresso such as equao 5.. However hese specalzed packages ma o have all of he feaures ha he aals wshes o dspla hs or her aalss. As poed ou b Sger 998EP a alerave or supplemear approach s o use a geeral purpose mxed lear effecs package such as SAS PROC MIXED ad rel drecl o he fudameal mxed lear model heor. 5..3 Mullevel modelg oher felds The feld of educaoal research has bee a area of acve developme of crosssecoal mullevel modelg alhough b o meas has a corer o he marke. Ths subseco descrbes examples where hese models have bee used oher felds of sud. Oe pe of sud ha s popular ecoomcs s daa based o a mached pars sample. For example we mgh selec a se of famles for level- sample ad for each faml observe he behavor of sblgs or ws. The dea uderlg hs desg s ha b observg more ha oe faml member we wll be able o corol for uobserved faml characerscs. See Wooldrdge 00E ad Exercse 3.0 for furher dscusso of hs desg. I surace ad acuaral scece s possble o model clams dsrbuos usg a herarchcal framework. Tpcall he level- u of aalss s based o a surace cusomer ad explaaor varables ma clude characerscs of he cusomer. The level- model uses clams amous as he respose pcall over me ad pcal me-varg explaaor varables clude me reds. For example Klugma 99O gves a Baesa perspecve of hs problem. For a freques perspecve see Frees Youg ad Luo 999O. 5. Logudal mullevel models Ths seco shows how o use he codoal modelg framework o represe logudal me-ordered daa. The ke chage he modelg se-up s ha we ow wll pcall cosder he dvdual as he level- u of aalss ad observaos a dffere me pos as he level- us. The goal s ow also subsaall dffere; pcall logudal sudes he assessme of chage s he ke research eres. As wh Seco 5. we beg wh he wo-level model ad he dscuss geeral mullevel exesos. 5.. Two-level models Followg he oao esablshed Seco 5. we cosder level- models of he form z β + x β + ε. 5. Ths s a model of T resposes over me for he h dvdual. The u of aalss for he level- model s a observao a a po me o he dvdual as Seco 5.. Thus we use he subscrp as a dex for me. Mos oher aspecs of he model are as Seco 5..; z ad x represe ses of level- explaaor varables. The assocaed parameers ha ma deped o he h dvdual appear as par of he β vecor whereas parameers ha are cosa appear he β vecor. Codoal o he subjec he dsurbace erm ε s mea zero radom varable ha s ucorrelaed wh β. A mpora feaure of he logudal mullevel model ha dsgushes from s cross-secoal couerpar s ha me geerall eers he level- specfcao. There are a umber of was ha hs ca happe. Oe wa s o le oe or more of he explaaor varables be a fuco of me. Ths s he approach hsorcall ake growh curve modelg descrbed below. Aoher approach s o le oe of he explaaor varables be a lagged respose varable.

Chaper 5. Mullevel Models / 5-7 Ths approach s parcularl prevale ecoomcs ad wll be furher explored Chaper 6. Ye aoher approach s o model he seral correlao hrough he varace covarace-marx of he vecor of dsurbace ε ε ε T. Specfcall Secos.5. ad 3.3. we developed he oao Var ε R o represe he seral covarace srucure. Ths approach s wdel adoped bosascs ad educaoal research ad wll be furher developed here. Lke he cross-secoal model he level- model ca be represeed as β X β + α ; see equao 5.6. Now however we erpre he uobserved β o be he radom coeffces assocaed wh he h dvdual. Thus alhough he mahemacal represeao s smlar o he cross-secoal seg our erpreaos of dvdual compoes of he model are que dffere. Ye as wh equao 5.7 we ma sll combe level- ad level- models o ge z X β + α + x β + ε z α + x β + ε 5.3 usg he oao x x z X z z ad β β β. Ths s he lear mxed effecs model roduced Seco 3.3.. Growh curve models To develop uo we ow cosder growh curve models models ha have a log hsor of applcaos. The dea behd growh curve models s ha we seek o moor he aural developme or agg of a dvdual. Ths developme s pcall moored whou erveo ad he goal s o assess dffereces amog groups. I growh curve modelg oe uses a polomal fuco of age or me o rack growh. Because growh curve daa ma reflec observaos from a developme process s uvel appealg o hk of he expeced respose as a fuco of me. Parameers of he fuco var b dvdual so ha oe ca summarze a dvdual s growh hrough he parameers. To llusrae we ow cosder a classc example. Example - Deal Daa Ths example s orgall due o Pohoff ad Ro 964B; see also Rao 987B. Here s he dsace measured mllmeers from he ceer of he puar o he peromaxllar fssure. Measuremes were ake o grls ad 6 bos a ages 8 0 ad 4. The eres s he relao bewee he dsace ad age specfcall how he dsace grows wh age ad wheher here s a dfferece bewee males ad females. Table 5. shows he daa ad Fgure 5. gves a graphcal mpresso of he growh over me. From Fgure 5. we ca see ha he measureme legh grows as each chld ages alhough s dffcul o deec dffereces bewee bos ad grls. I Fgure 5. we use ope crcular plog smbols for grls ad opaque plog smbols for bos. Fgure 5. does show ha he h bo has a uusual growh paer; hs paer ca also be see Table 5..

5-8 / Chaper 5. Mullevel Models Table 5. Deal measuremes of grls ad 6 bos. Measuremes are mllmeers. Grls Bos Age ears Age ears Number 8 0 4 8 0 4 0.5 3 6 5 9 3.5 4 5.5.5.5 3 6.5 3 0.5 4 4.5 6 3.5 4 7.5 4 3.5 4.5 5 6.5 5.5 7.5 6.5 7 5.5 3.5 3.5 0 3.5.5 6 6 0.5 4.5 5.5 7 8.5 7.5.5 3 5 4.5 6.5 8 3 3 3.5 4 4.5 4.5 5.5 9 0.5 3 0.5 3 6 0 6.5 9 9 9.5 7.5 8 3 3.5 4.5 5 8 8 3 3 3.5 5.5 3.5 4 8 3 7 4.5 6 9.5 4.5 5.5 5.5 6 5 3 4.5 6 30 6.5 3.5 5 Source: Pohoff ad Ro 964B Rao 987B Measure 3 30 8 6 4 0 8 6 8 0 4 Age Fgure 5. Mulple Tme Seres Plo of Deal Measuremes. Ope crcles represe grls; opaque crcles represe bos.

Chaper 5. Mullevel Models / 5-9 A level- model s β 0 + β z + ε where z s he age of he chld o occaso. Ths model relaes he deal measureme o he age of he chld wh parameers ha are specfc o he chld. Thus we ma erpre he qua β o be he growh rae for he h chld. A level- model s β 0 β 00 + β 0 GENDER + α 0 ad β β 0 + β GENDER + α. Here β 00 β 0 β 0 ad β are fxed parameers o be esmaed. Suppose ha we use a bar varable for geder sa codg he GENDER varable for females ad 0 for males. The we ma erpre β 0 o be he expeced male growh rae ad β o be he dfferece growh raes bewee females ad males. Table 5. shows he parameer esmaes for hs model. Here we see ha he coeffce assocaed wh lear growh s sascall sgfca over all models. Moreover he rae of crease for grls s lower ha bos. The esmaed covarace bewee α 0 ad α whch s also he esmaed covarace bewee β 0 ad β urs ou o be egave. Oe erpreao of he egave covarace bewee al saus ad growh rae s ha subjecs who sar a a low level ed o grow more quckl ha hose who sar a hgher levels ad vce versa. Table 5.. Deal daa growh curve model parameer esmaes Error Compoes Model Growh Curve Model Growh Curve Model deleg he 9 h bo Varable Parameer Esmaes - sasc Parameer Esmaes - sasc Parameer Esmaes - sasc β 00 6.34 6.65 6.34 6.04 6.470 5.4 Age β 0 0.784 0. 0.784 9. 0.77 8.57 GENDER β 0.03 0.67.03 0.65 0.903 0.55 AGE*GENDER -0.305 -.5-0.305 -.6-0.9 -. β Var ε.9.76 0.97 Var α 0 3.99 5.786.005 Var α 0.033 0.073 Cov α 0 α -0.90-0.734 - Log 433.8 43.6 388.5 Lkelhood AIC 445.8 448.6 404.5 For comparso purposes Table 5. shows he parameer esmaes wh he 9 h bo deleed. The effecs of hs subjec deleo o he parameer esmaes are small. Table 5. also shows parameer esmaes of he error compoes model. Ths model emplos he same level- model bu wh level- models β 0 β 00 + β 0 GENDER + α 0 ad β β 0 + β GENDER. Wh parameer esmaes calculaed usg he full daa se here aga s lle chage he parameer esmaes. Because he resuls appear o be robus o boh uusual subjecs ad model seleco we have greaer cofdece our erpreaos.

5-0 / Chaper 5. Mullevel Models 5.. Mulple level models Logudal versos of mulple level models follow he same oao as he crosssecoal models Seco 5.. excep ha he level- replcaos are over me. To llusrae we cosder a 3-level model he coex of a socal work applcao b Guo ad Husse 999EP. Guo ad Husse examed subjecve assessmes of chldre s behavor made b mulple raers a wo or more me pos. Tha s he level- repeaed measuremes are over me where he assessme was made b raer j o chld. Raers assessed 44 serousl emooall dsurbed chldre recevg servces hrough a large chld meal healh reame agec locaed Clevelad Oho. For hs sud he assessme s he respose of eres ; hs respose s he Deveroux Scale of Meal Dsorders a score made up of ems. Rags were ake over a wo-ear perod b pares ad eachers; a each me po assessmes ma be made eher b he pare eacher or boh. The me of he assessme was recorded as TIME j measured das sce he cepo of he sud. The varable PROGRAM j was recorded as a f he chld was program resdece a he me of he assessme ad 0 f he chld was da reame or da reame combed wh reame foser care. The varable RATER j was recorded as a f raer was a eacher ad 0 f he raer was a careaker. Aalogous o equao 5.8 he level- model s j z j β j + x j β + ε j 5.4 where here are chldre j J raers ad T j evaluaos. Specfcall Guo ad Husse 999EP used x j PROGRAM j ad z j TIME j. Thus her level- model ca be wre as j β 0j + β j TIME j + β PROGRAM j + ε j. The varables assocaed wh he ercep ad he coeffce for me ma var over chld ad raer whereas he program coeffce s cosa over all observaos. The level- model s he same as equao 5.9 β j Z j γ + X j β + ε j where here are chldre ad j J raers. The level- model of Guo ad Husse ca be wre as β 0j β 00 + β 00 RATER j + ε j ad β j β 0 + β RATER j. Aga we leave as a exercse for he reader o show how hs formulao s a specal case of equao 5.9. The level-3 model s he same as equao 5.0 γ X 3 β 3 + ε 3. To llusrae he level-3 model of Guo ad Husse ca be wre as β 00 β 000 + β 00 GENDER + ε 3. where GENDER s a bar varable dcag he geder of he chld. As wh he cross-secoal models Seco 5.. oe combes he hree levels o form a sgle equao represeao as equao 5.4. The herarchcal framework allows aalss o develop hpoheses ha are eresg o es. The combed model allows for smulaeous over all levels esmao of parameers ha s more effce ha esmag each level solao of he ohers.

Chaper 5. Mullevel Models / 5-5.3 Predco I Chaper 4 we dsgushed bewee he coceps of esmag model parameers as compared o predcg radom varables. I mullevel models he depede varables a secod ad hgher levels are uobserved radom coeffces. Because s ofe desrable o udersad her behavor we wsh o predc hese radom coeffces. To llusrae f he u of aalss a he secod level s a school we ma wsh o use predcos of secod level coeffces o rak schools. I ma also be of eres o use predcos of secod or hgher level coeffces for predco a frs level model. To llusrae f we are sudg a chld s developme over me we ma wsh o make predcos abou he fuure saus of a chld s developme. Ths subseco shows how o use he bes lear ubased predcors BLUPs developed Chaper 4 for hese predco problems. Bes lear ubased predcors b defo have he smalles varace amog all ubased predcors. I Chaper 4 we showed ha hese predcors ca also be erpreed as emprcal Baes esmaors. Moreover he ofe have desrable erpreaos as shrkage esmaors. Because we have expressed mullevel models erms of lear mxed effecs models we wll o eed o develop ew heor bu wll be able o rel drecl o he Chaper 4 resuls. Two-level models We beg our predco dscusso wh he wo-level model roduced equaos 5.5-5.7. To make he mullevel model oao cosse wh Chapers 3 ad 4 use D Var α Var β ad R Var ε where ε ε ε T. Suppose ha we wsh o predc β. Usg he resuls Seco 4.3. s eas o check ha he bes lear ubased predcor BLUP of β s b BLUP a BLUP + X b GLS where b GLS s he geeralzed leas squares esmaor of β ad from equao 4. a BLUP D Z V - - X b GLS. Recall ha z z so ha Z z z z J. Furher b GLS b GLS b GLS V R + Z D Z ad X x x x T where x x z X. Thus s eas o compue hese predcors. Chaper 4 dscussed erpreao some specal cases he error compoes ad he radom coeffces models. Suppose ha we have he error compoes model so ha z j z j ad R s a scalar mes he de marx. Furher suppose ha here are o level- explaaor varables. The oe ca check ha he BLUP of he codoal mea of he level- respose E α α + X β s a BLUP + X b GLS ζ - X b GLS + X b GLS ζ + - ζ X b GLS T where ζ. Thus he predcor s a weghed average of he level- h u s T + Varε /Varα average ad he regresso esmaor whch s a esmaor derved from all level- us. As oed Seco 5.. Raudebush ad Brk 00EP refer o hs as he meas-as-oucomes model. As descrbed Seco 4.3.4 oe ca also use he BLUP echolog o predc he fuure developme of a level- respose. From equao 4.4 we have ha he forecas L lead mes he fuure of T s ˆ T + L z T + Lb BLUP + x T + Lb GLS + Cov T + L ε R e BLUP ε

5- / Chaper 5. Mullevel Models where e.blup s he vecor of BLUP resduals gve equao 4.3a. As we saw Seco 4.3.4 he case where he dsurbaces follow a auoregressve model of order AR wh parameer ρ we have L ˆ z b + x + ρ e. T + L T + L BLUP T + L b GLS T BLUP To llusrae cosder he Seco 5.. Deal example. Here here s o seral correlao so ha R s a scalar mes he de marx o level- fxed parameers ad T 4 observaos for all chldre. Thus he L sep forecas for he h chld s ˆ b b z where z 4+L s he age of he chld a me 4+L. 4+ L 0 BLUP + BLUP 4+ L Mulple level models For hree ad hgher level models he approach s he same as wh wo-level models alhough becomes more dffcul o erpre he resuls. Noeheless for appled work he dea s sraghforward. Procedure for forecasg fuure level- resposes. Hpohesze a model a each level.. Combe all level models o a sgle model. 3. Esmae he parameers of he sgle model usg geeralzed leas squares ad varace compoes esmaors as descrbed Secos 3.4 ad 3.5 respecvel. 4. Deerme bes lear ubased predcors of each uobserved radom coeffce for levels wo ad hgher as descrbed Seco 4.3. 5. Use he parameer esmaors ad radom coeffce predcors o form forecass of fuure level- resposes. model. To llusrae le s see how hs procedure works for he hree-level logudal daa Sep. We wll use he level- model descrbed equao 5.4 ogeher wh he level- ad level-3 models equaos 5.9 ad 5.0 respecvel. For he level- model le R j Var ε j where ε j ε j ε jtj. Sep. The combed model s equao 5. excep ha ow we use a subscrp for me leu of he k subscrp. Assumg he level- ad 3 radom quaes are ucorrelaed wh oe aoher we defe ε j Var ε j 0 D 0 Var α j Var DV ε 0 ε 0 D 3 Var 3 3 ad Cov ε j ε k Cov ε j ε 3 0 0 Cov α j α k DC ε ε k ε 0 D. Cov 3 Var 3 3

Chaper 5. Mullevel Models / 5-3 Sackg vecors we wre j j jtj J ε ε ε J ad α α α J. Sackg marces we have X j x j x jtj Z j z j z jtj X X X J ad Z 0 L 0 0 Z L 0 Z. M M O M 0 0 L Z J Wh hs oao we ma wre equao 5. a lear mxed effecs model form as Z α + X β + ε. Noe he form of R Var ε blockdagoalr R J ad DV DC L DC DC DV L DC D Var α. M M O M DC DC L DV Sep 3. Havg coded he explaaor varables ad he form of he varace marces D ad R parameer esmaes follow drecl from he Secos 3.4 ad 3.5 resuls. Sep 4. The BLUP predcors are formed begg wh predcors for α of he form a BLUP D Z V - - X b GLS. Ths elds he BLUPs for α j ε j ε 3 sa a jblup e jblup e 3BLUP. These BLUPs allow us o predc he secod ad hgher level radom coeffces hrough he relaos g BLUP X 3 b 3GLS + e 3BLUP ad b jblup Z j g BLUP + X j b GLS + e jblup correspodg o equaos 5.0 ad 5.9 respecvel. Sep 5. If desred we ma forecas fuure level- resposes. From equao 4.4 for a L-sep forecas we have ˆ z b + x b Cov ε ε R e j T + L j T + L j BLUP j T + L GLS + j T + L j j j BLUP j j For AR level- dsurbaces hs smplfes o L ˆ z b + x + ρ e. j j T + L j T + L j BLUP j T + L b GLS j T BLUP j j j j j 5.4 Tesg varace compoes Mullevel models mplcl provde a represeao for he varace as a fuco of explaaor varables. To llusrae cosder he cross-secoal wo-level model summarzed equaos 5.5-5.7. Wh equao 5.7 we have Var j z j Var α z j + Var ε j ad Cov j k z j Var α z k. Thus eve f he radom quaes α ad ε j are homoscedasc he varace s a fuco of he explaaor varables z j. Parcularl educao ad pscholog researchers wsh o es heores b examg hpoheses cocerg hese varace fucos.

5-4 / Chaper 5. Mullevel Models Uforuael he usual lkelhood rao esg procedure s o vald for esg ma varace compoes of eres. I parcular he cocer s for esg parameers where he ull hpohess s o he boudar of possble values. As a geeral rule he sadard hpohess esg procedures favors he smpler ull hpohess more ofe ha should. To llusrae he dffcules wh boudar problems le s cosder he classc example of..d. radom varables where each radom varable s dsrbued ormall wh kow mea zero ad varace σ. Suppose ha we wsh o es he ull hpohess H 0 : σ σ 0 where σ 0 s a kow posve cosa. I s eas o check ha he maxmum lkelhood esmaor of σ s. As we have see a sadard mehod of esg hpoheses s he lkelhood rao es procedure descrbed more deal Appedx A.7. Here oe compues he lkelhood rao es sasc whch s wce he dfferece bewee he ucosraed maxmum loglkelhood ad he maxmum log-lkelhood uder he ull hpohess ad compares hs sasc o a ch-square dsrbuo wh oe degree of freedom. Uforuael hs procedure s o avalable whe σ 0 0 because he log-lkelhoods are o well defed. Because σ 0 0 s o he boudar of he parameer space [0 he regular codos of our usual es procedures are o vald. However H 0 : σ 0 s sll a esable hpohess; a smple es s o rejec H 0 f he maxmum lkelhood esmaor exceeds zero. Ths procedure wll alwas rejec he ull hpohess whe σ > 0 ad accep whe σ 0. Thus hs es procedure has power versus all aleraves ad a sgfcace level of zero a ver good es! For a example closer o logudal daa models cosder he Seco 3. error compoes model wh varace parameers σ ad σ α. I he Exercse 5.4 we oule he proof o esablsh ha he lkelhood rao es sasc for assessg H 0 : σ α 0 s ½ χ where χ s a ch-square radom varable wh degree of freedom. I he usual lkelhood rao procedure for esg oe varable he lkelhood rao es sasc has a χ dsrbuo uder he ull hpohess. Ths meas ha usg omal values we wll accep he ull hpohess more ofe ha we should; hus we wll somemes use a smpler model ha suggesed b he daa. The crcal po of hs exercse s ha we defe maxmum lkelhood esmaors o be o-egave argug ha a egave esmaor of varace compoes s o vald. Thus he dffcul s ha he usual regular codos see for example Serflg 980G requre ha he hpoheses ha we es le o he eror of a parameer space. For mos varaces he parameer space s [0. B esg ha he varace equals zero we are o he boudar ad he usual asmpoc resuls are o vald. Ths does o mea ha ess for all varace compoes are o vald. For example for esg mos correlaos ad auocorrelaos he parameer space s [-]. Thus for esg correlaos ad covaraces equal o zero we are he eror of he parameer space ad so he usual es procedures are vald. I coras Exercse 5.3 we allow egave varace esmaors. I hs case b followg he oule of he proof ou wll see ha he usual lkelhood rao es sasc for assessg H 0 : σ α 0 s χ he cusomar dsrbuo. Thus s mpora o kow he cosras uderlg he sofware package ha ou are usg. A complee heor for esg varace compoes has e o be developed. Whe ol oe varace parameer eeds o be assessed for equal o zero resuls smlar o he error compoes model dscussed above have bee worked ou. For example Balag ad L 990E developed a es for a secod depede error compoe represeg me; hs model wll be descrbed Chaper 8. More geerall checkg for he presece of a addoal radom effec he model mplcl meas checkg ha o ol he varace bu also he covaraces are

Chaper 5. Mullevel Models / 5-5 equal o zero. For example for he lear mxed effecs model wh a q vecor of varace compoes α we mgh wsh o assess he ull hpohess Var α... α q 0 H 0 : D Var α. 0 0 I hs case based o he work of Self ad Lag 987S Sram ad Lee 994S showed ha he usual lkelhood rao es sasc has asmpoc dsrbuo χ q + χ where q χ ad q χ are depede ch-square radom varables wh q- ad q degrees of freedom q respecvel. The usual procedure for esg meas comparg he lkelhood rao es sasc o χ because we are esg a varace parameer ad q- covarace parameers. Thus f oe q rejecs usg he usual procedure oe wll rejec usg he mxure dsrbuo correspodg o χ q + χ. Pu aoher wa he acual p value compued usg he mxure dsrbuo q s less ha he omal p value compued usg he sadard dsrbuo. Based o hs we see ha he sadard hpohess esg procedures favors he smpler ull hpohess more ofe ha should. No geeral rules for checkg for he presece of several addoal radom effecs are avalable alhough smulao mehods are alwas possble. The mpora po s ha aalss should o quckl quoe p-values assocaed wh esg varace compoes whou carefull cosderg he model ad esmaor. Furher readg There are ma roducos o mullevel modelg avalable he leraure. Two of he more echcal ad wdel ced refereces are Raudebush ad Brk 00EP ad Goldse 995EP. If ou would lke a roduco ha emplos he mmal amou of mahemacs cosder Too 000EP. A revew of mullevel sofware s de Leeuw ad Kref 00EP. Adrews 00E provdes rece resuls o esg whe a parameer s o he boudar of he ull hpohess.

5-6 / Chaper 5. Mullevel Models Appedx 5A Hgh Order Mullevel Models Despe her wdespread applcao sadard reames ha roduce he mullevel model use a mos ol hree levels acpag ha users wll be able o u paers ad hopefull equao srucures o hgher levels. I coras hs appedx descrbes a hgh order mullevel model usg k levels. To movae he exesos beg wh he hree-level model Seco 5... Exedg equao 5.8 he level- model s expressed as Z β + X β + ε.... k... k... k... k... k Here we mgh use k as a me dex k- s a sude dex k- s classroom dex ad so o. We deoe he observao se b k {... k :... s observed}. More geerall defe k {... : s observed for some j j } k s k s... k s j k s... j k k-s... + + for s 0 k-. We wll le k { k } be a pcal eleme of k ad use k-s { k-s } for he correspodg eleme of k-s. Wh hs addoal oao we are ow a poso o provde a recursve specfcao of hgh order mullevel models. k Recursve specfcao of hgh order mullevel models. The level- model s Z β + X β + ε k k. 5A. k k k k k The level- fxed parameer vecor β has dmeso K ad he level- vecor of parameers ha ma var over hgher levels β k has dmeso q.. For g k- he level-g model s g g g g g β Z β + X β ε for g k. 5A. k + g k + g k g k + g g + k + g Here he level- fxed parameer vecor β g has dmeso K g ad he level-g varg g g parameer vecor β has dmeso q g. Thus for he covaraes Z has kg g k+ g k + g dmeso q g- q g ad X has dmeso q g- K g. 3. The level-k model s k k k β X β + ε. 5A.3 k We assume ha all dsurbace erms ε are mea zero ad are ucorrelaed wh oe aoher. g Furher defe D g Var ε k+ g σ gi q for g. g We ow show how o wre he mullevel model as a lear mxed effecs model. We do hs b recursvel serg he hgher level models from equao 5A. o he level- equao 5A.. Ths elds ε + X β + Z Z β + X β ε k k k k k k k + k

Chaper 5. Mullevel Models / 5-7 β X ε Z β X + + + k k k k k ε 3 3 3 3 3 3 + + + k k k k k k ε β X β Z Z Z + + + + + + + k s s s s k s s k s j j j k k k β X ε Z β X ε L. To smplf oao defe he q s vecor + s j j j k k s Z Z. 5A.4 Furher defe K K + + K k he K vecor β β β β k ad he K vecor k k k k k k k X Z X Z X X L. Ths elds + + + k s s s s k k s k k β X Z β X β X. Thus we ma express he mullevel model as + + + k s s s k k s k k k ε Z β X ε. 5A.5 To wre equao 5A.5 as a mxed lear model we requre some addoal oao. For a fxed se { k- } k- le k- deoe he umber of observed resposes of he form j k... for some j. Deoe he se of observed resposes as...... k k k k k k k M M. For each s k- cosder a se { k-s } k-s ad le k-s deoe he umber of observed resposes of he form j s k for some j. Thus we defe s k s k s k s k M. Fall le. Use a smlar sackg scheme for X ad ε s for s k. We ma also use hs oao whe sackg over he frs level of Z. Thus defe k k s k s k s Z Z Z M for s k-. Wh hs oao whe sackg over he frs level we ma express equao 5A.5 as + + + k s s s k k s k k k ε Z ε β X.

/ Chaper 5. Mullevel Models 5-8 For he ex level defe k k s k s k s Z Z Z M for s k-. ad k k k k blkdag Z Z Z L. Wh hs oao we have k k k k k k k k k k ε ε ε Z 0 0 0 Z 0 0 0 Z ε Z M L M O M M L L k k k k k k k k ε Z ε Z ε Z M. Thus we have + + + + k s s s k k s k k k k k ε Z ε Z ε β X. Coug a he gh sage we have < g s blkdag g s g k g k s g k s g k g k s g k s g k s for for Z Z Z Z Z L M. Ths elds + + + + + + k g s s s k g k s g s s g k g k s g k g k g k ε Z ε Z ε β X. Takg g k we have + + + k s s s ε Z ε β X 5A.6 a expresso for he usual lear mxed effecs model. The ssem of oao akes us drecl from he mullevel model equaos 5A.- 5A.3 o he lear mxed effecs model equao 5A.6. Properes of parameer esmaes for lear mxed effecs model are well esablshed. Thus parameer esmaors of he mullevel model also ejo hese properes. Moreover b showg how o wre mullevel models as lear mxed effecs model o specal sascal sofware s requred. Oe ma smpl use sofware wre for lear mxed effecs models for mullevel modelg.

Chaper 5. Mullevel Models / 5-9 5. Exercses ad Exesos Seco 5.3 5.. Two-level model Cosder he wo-level model descrbed Seco 5.. ad suppose ha we have he error compoes model so ha z j z j ad R s a scalar mes he de marx. Furher suppose ha here are o level- explaaor varables. Show ha he BLUP of he codoal mea of he level- respose E α α + X β s ζ + - ζ X b GLS where ζ T T. + Varε /Varα 5.. Radom erceps hree-level model Assume ha we observe school dsrcs. Wh each school dsrc we observe j J sudes. For each sude we have T j observaos. Assume ha he model s gve b j x j β + α + ν j + ε j. Here assume ha each of {α } {ν } {ν J } ad {ε j } are depedel ad decall dsrbued as well as depede of oe aoher. Also assume ha {α } {ν } {ν J } ad {ε j } are mea zero wh varaces σ α σ υ σ υj ad σ ε respecvel. Defe z j o be a T j J+ marx wh oes he frs ad j+ s colums ad zeroes elsewhere. For example we have 0 L 0 0 L 0 z. M M M M M 0 L 0 Furher defe Z z z L z J α α υ L υ D J σ α 0 Varα where D υ dag σ υ σ υ... σ υj. 0 D υ a. Defe X ad ε erms of { j } { x j } ad { ε j } so ha we ma wre Z α + X β + ε usg he usual oao. b. For he approprae choce of R show ha where e j j - x j b GLS e c. Show ha GLS σ ε ad T e T e L T e Z R X b j T T j j ej ad T e J T j j C Cζυ D + Z R Z Cζυ C e j. J J

5-0 / Chaper 5. Mullevel Models σ αt where ζ ζ υj σ + σ T ε α σ σ T ε υj j συ jtj ζ ζ ζ + ζ υ υ υ... υj σ ε ζ T dag T T... TJ C ad J T ζ ζ j j υj C Dυ + σ ε T + C ζυζ υ. d. Wh he oao ad a BLUP a BLUP υ BLUP L υj BLUP a J j BLUP ζ J j T ζ j j υj T ζ ζ e υj e a υ j BLUP ζ j j BLUP υ. j show ha Seco 5.4 5.3. MLE varace esmaors whou boudar codos Cosder he basc radom effecs model ad suppose ha T T K ad ha x. Pars a ad b are he same as Exercse 3.0 a ad b. As here we ow gore boudar codos so ha he esmaor ma become egave wh posve probabl. a. Show ha he maxmum lkelhood esmaor of σ ε ma be expressed as: T σ ˆε ML. T b. Show ha he maxmum lkelhood esmaor of σ α ma be expressed as: ˆ σ ˆ α ML σ ε ML. T c. Show ha he maxmum lkelhood ma be expressed as: L ˆ σ ˆ α ML σ ε ML { T lπ + T + T l ˆ σ l ˆ ˆ ε ML + Tσ α ML + σ ε ML }. d. Cosder he ull hpohess H 0 : σ α 0. Uder hs ull hpohess show ha he maxmum lkelhood esmaor of σ ε ma be expressed as: T σ. ˆε Reduced T e. Uder he ull hpohess H 0 : σ α 0 show ha he maxmum lkelhood ma be expressed as: T L 0 ˆ σ ε Re duced T lπ + T + T l. T f. Use a secod order approxmao of he logarhm fuco o show ha wce he dfferece of log-lkelhoods ma be expressed as: L ˆ σ ˆ L0 ˆ α ML σ ε ML σ ε Reduced { SSW T SSB} 4 T T σ ε

Chaper 5. Mullevel Models / 5- T where SSW ad SSB T f. Assumg ormal of he resposes ad he ull hpohess H 0 : σ α 0 show ha L ˆ σ ˆ L0 ˆ α ML σ ε ML σ ε Reduced D χ as. 5.4. MLE varace esmaors wh boudar codos Cosder he basc radom effecs model ad suppose ha T T K ad ha x. Ulke problem 5.3 we ow mpose boudar codos so ha varace esmaors mus be oegave. a. Usg he oao of Exercse 5.3 show ha he maxmum lkelhood esmaors of σ ε ad σ α ma be expressed as: ˆ ˆ σ α ML f σ > 0 α ML > ˆ σ α CML ad 0 f ˆ σ 0 ˆ ˆ ˆ σ ε ML f σ α ML 0 σ ε CML. ˆ σ f ˆ σ 0 α ML A earl referece for hs resul s Herbach 959G. b. Show ha he maxmum lkelhood ma be expressed as: L ˆ σ ˆ σ L ˆ σ α ˆ σ CML ε CML α ML L0 ˆ σ ε ε ML Reduced. ε Reduced f ˆ σ f ˆ σ α ML α ML > 0. 0 SSB c. Defe he cu-off c T. Check ha c > 0 f ad ol f ˆ σ α ML > 0. Cofrm SSW ha we ma express he lkelhood rao sasc as L ˆ σ L0 ˆ α CML σ ε CML σ ε Reduced > c + T l l + c f c 0 T. 0 f c 0 α ML d. Assumg ormal of he resposes ad he ull hpohess H 0 : σ α 0 show ha he cu-off c p 0 as. e. Assumg ormal of he resposes ad he ull hpohess H 0 : σ α 0 show ha T c D N0 as T where Φ s he sadard ormal dsrbuo fuco. f. Assume ormal of he resposes ad he ull hpohess H 0 : σ α 0. Show for a > 0 ha Prob L ˆ σ L0 ˆ σ a Φ a as. [ σ ] α CML ε CML ε Reduced > g. Assume ormal of he resposes ad he ull hpohess H 0 : σ α 0. Summarze he resuls above o esablsh ha he lkelhood rao es sasc asmpocall has a dsrbuo ha s 50% equal o 0 ad 50% a ch-square dsrbuo wh oe degree of freedom. D

5- / Chaper 5. Mullevel Models Emprcal Exercse 5.5. Sude Acheveme These daa were gahered o assess he relaoshp bewee sude acheveme ad educao aves. Moreover he ca also be used o address relaed eresg quesos such as how oe ca rak he performace of schools or how oe ca forecas a chld s fuure performace o acheveme ess based o her earl es scores. Webb e al. 00EP vesgaed relaoshps bewee sude acheveme ad Texas school dsrc parcpao he Naoal Scece Foudao Saewde Ssemc Iaves program bewee 994 ad 000. The focused o he effecs of ssemc reform o performace o a sae mahemacs es. We cosder here a subse of hese daa o model rajecores of sudes mahemacs acheveme over me. Ths subse cosss of a radom sample of 0 elemear schools Dallas wh 0 sudes radoml seleced from each school. All avalable records for hese 400 sudes durg elemear school are cluded. I Dallas Grades 3 hrough 6 correspod o elemear school. Alhough here exss a aural herarch a each me po sudes are esed wh schools hs herarch was o maaed compleel over me. Several sudes swched schools see varable SWITCH_SCHOOLS ad ma sudes were o promoed see varable RETAINED. To maa he herarch of sudes wh schools a sude was assocaed wh a school a he me of seleco. To maa a herarch over me a cohor varable was defed as 3 4 for hose grades 6 5 4 ad 3 respecvel 994 ad a 5 for hose grade 3 995 ad so o up o a 0 for hose grade 3 000. The varable FIRST_COHORT aaches a sude o a cohor durg he frs ear of observao whereas he varable LAST_COHORT aaches a sude o a cohor durg he las ear of observao. Varable Descrpo Level- varables replcaos over me GRADE Grade whe assessme was made 3-6 YEAR Year of assessme 994-000 TIME Observed repeaed occasos for each sude RETAINED Reaed grade for a parcular ear es 0o SWITCH_SCHOOLS Swched schools a parcular ear es 0o DISADVANTAGED Ecoomcall dsadvaaged free/reduced luch 0o TLI_MATH Texas Learg Idex o mahemacs assessme measure Level- varables replcaos over chld CHILDID Sude defcao umber MALE Geder of sudes male 0female ETHNICITY Whe black hspac oher oher cludes asa as well as mxed races FIRST_COHORT Frs observed cohor membershp LAST_COHORT Las observed cohor membershp Level-3 varables replcaos over school SCHOOLID School defcao umber USI Urba Ssem Iave cohor 993 994 3995 MATH_SESSIONS Number of eachers aeded mahemacs sessos N_TEACHERS Toal umber of eachers he school Source: N.L. Webb W. H. Clue D. Bol A. Gamora R. H. Meer E. Oshoff ad C. Thor 00EP.

Chaper 5. Mullevel Models / 5-3 a b c Basc summar sascs Summarze he school level varables. Produce a able o summarze he frequec of he USI varable. For MATH_SESSIONS ad N_TEACHERS provde he mea meda sadard devao mmum ad he maxmum. Summarze he chld level varables. Produce ables o summarze he frequec of geder ehc ad he cohor varables. Provde basc relaoshps amog level ad 3 varables. Summarze he umber of eachers b geder. Exame ehc b geder. v Summarze he level varables. Produce meas for he bar varables RETAINED SWITCH_SCHOOLS ad DISAVANTAGED. For TLI_MATH provde he mea meda sadard devao mmum ad he maxmum. v Summarze umercall some basc relaoshps bewee TLI_MATH ad he explaaor varables. Produce ables of meas of TLI_MATH b GRADE YEAR RETAINED SWITCH_SCHOOLS ad DISAVANTAGED. v Summarze graphcall some basc relaoshps bewee TLI_MATH ad he explaaor varables. Produce boxplos of TLI_MATH b GRADE YEAR RETAINED SWITCH_SCHOOLS ad DISAVANTAGED. Comme o he red over me ad grade. v Produce a mulple me seres plo of TLI_MATH. Comme o he dramac decles of some sudes ear-o-ear es scores. Two-level - Error compoes model Igorg he school level formao ru a error compoes model usg chld as he secod level u of aalss. Use he level caegorcal varables GRADE ad YEAR ad bar varables RETAINED ad SWITCH_SCHOOLS. Use he level caegorcal varables ETHNICITY ad he bar varable MALE. Repea our aalss par b bu clude he varable DISAVANTAGED. Descrbe he advaages ad dsadvaages of cludg hs varable he model specfcao. Repea our aalss par b bu clude a AR specfcao of he error. Does hs mprove he model specfcao? v Repea our aalss par b bu clude a fxed school level caegorcal varable. Does hs mprove he model specfcao? Three-level model Now corporae school level formao o our model b. A he frs level he radom ercep vares b chld ad school. We also clude GRADE YEAR RETAINED ad SWITCH_SCHOOLS as level explaaor varables. For he secod level model he radom ercep vares b school ad cludes ETHNICITY ad MALE as level explaaor varables. A he hrd level we clude USI MATH_SESSIONS ad N_TEACHERS as level 3 explaaor varables. Comme o he approprae of hs f. Is he USI caegorcal varable sascall sgfca? Re-ru he par c model whou USI ad use a lkelhood rao es sasc o respod o hs queso. Repea our aalss par c bu clude a AR specfcao of he error. Does hs mprove he model specfcao?

5-4 / Chaper 5. Mullevel Models Appedx 5A 5.6. BLUP predcors for a geeral mullevel model Cosder he geeral mullevel model developed Appedx 5A ad he mxed lear model represeao equao 5A.6. Le V Var. a. Usg bes lear ubased predco roduced Seco 4. show ha we ca express he BLUP predcors of he resduals as g g g e k+ g BLUP Cov ε ε k+ g Z g V - X bgls for g k ad for g e k BLUP Cov ε ε k V - X bgls. g b. Show ha he BLUP predcor of β s k+ g b e. g g g g k+ g BLUP X k+ g b g GLS + Z k+ g b kg BLUP + g k+ g BLUP + c. Show ha he BLUP forecas of... k L s ˆ... k + L Z... k + Lb k BLUP + X... k + Lb GLS... k + L GLS + Cov ε ε V - X b.

Chaper 6. Sochasc Regressors / 6-003 b Edward W. Frees. All rghs reserved Chaper 6. Sochasc Regressors Absrac. I ma applcaos of eres explaaor varables or regressors cao be hough of as fxed quaes bu raher are modeled sochascall. I some applcaos ca be dffcul o deerme wha varables are beg predced ad wha varables are dog he predco! Ths chaper summarzes several models ha corporae sochasc regressors. The frs cosderao s o def uder wha crcumsaces we ca safel codo o sochasc regressors ad use he resuls from pror chapers. We he dscuss exogee formalzg he dea ha a regressor flueces he respose varable ad o he oher wa aroud. Fall hs chaper roduces suaos where more ha oe respose s of eres hus permg us o vesgae complex relaoshps amog resposes. Seco 6.. Sochasc regressors o-logudal segs Up o hs po we have assumed ha he explaaor varables X ad Z are osochasc. Ths coveo follows a log-sadg rado he sascs leraure. Pedagogcall hs rado allows for smpler verfcao of properes of esmaors ha he sochasc coveo. Moreover classcal expermeal or laboraor segs reag explaaor varables as o-sochasc allows for uve erpreaos such as whe X s uder he corol of he aals. However for oher applcaos such as he aalss of surve daa ha are draw as a probabl sample from a populao he assumpo of o-sochasc varables s more dffcul o erpre. For example whe drawg a sample of dvduals o udersad each dvdual s healh care decsos we ma wsh o expla her healh care servces ulzao erms of her age geder race ad so o. These are plausble explaaor varables ad seems sesble o model hem as sochasc ha he sample values are deermed b a radom draw from a populao. I some was he sud of sochasc regressors subsumes ha of o-sochasc regressors. Frs wh sochasc regressors we ca alwas adop he coveo ha a sochasc qua wh zero varace s smpl a deermsc or o-sochasc qua. Secod we ma make fereces abou populao relaoshps codoal o values of sochasc regressors esseall reag hem as fxed. However he choce of varables o whch we codo depeds o he scefc eres of he problem makg he dfferece bewee fxed ad sochasc regressors dramacall dffere some cases. Udersadg he bes was o use sochasc regressors logudal segs s sll a developg research area. Thus before preseg echques useful for logudal daa hs seco revews kow ad prove mehods ha are useful o-logudal segs eher for he cross-seco or he me dmeso. Subseque secos hs chaper focus o logudal segs ha corporae boh he cross-seco ad me dmesos. Seco 6.. Edogeous sochasc regressors A exogeous varable s oe ha ca be ake as gve for he purposes a had. As we wll see exogee requremes var depedg o he coex. A edogeous varable s oe ha fals he exogee requreme. I coras s cusomar ecoomcs o use he erm

6- / Chaper 6. Sochasc Regressors edogeous o mea a varable ha s deermed wh a ecoomc ssem whereas a exogeous varable s deermed ousde he ssem. Thus he acceped ecoomerc/sascal usage dffers from he geeral ecoomc meag. To develop exogee ad edogee coceps we beg b hkg of {x } as a se of observaos from he same dsrbuo. For example hs assumpo s approprae whe he daa arse from a surve where formao s colleced usg a smple radom samplg mechasm. We suppress he subscrp because we are cosderg ol oe dmeso hs seco. Thus ma represe eher he cross-secoal defer or me perod. For ologudal daa we do o cosder he z varables. For depede observaos we ca wre he assumpos of he lear model as Chaper addg ol ha we are codog o he sochasc explaaor varables whe wrg dow he momes of he respose. To beg for he regresso fuco we assume ha E x x β ad he codoal varace s Var x σ. To hadle addoal samplg mechasms we ow roduce a more geeral seg. Specfcall we codo o all of he explaaor varables he sample o jus he oes assocaed wh he h draw. Defe X x x ad work wh he followg assumpos. Assumpos of he Lear Regresso Model wh Srcl Exogeous Regressors SE. E X x β. SE. {x x } are sochasc varables. SE3. Var X σ. SE4. { X} are depede radom varables. SE5. { } s ormall dsrbued codoal o {X}. Assumg for he mome ha {x } are muuall depede he E X E x Var X Var x ad x are depede. Thus he assumpos SE-4 are ceral useful he radom samplg coex. Moreover he assumpos SE-4 are he approprae sochasc regressor geeralzao of he fxed regressors model assumpos ha esure ha we rea mos of he desrable properes of he ordar leas squares esmaors of β. For example he ubasedess ad he Gauss-Markov properes of ordar leas squares esmaors of β hold uder SE-SE4. Moreover assumg codoal ormal of he resposes he he usual ad F sascs have her cusomar dsrbuos regardless as o wheher or o X s sochasc; see for example Greee 00E or Goldberger 99E. I urs ou ha he usual ordar leas squares esmaors also have desrable asmpoc properes uder assumpos SE-4. For ma socal scece applcaos daa ses are large ad researchers are prmarl eresed asmpoc properes of esmaors. If achevg desrable asmpoc properes s he goal he he sochasc regressor model assumpos SE-4 ca be relaxed hus permg a wder scope of applcaos. For dscusso purposes we ow focus o he frs assumpo. Usg he lear model framework we defe he dsurbace erm o be ε - x β ad wre SE as E ε X 0. Ths assumpo o he regressors s kow as src exogee he ecoomercs leraure see for example Haash 000E. If he dex represes depede cross-secoal draws such as wh smple radom samplg he src exogee s a approprae assumpo. However f he dex as represes me he he src exogee s o useful for ma applcaos; assumes ha he me dsurbace erm s orhogoal o all regressors he pas coemporaeous ad he fuure. A alerave assumpo s SEp. E ε x E - x β x 0.

Chaper 6. Sochasc Regressors / 6-3 If SEp holds he he regressors are sad o be predeermed. Because SEp mples zero covarace bewee he regressors ad he dsurbaces we sa ha predeermed regressors are ucorrelaed wh coemporaeous dsurbaces. Aoher wa of expressg assumpo SEp s hrough he lear projeco L ε x 0. See Appedx 6A for defos ad properes of lear projecos. Ths alerave mehod wll be useful as we explore logudal exesos of he oo of edogee Seco 6.3. The assumpo SEp s weaker ha SE. Ol he weaker assumpo SEp ad codos aalogous o hose SE-4 s requred for he asmpoc proper of cossec of he ordar leas squares esmaors of β. We wll be more specfc our dscusso of logudal daa begg Seco 6.. For specfcs regardg o-logudal daa segs see for example Haash 000E. To reerae he src exogee assumpo SE s suffce for he ordar leas squares esmaors of β o rea fe sample properes such as ubasedess whereas ol he weaker predeermed assumpo SEp s requred for cossec. For asmpoc ormal we requre a assumpo ha s somewha sroger ha SEp. A suffce codo s: SEm. E ε ε - ε x x 0 for all. Whe SEm holds he {ε } sasfes he requremes for a margale dfferece sequece. We oe usg he law of eraed expecaos ha SEm mples SEp. For me seres daa where he dex represes me we see ha boh Assumpos SEp ad SEm do o rule ou he possbl ha he curre error erm ε wll be relaed o fuure regressors as does he src exogee assumpo SE. Seco 6.. Weak ad srog exogee We bega Seco 6.. b expressg wo pes of exogee src exogee ad predeermedess erms of codoal meas. Ths s approprae for lear models because gves precsel he codos eeded for ferece ad s drecl esable. We beg hs subseco b geeralzg hese coceps o assumpos regardg he ere dsrbuo o jus he mea fuco. Alhough sroger ha he codoal mea versos hese assumpos are drecl applcable o olear models. Moreover we use hs dsrbuo framework o roduce wo ew pes of exogee weak ad srog exogee. A sroger verso of src exogee SE s SE`: ε s depede of X. Here we are usg he coveo ha he zero mea dsurbaces are defed as ε - x β. Noe ha SE` mples SE; SE` s a requreme o he jo dsrbuo of ε ad X o jus he codoal mea. Smlarl a sroger verso of SEp s SEp`. ε s depede of x. A drawback of SE ad SEp s ha he referece o parameer esmabl s ol mplc. A alerave se of defos roduced b Egle Hedr ad Rchard 983E explcl defes exogee erms of paramerc lkelhood fucos. Iuvel a se of varables are sad o be weakl exogeous f whe we codo o hem here s o loss of formao for he parameers of eres. If addo he varables are o caused b he edogeous varables he he are sad o be srogl exogeous. Weak exogee s suffce for effce esmao. Srog exogee s requred for codoal predcos forecasg of edogeous varables.

6-4 / Chaper 6. Sochasc Regressors Specfcall suppose ha we have radom varables x x T T wh jo probabl des or mass fuco for f T x x T. Usg for he me dex we ca alwas wre hs codoall as f T... x... x f x... x x T T... T { f...... x f x... x... x } x. Here whe he codoal dsrbuos are he margal dsrbuos of ad x as approprae. Now suppose ha hs jo dsrbuo s characerzed b vecors of parameers θ ad ψ such ha SEw.... x... x f T T T T f... x... x θ f x... x... x ψ. I hs case we ca gore he secod erm for ferece abou θ reag he x varables as esseall fxed. If he relaoshp SEw holds he we sa ha he explaaor varables are weakl exogeous. Suppose addo ha x... x... x ψ f x x... x ψ 6.a f ha s codoal o x x - ha he dsrbuo of x does o deped o pas values of -. The we sa ha { - } does o Grager-cause x. Ths codo ogeher wh SEw suffces for srog exogee. We oe ha Egle e al. 983E also roduce a so-called super exogee assumpo for polc aalss purposes; we wll o cosder hs pe of exogee. Seco 6..3 Causal effecs Issues of whe varables are edogeous or exogeous are mpora o researchers ha use sascal models as par of her argumes for assessg wheher or o causal relaoshps hold. Researchers are eresed causal effecs ofe more so ha measures of assocao amog varables. Tradoall sascs has corbued o makg causal saemes prmarl hrough radomzao. Ths rado goes back o he work of Fsher ad Nema he coex of agrculural expermes. I Fsher s work reames were radoml allocaed o expermeal us plos of lad. Because of hs radom assgme dffereces resposes crop elds from he lad could be reasoabl ascrbed o reames whou fear of uderlg ssemac flueces from ukow facors. Daa ha arse from hs radom assgme mechasm are kow as expermeal. I coras mos daa from he socal sceces are observaoal where s o possble o use radom mechasms o radoml allocae observaos accordg o varables of eres. However s possble o use radom mechasms o gaher daa hrough probabl samples ad hus o esmae sochasc relaoshps amog varables of eres. The prmar example of hs s o use a smple radom sample mechasm o collec daa ad esmae a codoal mea hrough regresso mehods. The mpora po s ha hs regresso fuco measures

Chaper 6. Sochasc Regressors / 6-5 relaoshps developed hrough he daa gaherg mechasm o ecessarl he relaoshps of eres o researchers. I he ecoomcs leraure Goldberger 97E defes a srucural model as a sochasc model represeg a causal relaoshp o a relaoshp ha smpl capures sascal assocaos. I coras a samplg based model s derved from our kowledge of he mechasms used o gaher he daa. The samplg based model drecl geeraes sascs ha ca be used o esmae quaes of eres ad hus s also kow as a esmable model. To llusrae suppose ha {x } represes a radom sample from a populao. The we ca alwas esmae E x oparamercall. Moreover we mgh assume ha E x x β for some vecor β. Ths requres o appeal o he heor from a uderlg fucoal feld. We use ol he assumpo of he daa geerag mechasm ad hus refer o hs as a samplg based model. As a example of a srucural model Duca 969EP cosders he followg model equaos ha relae oe s self-eseem o delquec x : β 0 + β + β x + ε x γ 0 + γ + γ x + ε. I hs model curre perod self-eseem ad delquec are affeced b he pror perod s self-eseem ad delquec. Ths model specfcao reles o heor from he fucoal feld. Ths s a example of a srucural equaos model ha Secos 6.4 ad 6.5 wll dscuss more deal. Parcularl for observaoal daa causal saemes are based prmarl o subsave hpoheses whch he researcher carefull develops. Causal ferece s heorecall drve. Causal processes geerall cao be demosraed drecl from he daa; he daa ca ol prese releva emprcal evdece servg as a lk a cha of reasog abou causal mechasms. Logudal daa are much more useful esablshg causal relaoshps ha crosssecoal regresso daa. Ths s because for mos dscples he causal varable mus precede he effec varable me. To llusrae Lazarsfeld ad Fske 938O cosdered he effec of rado adversg o produc sales. Tradoall hearg rado adversemes was hough o crease he lkelhood of purchasg a produc. Lazarsfeld ad Fske cosdered wheher hose ha bough he produc would be more lkel o hear he adverseme hus posg a reverse he dreco of causal. The proposed repeaedl ervewg a se of people he pael o clarf he ssue. Noos of radomzao have bee exeded b Rub 976G 978G 990G o observaoal daa hrough he cocep of poeal oucomes. Ths s a area ha s rapdl developg; we refer o Agrs Imbes ad Rub 996G for furher dscussos. Seco 6..4 Isrumeal varable esmao Accordg o Wooldrdge 00E page 83 srumeal varable esmao s probabl secod ol o ordar leas squares erms of mehods used emprcal ecoomcs research. Isrumeal varable esmao s a geeral echque ha s wdel used ecoomcs ad relaed felds o hadle problems assocaed wh he dscoec bewee he srucural model ad a samplg based model. To roduce srumeal varable esmao hs subseco assumes ha he dex represes cross-secoal draws ad ha hese draws are depede. The srumeal varable echque ca be used saces where he srucural model s specfed b a lear equao of he form x β + ε 6.

6-6 / Chaper 6. Sochasc Regressors e o all of he regressors are predeermed ha s E ε x 0. The srumeal varable echque emplos a se of predeermed varables w ha are correlaed wh he regressors specfed he srucural model. Specfcall we assume IV. E ε w E - x β w 0 ad IV. E w w s verble. Wh hese addoal varables a srumeal varable esmaor of β s b IV X P W X - X P W where P W W W W - W s a projeco marx ad W w w s he marx of srumeal varables. Isrumeal varable esmaors ca be expressed as specal cases of geeralzed mehod of mome esmaors; see Appedx C.6 for furher deals. To llusrae we ow descrbe hree commol ecouered suaos where he srumeal varable echque has prove o be useful. The frs suao cocers suaos where mpora varables have bee omed from he samplg model. I hs suao we wre he srucural regresso fuco as E x u x β + γ u where u represes mpora uobserved varables. However he samplg based model uses ol E x x β hus omg he uobserved varables. For example hs dscusso of omed varable bas Wooldrdge 00E dscusses a applcao b Card 995E cocerg a cross-seco of me where he eres s sudg logarhmc wages relao o ears of educao. Addoal corol varables clude ears of experece ad s square regoal dcaors racal dcaors ad so forh. The cocer s ha he srucural model oms a mpora varable he ma s abl u ha s correlaed wh ears of educao. Card roduces a varable o dcae wheher a ma grew up he vc of a four-ear college as a srume for ears of educao. The movao behd hs choce s ha hs varable should be correlaed wh educao e ucorrelaed wh abl. I our oao we would defe w o be he same se of explaaor varables used he srucural equao model bu wh he vc varable replacg he ears of educao varable. Assumg posve correlao bewee he vc ad ears of educao varables we expec assumpo IV o hold. Moreover assumg ha vc o be ucorrelaed wh abl we expec assumpo IV o hold. The secod suao where he srumeal varable echque has prove useful cocers mpora explaaor varables ha have bee measured wh error. Here he srucural model s gve as equao 6. bu esmao s based o he model x * β + ε 6. where x * x + η ad η s a error erm. Tha s he observed explaaor varables x * are measured wh error e he uderlg heor s based o he rue explaaor varables x. Measureme error causes dffcules because eve f he srucural model explaaor varables are predeermed such ha E - x β x 0 hs does o guaraee ha he observed varables wll be because E - x * β x * 0. For example Card s 995E reurs o schoolg example descrbed above s ofe maaed ha ears of educao records are fraugh wh errors due o lack of recall ad oher reasos. Oe sraeg s o replace ears of educao b a more relable srume such as compleo of hgh school or o. As wh omed varables he goal s o selec srumes ha are hghl relaed o he suspec edogeous varables e are urelaed o model devaos. A hrd mpora applcao of srumeal varable echques regards he edogee duced b ssems of equaos. We wll dscuss hs opc furher Seco 6.4.

Chaper 6. Sochasc Regressors / 6-7 I ma suaos srumeal varable esmaors ca be easl compued usg wosage leas squares. I he frs sage oe regresses each edogeous regressor o he se of exogeous explaaor varables ad calculaes fed values of he form Xˆ PW X. I he secod sage oe regresses he depede varable o he fed values usg ordar leas squares o ge he srumeal varable esmaor ha s X X ˆ ˆ Xˆ b. However Wooldrdge 00E IV page 98 recommeds for emprcal work ha researchers use sascal packages ha explcl corporae a wo-sage leas squares roue; some of he sums of squares produced he secod sage ha would ordarl be used for hpohess esg are o approprae he wo-sage seg. The choce of srumes s he mos dffcul decso faced b emprcal researchers usg srumeal varable esmao. Theorecal resuls are avalable cocerg he opmal choce of srumes Whe 984E. For praccal mplemeao of hese resuls emprcal researchers should esseall r o choose srumes ha are hghl correlaed wh he edogeous explaaor varables. Hgher correlao meas ha he bas as well as sadard error of b IV wll be lower Boud Jaeger ad Baker 995E. For addoal backgroud readg we refer he reader o vruall a graduae ecoomercs ex see for example Greee 00E Haash 000E Wooldrdge 00E. Seco 6.. Sochasc regressors logudal segs Ths seco descrbes esmao logudal suaos ha ca be readl hadled usg echques alread descrbed he ex. Seco 6.3 follows b cosderg more complex models ha requre specalzed echques. Seco 6.. Logudal daa models whou heerogee erms As we wll see he followg subsecos he roduco of heerogee erms α complcaes he edogee quesos logudal ad pael daa models cosderabl. Coversel whou heerogee erms logudal ad pael daa models carr few feaures ha would o allow us o appl he Seco 6. echques drecl. To beg we ma wre a model wh srcl exogeous regressors as: Assumpos of he Logudal Daa Model wh Srcl Exogeous Regressors SE. E X x β. SE. {x } are sochasc varables. SE3. Var X R. SE4. { X} are depede radom vecors. SE5. { } s ormall dsrbued codoal o {X}. Recall ha X {X X } s he complee se of regressors over all subjecs ad me perods. Because hs se of assumpos cludes hose he Seco 6.. o-logudal seg we sll refer o he se as Assumpos SE-5. Wh logudal daa we have repeaedl oed he mpora fac ha observaos from he same subjec ed o be relaed. Ofe we have used he heerogee erm α o accou for hs relaoshp. However oe ca also use he covarace srucure of he dsurbaces R o accou for hese depedeces; see Seco 7.. Thus SE3 allows aalss o choose a correlao srucure such as arses from a auoregressve or compoud smmer srucure o accou for hese ra-subjec correlaos. Ths formulao emplog srcl exogeous

6-8 / Chaper 6. Sochasc Regressors varables meas ha he usual leas squares esmaors have desrable fe as well as asmpoc properes. As we saw Seco 6.. he src exogee assumpo does o perm lagged depede varables aoher wdel used approach for corporag ra-subjec relaoshps amog observaos. Sll whou heerogee erms we ca weake he assumpos o he regressors o he assumpo of predeermed regressors as Seco 6.. ad sll acheve cosse regresso esmaors. Wh he logudal daa oao hs assumpo ca be wre as: SEp. E ε x E - x β x 0. Usg lear projeco oao Appedx 6A we ca also express hs assumpo as L ε x 0 assumg E x x s verble. Wrg he correspodg margale dfferece sequece assumpo ha allows for asmpoc ormal s slghl more cumbersome because of he wo dces for he observaos he codog se. We leave hs as a exercse for he reader. The mpora po of hs subseco s o emphasze ha logudal ad pael daa models have he same edogee cocers as he cross-secoal models. Moreover ofe he aals ma use well-kow echques for hadlg edogee developed cross-secoal aalss for logudal daa. However whe emplog hese echques he logudal daa models should o possess heerogee erms. Isead devces such as a correlao srucure for he codoal respose or laggg he depede varable ca be used o accou for heerogee logudal daa hus allowg he aals o focus o edogee cocers. Seco 6.. Logudal daa models wh heerogee erms ad srcl exogeous regressors As we saw Seco 6. for o-logudal daa he remedes ha accou for edogeous sochasc regressors requre kowledge of a fucoal feld. The formulao of a uderlg srucural model s b defo feld-specfc ad hs formulao affecs he deermao of he bes model esmaors. For logudal daa ma dscples s cusomar o corporae a subjec-specfc heerogee erm o accou for ra-subjec correlaos eher from kowledge of he uderlg daa geerag process beg suded or b rado wh he feld of sud. Thus s ofe mpora o udersad he effecs of regressors whe a heerogee erm α s prese he model. To defe edogee he pael ad logudal daa coex we aga beg wh he smpler cocep of src exogee. Recall he lear mxed effecs model z α + x β + ε ad s vecor verso Z α + X β + ε. To smplf he oao le X* {X Z X Z } be he colleco of all observed explaaor varables ad α α α be he colleco of all subjec-specfc erms. We ow cosder:

Chaper 6. Sochasc Regressors / 6-9 Assumpos of he Lear Mxed Effecs Model wh Srcl Exogeous Regressors Codoal o he Uobserved Effec SEC. E α X* Z α + X β. SEC. {X*} are sochasc varables. SEC3. Var α X* R. SEC4. { } are depede radom vecors codoal o {α} ad {X*}. SEC5. { } s ormall dsrbued codoal o {α} ad {X*}. SEC6. E α X* 0 ad Var α X* D. Furher {α α } are muuall depede codoal o {X*}. SEC7. {α } s ormall dsrbued codoal o {X*}. These assumpos are readl suppored b a radom samplg scheme. For example suppose ha x z x z represes a radom sample from a populao. Each draw x z has assocaed wh a uobserved lae vecor α ha s par of he codoal regresso fuco. The because {α x z } are decall ad depedel dsrbued we mmedael have SEC ad SEC4 as well as he codoal depedece of {α } SEC6. Assumpos SEC ad SEC ad he frs par of SEC6 are mome codos ad hus deped o he codoal dsrbuos of he draws. Furher assumpos SEC5 ad SEC7 are also assumpos abou he codoal dsrbuo of a draw. Assumpo SEC s a sroger assumpo ha src exogee SE. Usg he dsurbace erm oao we ma re-wre hs as E ε α X* 0. B he law of eraed expecaos hs mples ha boh E ε X* 0 ad E ε α 0 hold. Tha s hs codo requres boh ha he regressors are srcl exogeous ad ha he uobserved effecs are ucorrelaed wh he dsurbace erms. I he coex of a error compoes model wh radom samplg he case where q z ad radom varables from dffere subjecs are depede SEC ma be expressed as: E α x x T α + x β for each. Chamberla 98E 984E roduced codoal src exogee hs coex. The frs par of Assumpo SEC6 E α X* 0 also mples ha he uobserved me-cosa effecs ad he regressors are ucorrelaed. Ma ecoomerc pael daa applcaos use a error compoes model such as Seco 3.. I hs case s cusomar o erpre α o be a uobserved me-cosa effec ha flueces he expeced respose. Ths s movaed b he relao E α X* α + x β. I hs case we erpre hs par of Assumpo SEC6 o mea ha hs uobserved effec s o correlaed wh he regressors. Secos 7. ad 7.3 wll dscuss was of esg ad relaxg hs assumpo. Example Tax labl Coued Seco 3. descrbes a example where we use a radom sample of axpaers ad exame her ax labl erms of demographc ad ecoomc characerscs summarzed Table 3.. Because he daa were gahered usg a radom samplg mechasm we ca erpre he regressors as sochasc ad assume ha observable varables from dffere axpaers are muuall depede. I hs coex he assumpo of src exogee mples ha we are assumg ha ax labl wll o affec a of he explaaor varables. For example he demographc characerscs such as umber of depedes ad maral saus ma affec he ax labl bu ha he reverse mplcao s o rue. I parcular oe ha he oal persoal come s based o posve come ems from he ax reur; exogee cocers dcaed usg hs varable coras o a alerave such as e come a varable ha ma be affeced b pror ear s ax labl.

6-0 / Chaper 6. Sochasc Regressors Oe poeall roublg varable s he use of he ax preparer; ma be reasoable o assume ha he ax preparer varable s predeermed alhough o srcl exogeous. Tha s we ma be wllg o assume ha hs ear s ax labl does o affec our decso o use a ax preparer because we do o kow he ax labl pror o hs choce makg he varable predeermed. However seems plausble ha he pror ear s ax labl wll affec our decso o rea a ax preparer hus falg he src exogee es. I a model whou heerogee erms cossec ma be acheved b assumg ol ha he regressors are predeermed. For a model wh heerogee erms cosder he error compoes model Seco 3.. Here we erpre he heerogee erms o be uobserved subjec-specfc axpaer characerscs such as aggressveess ha would fluece he expeced ax labl. For src exogee codoal o he uobserved effecs oe eeds o argue ha he regressors are srcl exogeous ad ha he dsurbaces represeg uexpeced ax lables are ucorrelaed wh he uobserved effecs. Moreover Assumpo SEC6 emplos he codo ha he uobserved effecs are ucorrelaed wh he observed regressor varables. Oe ma be cocered ha dvduals wh hgh eargs poeal who have hsorcall hgh levels of ax labl relave o her corol varables ma be more lkel o use a ax preparer hus volag hs assumpo. As Chaper 3 he assumpos based o dsrbuos codoal o uobserved effecs lead o he followg codos ha are he bass of sascal ferece. Observables Represeao of he Lear Mxed Effecs Model wh Srcl Exogeous Regressors Codoal o he Uobserved Effec SE. E X* X β. SE. {X*} are sochasc varables. SE3a. Var X* Z D Z + R. SE4. { } are depede radom vecors codoal o {X*}. SE5. { } s ormall dsrbued codoal o {X*}. These codos are vruall decal o he assumpos of he logudal daa mxed model wh srcl exogeous regressors ha does o coa heerogee erms. The dfferece s he codoal varace compoe SE3. I parcular he ferece procedures descrbed Chapers 3 ad 4 ca be readl used hs suao. Fxed effecs esmao As we saw he above example ha dscussed exogee erms of he come ax labl here are mes whe he aals s cocered wh Assumpo SEC6. Amog oher hgs hs assumpo mples ha he uobserved effecs are ucorrelaed wh he observed regressors. Alhough readl acceped as he orm he bosascs leraure hs assumpo s ofe quesoed he ecoomcs leraure. Foruael Assumpos SEC-4 ad SEC5 as eeded are suffce o allow for cosse as well as asmpoc ormal esmao usg he fxed effecs esmaors descrbed Chaper. Iuvel hs s because he fxed effecs esmao procedures sweep ou he heerogee erms ad hus do o rel o he assumpo ha he are ucorrelaed wh observed regressors. See Mudlak 978aE for a earl corbuo; Seco 7. provdes furher deals. These observaos sugges a sraeg ha s commol used b aalss. If here s o cocer ha uobserved effecs ma be correlaed wh observed regressors use he more effce ferece procedures Chaper 3 based o mxed models ad radom effecs. If here s a cocer use he more robus fxed effecs esmaors. Some aalss prefer o es he

Chaper 6. Sochasc Regressors / 6- assumpo of correlao bewee uobserved ad observed effecs b examg he dfferece bewee hese wo esmaors. Ths s he subjec of Secos 7. ad 7.3 where we wll exame ferece for he uobserved or omed varables. I some applcaos researchers have paral formao abou he frs par of Assumpo SEC6. Specfcall we ma re-arrage he observables o wo peces o o o so ha Covα o 0 ad Covα o 0. Tha s he frs pece of o s correlaed o he uobservables whereas he secod pece s o. I hs case esmaors ha are eher fxed or radom effecs esmaors have bee developed he leraure. Ths dea due o Hausma ad Talor 98E s furher pursued Seco 7.3. Seco 6.3 Logudal daa models wh heerogee erms ad sequeall exogeous regressors For some ecoomc applcaos such as produco fuco modelg Keae ad Rukle 99E he assumpo of src exogee eve whe codog o uobserved heerogee erms s lmg. Ths s because src exogee rules ou curre values of he respose feedg back ad fluecg fuure values of he explaaor varables such as x +. A alerave assumpo roduced b Chamberla 99E allows for hs feedback. To roduce hs assumpo we follow Chamberla ad assume radom samplg so ha radom varables from dffere subjecs are depede. Followg hs ecoomerc leraure we assume q ad z so ha he heerogee erm s a ercep. We sa ha he regressors are sequeall exogeous codoal o he uobserved effecs f E ε α x x 0. 6.3 Ths mples E α x x α + x β. Afer corollg for α ad x pas values of regressors do o affec he expeced value of. Lagged depede varable model I addo o feedback models hs formulao allows us o cosder lagged depede varables as regressors. For example he equao α + γ - + x β + ε 6.4 sasfes equao 6.3 b usg he se of regressors o - x ad E ε α - x x 0. The explaaor varable - s o srcl exogeous so ha he Seco 6.. dscusso does o appl. As wll be dscussed Seco 8. hs model dffers from he auoregressve error srucure he commo approach he logudal bomedcal leraure. Judged b he umber of applcaos hs s a mpora damc pael daa model ecoomercs. The model s appealg because s eas o erpre he lagged depede varable he coex of ecoomc modelg. For example f we hk of as he demad of a produc s eas o hk of suaos where a srog demad he pror perod - has a posve fluece o he curre demad suggesg ha γ be a posve parameer. Esmao dffcules Esmao of he model equao 6.4 s dffcul because he parameer γ appears boh he mea ad varace srucure. I appears he varace srucure because Cov - Cov α + γ - + x β + ε - Cov α - + γ Var -.

6- / Chaper 6. Sochasc Regressors To see ha appears he mea srucure cosder equao 6.4. B recursve subsuo we have: E γ E - + x β γ γ E - + x - β + x β x + γ x - + + γ - x β + γ - E. Thus E clearl depeds o γ. Specal esmao echques are requred for he model equao 6.4; s o possble o rea he lagged depede varables as explaaor varables eher usg a fxed or radom effecs formulao for he heerogee erms α. We frs exame he fxed effecs form begg wh a example from Hsao 00E. Specal case Hsao 00E. Suppose ha α s reaed as fxed parameer effec ad for smplc ake K ad x so ha equao 6.4 reduces o α * + γ - + ε 6.5 where α * α +β. The ordar leas squares esmaor of γ urs ou o be T T ˆ ε γ + T T T / T where γ. Now we ca argue ha E ε - 0 b codog o formao avalable a me -. However s o rue ha E ε 0 suggesg haγˆ s based. I fac Hsao demosraes ha he asmpoc bas s T + γ γ T T γ lm E ˆ γ γ. T γ γ γ T T γ Ths bas s small for large T ad eds o zero as T eds o f. Furher s eresg ha he bas s ozero eve whe γ 0. To see he esmao dffcules he coex of he radom effecs model ow cosder he model equao 6.4 where α are reaed as radom varables ha are depede of he error erms ε. I s empg o rea lagged respose varables as explaaor varables ad use he usual geeralzed leas squares GLS esmaors. However hs procedure also duces bas. To see hs we oe ha s clear ha s a fuco of α ad hus so s -. However GLS esmao procedures mplcl assume depedece of he radom effecs ad explaaor varables. Thus hs esmao procedure s o opmal. Alhough he usual geeralzed leas squares esmaors are o desrable alerave esmaors are avalable. To llusrae akg frs dffereces of he model equao 6.5 elds - - γ - - - + ε - ε - elmag he heerogee erm. Noe ha - ad ε - are clearl depede; hus usg ordar leas squares wh regressors - - - - produces based esmaors of γ. We ca

Chaper 6. Sochasc Regressors / 6-3 however use - as a srume for - because - s depede of he dffereced dsurbace erm ε - ε -. Ths approach of dfferecg ad usg srumeal varables s due o Aderso ad Hsao 98E. Of course hs esmaor s o effce because he dffereced error erms wll usuall be correlaed. Thus frs dfferecg proves o be a useful devce for hadlg he heerogee erm. To llusrae how frs dfferecg b self ca fal cosder he followg specal case. Specal case Feedback. Cosder he error compoes α + x β + ε where {ε } are..d. wh varace σ. Suppose ha he curre regressors are flueced b he feedback from he pror perod s dsurbace hrough he relao x x - + υ ε - where {υ } s a..d. radom vecor ha s depede of {ε }. Takg dffereces of he model we have - - x β + ε where ε ε - ε - ad x x - x - υ ε -. Usg frs dffereces he ordar leas squares esmaor of β s b FD X X X β + X X X ε where T ε ε ε T ad X x x T υ ε υ ε T. Sraghforward calculaos show ha T lm X X lm υυ ε T σ E υυ ad ε ε lm X ε lm υ υ T T E υ ε L ε M σ ε T ε T boh wh probabl oe. Wh probabl oe hs elds he asmpoc bas lm b β E υ υ E υ. FD Oe sraeg for hadlg sequeall exogeous regressors wh heerogee erms s o use a rasform such as frs dfferecg or fxed effecs o sweep ou he heerogee ad he use srumeal varable esmao. Oe such reame has bee developed b Arellao ad Bod 99E. For hs reame we assume ha he resposes follow he model equao α + x β + ε e he regressors are poeall edogeous. We also assume ha here exs wo ses of srumeal varables. The frs se of he form w are srcl exogeous so ha L ε w w T 0 T. 6.6 The secod se of he form w sasfes he followg sequeal exogee codos L ε w w 0 T. 6.7 The dmesos of {w } ad {w } are p ad p respecvel. Because we wll remove he heerogee erm va rasformao we eed o specf hs our lear projecos. Noe

/ Chaper 6. Sochasc Regressors 6-4 ha equao 6.7 mples ha curre dsurbaces are ucorrelaed wh curre as well as pas srumes. Tme-cosa heerogee parameers are hadled va sweepg ou her effecs so le K be a T T upper ragular marx such ha K 0. For example Arellao ad Bover 995E recommed he marx suppressg he subscrp 0 0 0 0 0 0 0 0 dag L L M M M O M M M L L L T T T T T T T T T T T K FOD. Defg ε FOD K FOD ε he h row s + + + + T FOD T T T... ε ε ε ε for T-. These are kow as forward orhogoal devaos. If he orgal dsurbaces are serall ucorrelaed ad cosa varace he so are orhogoal devaos. Preservg hs srucure s he advaage of forward orhogoal devaos whe compared o smple dffereces. To defe he srumeal varable esmaor le W * be a block dagoal marx wh he h block gve b w w w. Tha s defe * T w w w w 0 0 0 w w w 0 0 0 w w W L L M O M M L L. Wh hs oao ca be show ha he sequeall exogee assumpo equaos 6.6 ad 6.7 mples E W * K ε 0. Le W W * : 0 where W has dmesos T - p T + p T T +/ ad T max T T. Ths zero marx augmeao s eeded whe we have ubalaced daa; s o eeded f T T. Specal case - Feedback Coued. A aural se of srumes s o choose w x. For smplc also use frs dffereces our choce of K. Thus 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 T T T FD ε ε ε ε ε ε M M L L M M M O M M M L L ε K. Wh hese choces he h block of E W K FD ε s + 0 0 x x M M E ε ε

Chaper 6. Sochasc Regressors / 6-5 so he exogee assumpo equao 6.6 s sasfed. Specal case Lagged depede varable. Cosder he model equao 6.5. I hs case oe ca choose w - ad for smplc use frs dffereces our choce of K. Wh hese choces he h block of E W K FD ε s 0 E M ε + ε M 0 so he sequeall exogee assumpo equao 6.7 s sasfed. Now defe he marces MWX W K X ad M W W K oao we defe he srumeal varable esmaor as where Σ W K ε ε K W b IV M WX Σ IV MWX M WX Σ IV MW. Wh hs 6.8 IV E. Ths esmaor s cosse ad asmpocall ormal wh asmpoc covarace marx M Σ M IV WX IV WX Var b. Boh he esmaor ad he asmpoc varace rel o he ukow marx Σ IV. To compue a frs sage esmaor we ma assume ha he dsurbaces are serall ucorrelaed ad homoscedasc ad hus use b M Σˆ M M Σˆ M IV WX IV WX WX IV W where Σˆ IV W K K W. As s he usual case wh geeralzed leas squares esmaors hs esmaor s vara o he esmaor of he scale Var ε. To esmae hs scale parameer use he resduals e x b IV. A esmaor of Var b IV ha s robus o he assumpo of o seral correlao ad homoscedasc of he dsurbaces s Σ ˆ W K e e K W where e e. e T. IV Example Tax labl Coued I Seco 6.. we suggesed ha a heerogee erm ma be due o a dvdual s earg poeal ad ha hs ma be correlaed wh he varable ha dcaes use of a professoal ax preparer. Moreover here was cocer ha ax lables from oe ear ma fluece he choce subseque ax ear s choce of wheher or o o use a professoal ax preparer. Isrumeal varable esmaors provde proeco agas hese edogee cocers. Table 6. summarzes he f of wo damc models. Boh models use heerogee erms ad lagged depede varables. Oe model assumes ha all demographc ad ecoomc varables are srcl exogeous so are used as w ; he oher models assumes ha demographc varables are srcl exogeous so are used as w bu ha he ecoomc varables are

6-6 / Chaper 6. Sochasc Regressors sequeall exogeous so are used as w. For he secod model we also prese robus - sascs based o he varace-covarace marx Σ ˆ IV addo o he usual model-based -sascs based o he varace-covarace marx Σ ˆ IV. These models were f usg he sascal/ecoomerc package STATA. Table 6. shows ha he lagged depede varable was sascall sgfca for boh models ad mehods of calculag -sascs. Of he demographc varables ol he head of household varables was sascall sgfca ad hs was o eve rue uder he model reag ecoomc varables as sequeall exogeous ad usg robus -sascs. Of he ecoomc varables eher EMP or PREP was sascall sgfca whereas MR was sascall sgfca. The oher measure of come LNTPI was sascall sgfca whe reaed as srcl exogeous bu o whe reaed as sequeall exogeous. Because he ma purpose of hs sud was o sud he effec of PREP o LNTAX he effecs of LNTPI were o furher vesgaed. Varable Table 6.. Comparso amog Isrumeal Varable Esmaors Based o he Seco 3. Example. Demographc ad Demographc Varables are Srcl Ecoomc Varables are Exogeous Ecoomc Varables are Srcl Exogeous Sequeall Exogeous Parameer Esmaes Modelbased - sasc Parameer Esmaes Modelbased - sasc Robus - sasc Lag LNTAX 0.05 4.6 0.08 3.3.48 Demographc Varables MS -0.35-0.94-0.49-0.4-0.49 HH -.36 -.70 -.357-3. -.7 AGE -0.60-0.34 0.00 0.0 0.0 DEPEND 0.06 0. 0.084 0.73 0.68 Ecoomc Varables LNTPI 0.547 4.53 0.340.9.07 MR 0.6 8.40 0.43 7.34 5. EMP 0.43. 0.85 0.48 0.36 PREP -0.7 -.0-0.87-0.78-0.68 INTERCEPT 0.78 4. 0.5 4.90 3.4 As wh oher srumeal varable procedures a es of he exogee assumpo E W K ε 0 s avalable. The robus verso of he es sasc s TS IV e K W Σˆ IV W K e. Uder he ull hpohess of E W K ε 0 hs es sasc has a asmpoc ch-square dsrbuo wh p T + p T T +/ - K degrees of freedom see Arellao ad Hooré 00E. Moreover oe ca use cremeal versos of hs es sasc o assess he exogee of seleced varables he same maer as paral F-es. Ths s mpora because he umber of mome codos creases subsaall as oe cosders modelg a varable as srcl exogeous ha uses T mome codos compared o he less resrcve sequeal exogee assumpo ha uses T T +/ mome codos. For addoal dscusso o esg exogee usg srumeal varable esmaors we refer o Arellao 003E Balag 00E ad Wooldrdge 00E.

Chaper 6. Sochasc Regressors / 6-7 Seco 6.4 Mulvarae resposes As wh Seco 6. we beg b revewg deas from a o-logudal seg specfcall cross-secoal daa. Seco 6.4. roduces mulvarae resposes he coex of mulvarae regresso. Seco 6.4. descrbes he relao bewee mulvarae regresso ad ses of regresso equaos. Seco 6.4.3 roduces smulaeous equaos mehods. Seco 6.4.4 he apples hese deas o ssems of equao wh error compoes ha have bee proposed he ecoomerc leraure. Seco 6.4. Mulvarae regresso A classc mehod of modelg mulvarae resposes s hrough a mulvarae regresso model. We sar wh a geeral expresso Y X Γ + ε 6.9 where Y s a G marx of resposes X s a K marx of explaaor varables Γ s a G K marx of parameers ad ε s a G marx of dsurbaces. To provde uo cosder he raspose of he h row of equao 6.9 Γ x + ε. 6.0 Here G resposes are measured for each subjec G whereas he vecor of explaaor varables x s K. We all assume ha he dsurbace erms are decall ad depedel dsrbued wh varace-covarace marx Var ε Σ. Example 6. Suppl ad demad To llusrae cosder a sample of coures; for each cour we measure G resposes. Specfcall we wsh o relae he prce of a good provded b supplers ad he qua of hs good demaded o several exogeous measures x s such as come he prce of subsue goods ad so o. We assume ha prce ad qua ma be relaed hrough σ σ Var Var Σ. Specfcall σ measures he covarace bewee prce ad σ σ qua. Usg Γ β β he expresso for equao 6.0 s β x + ε prce β x + ε qua. The ordar leas squares regresso coeffce esmaor of Γ s G OLS x xx. 6. Somewha surprsgl urs ou ha hs esmaor s also he geeralzed leas squares esmaor ad hece he maxmum lkelhood esmaor. Now le β g be he gh row of Γ so ha Γ β β β G. Wh hs oao he gh row of equao 6.0 s g β g x + ε g. We ca calculae he ordar leas squares esmaor of β g as

6-8 / Chaper 6. Sochasc Regressors g OLS xx x b g g G. Thus he esmaor G OLS ca be calculaed o a row-b-row bass ha s usg sadard uvarae respose mulple lear regresso sofware. Noeheless he mulvarae model srucure has mpora feaures. To llusrae b cosderg ses of resposes smulaeousl equao 6.0 we ca accou for relaoshps amog resposes he covarace of he regresso esmaors. For example wh equao 6.0 s sraghforward o show ha Covb gols b kols σ gk xx. Mulvarae regresso aalss ca be appled drecl o logudal daa b reag each observao over me as oe of he G resposes. Ths allows for a vare of seral correlao marces hrough he varace marx Σ. However hs perspecve requres balaced daa G T. Seco 6.4. Seemgl urelaed regressos Eve for cross-secoal daa a drawback of he classc mulvarae regresso s ha he same se of explaaor varables s requred for each pe of respose. To uderscore hs po we reur o our suppl ad demad example. Example 6. Suppl ad demad - coued Suppose ow ha he expeced prce of a good depeds learl o x he purchasers come; smlarl qua depeds o x he supplers wage rae. Tha s we wsh o cosder β 0 + β x + ε prce β 0 + β x + ε qua so ha dffere explaaor varables are used dffere equaos. Le us reorgaze our observed varables so ha g g g s he vecor of he gh respose ad X g s he K g marx of explaaor varables. Ths elds wo ses of regresso equaos X β + ε prce X β + ε qua represeg equaos. Here Var σ I Var σ I ad Cov σ I. Thus here s zero covarace bewee dffere coures e a commo covarace wh a cour bewee prce ad qua of he good. If we ru separae regressos we ge regresso coeffce esmaors b gols X g X g - X g g g. These are ordar leas squares esmaors; he do o accou for he formao σ. The seemgl urelaed regressos echque arbued o Zeller 96E combes dffere ses of regresso equaos ad uses geeralzed leas squares esmao. Specfcall

Chaper 6. Sochasc Regressors / 6-9 suppose ha we sar wh G ses of regresso equaos of he form g X g β g + ε g. To see how o combe hese we work wh G ad defe X 0 β ε X β ad ε. 6. 0 X β ε Thus we have σ I σi Var Var ε σi σ I ad wh hs he geeralzed leas squares esmaor s b GLS X Var - X - X Var -. These are kow as he SUR for seemgl urelaed regresso esmaors. I s eas o check b OLS ha b GLS f eher σ 0 or X X holds. I eher case we have ha he b OLS GLS esmaor s equvale o he OLS esmaor. O oe had he seemgl urelaed regresso se-up ca be vewed as a specal case of mulple lear ad hece mulvarae regresso wh dspla 6.. O he oher had seemgl urelaed regressos ca be vewed as a wa of exedg mulvarae regressos o allow for explaaor varables ha deped o he pe of respose. As we wll see aoher wa of allowg pe specfc explaaor varables s o resrc he parameer marx. Seco 6.4.3 Smulaeous equaos models I our suppl ad demad example we assumed ha prce ad qua were poeall relaed hrough covarace erms. However s ofe he case ha researchers wsh o esmae alerave models ha allow for relaoshps amog resposes drecl hrough he regresso equaos. Example 6. Suppl ad demad - coued Cosder a demad-suppl model ha elds he followg wo ses of equaos: β + γ 0 + γ x + ε prce 6.3 β + γ 0 + γ x + ε. qua Here we assume ha qua learl affecs prce ad vce-versa. As before boh x s are assumed o be exogeous for he demad ad suppl equaos. I Seco 6..3 we saw ha usg ol ordar leas squares a sgle equao produced based esmaors due o he edogeous regressor varables. Tha s whe examg he prce equao he qua varable s clearl edogeous because s flueced b prce as s see he qua equao. Oe ca use smlar reasog o argue ha SUR esmaors also eld based ad cosse esmaors; seemgl urelaed regresso echques mprove upo he effcec of ordar leas squares bu do o chage he aure of he bas esmaors. To roduce esmaors for he equaos dspla 6.3 we collec he depede 0 β varables wh he marx B. Thus we ma express dspla 6.3 as B + Γ x β 0

6-0 / Chaper 6. Sochasc Regressors γ 0 γ 0 + ε where ε ε ε x x x ad Γ. Ths expresso looks ver γ 0 0 γ much lke mulvarae regresso model equao 6.0 he dfferece beg ha we ow have cluded a se of edogeous regressors B. As oed Seco 6.4. we have corporaed dffere regressors dffere equaos b defg a combed se of explaaor varables ad mposg he approprae resrcos o he marx of coeffces β. Ths subseco cosders ssems of regresso equaos where resposes from oe equao ma serve as edogeous regressors aoher equao. Specfcall we cosder model equaos of he form B + Γ x + ε. 6.4 Here we assume ha I-B s a G G o-sgular marx ha Γ has dmeso G K ad he vecor of explaaor varables x s K ad ha Var ε Σ. Wh hese assumpos we ma wre he so-called reduced form Π x + η where Π I-B - Γ η I-B - ε ad Var η Var [I-B - ε ] I-B - Σ I-B - Ω. For β example our suppl-demad example we have I B ad hus ββ β Π γ 0 + βγ 0 γ βγ ββ β γ 0 + γ 0 β γ γ. The reduced form s smpl a mulvarae regresso model as equao 6.0. We wll assume suffce codos o he observables o cossel esmae he reduced form coeffces Π ad he correspodg varace-covarace marx Ω. Thus we wll have formao o he GK elemes Π ad he GG+/ elemes of Ω. However hs formao ad of self wll o allow us o properl def all he elemes of Γ B ad Σ. There are GK G ad GG+/ elemes hese marces respecvel. Addoal resrcos geerall from ecoomc heor are requred. To llusrae our suppl demad example here are sx srucural parameers of eres Γ ad sx elemes of Π. Thus we eed o check o see ha hs provdes suffce formao o recover he releva srucural parameers. Ths process s kow as defcao. Dealed reames of hs opc are avalable ma sources; see for example Greee 00E Haash 000E ad Wooldrdge 00E. Esmaes of Π allow us o recover he srucural parameers Γ ad B. Ths mehod of esmag he srucural parameers s kow as drec leas squares. Aleravel s possble o esmae equao 6.4 drecl usg maxmum lkelhood heor. However hs becomes complex because he parameers B appear boh he mea ad varace. No surprsgl ma alerave esmao procedures are avalable. A commol used mehod s wo-sage leas squares roduced Seco 6..3. For he frs sage of hs procedure oe rus all he exogeous varables o f he resposes. Tha s usg equao 6. calculae ˆ G OLSx x xx x. 6.5 For he secod sage assume ha we ca wre he gh row of equao 6.4 as g B g g + β g x + ε g 6.6

Chaper 6. Sochasc Regressors / 6- where g s wh he gh row omed ad B g s he raspose of he gh row of B omg he dagoal eleme. The we ma calculae ordar leas squares esmaors correspodg o he equao g B g ˆ g + β g x + resdual. 6.7 ˆ g The fed values are deermed from ŷ equao 6.5 afer removg he gh row. Ordar leas squares equao 6.6 s approprae because of he edogeous regressors g. Because he fed values ˆ g are lear combaos of he exogeous varables here s o such edogee problem. Usg equao 6.7 we ca express he wo-sage leas squares esmaors of he srucural parameers as Bˆ g ˆ g ˆ g ˆ g g g G. 6.8 βˆ g x x x Noe ha for hs esmao mehodolog o work he umber of exogeous varables excluded from he gh row mus be a leas as large as he umber of edogeous varables ha appear B g g. Example 6. Suppl ad demad - coued To llusrae we reur o our demad-suppl example. The for g we have B g γ g β g γ 0 γ 0 ad x x x. We calculae fed values for as x xx x. ˆ Smlar calculaos hold for g. The sraghforward subsuo o equao 6.8 elds he wo-sage leas squares esmaors. Seco 6.4.4 Ssems of equaos wh error compoes Ths seco descrbes pael daa exesos o ssems of equaos volvg error compoes o model he heerogee. We frs exame seemgl urelaed regresso models ad he smulaeous equao models. Ol oe-wa error compoes are deal wh. Ieresed readers wll fd addoal deals ad summares of exesos o wo-wa error compoes Balg 00E ad Krshakumar Chaper 9 of Máás ad Sevesre 996E. Seemgl urelaed regresso models wh error compoes Lke seemgl urelaed regressos he cross-secoal seg we ow wsh o sud several error compoe models smulaeousl. Followg he oao of Seco 3. our eres s sudg suaos ha ca be represeed b g α g + x g β g + ε g g G. I our suppl demad example g represeed he prce ad qua equaos whereas represeed he cour. We ow assume ha we follow coures over me so ha T. Assumg ha he x s are he ol exogeous varables ad ha pe ad cour specfc radom

6- / Chaper 6. Sochasc Regressors effecs α g are depede of he dsurbaces erms oe ca alwas use ordar leas squares esmaors of β g ; hese are ubased ad cosse. To compue he more effce geeralzed leas squares we beg b sackg over he G resposes α x 0 0 β ε M M + 0 O 0 M + M G α G 0 0 xg βg ε G ha we wre as α + X β + ε. 6.9 Here we assume ha β has dmeso K so ha X has dmeso G K. Followg coveoal seemgl urelaed regressos we ma allow for covaraces amog resposes hrough he oao Var ε Σ. We ma also allow for covaraces hrough Var α D. Sackg over we have α X ε ha we wre as M M + M β + M T α XT εt α T + X β + ε. Wh hs oao oe ha Var ε blkdagoalvar ε Var ε T Σ I T ad ha Var α T E α T α T D J T. Thus he geeralzed leas squares esmaor for β s D J + Σ I X X D J + Σ I b GLS X T T T T. 6.0 Aver 977E ad Balag 980E cosdered hs model. Smulaeous equao models wh error compoes Exedg he oao of equao 6.4 we ow cosder α + B + Γ x + ε. 6. The subjec-specfc erm s α α α α G ha has mea zero ad varace-covarace marx Var α D. We ma re-wre equao 6. reduced form as Π x + η where Π I-B - Γ ad η I-B - α + ε. Wh hs formulao we see ha he pael daa mea effecs are he same as he model equao 6.4 whou subjec-specfc effecs. Specfcall as poed ou b Hausma ad Talor 98E whou addoal resrcos o he varace or covarace parameers he defcao ssues are he same wh ad whou subjec-specfc effecs. I addo esmao of he reduced form s smlar; deals are provded he revew b Krshakumar Chaper 9 of Máás ad Sevesre 996E. We ow cosder drec esmao of he srucural parameers usg wo-sage leas squares. To beg oe ha he wo-sage leas squares esmaors descrbed equao 6.8 sll provde ubased cosse esmaors. However he do o accou for he error compoes varace srucure ad hus ma be effce. Noeheless hese esmaors ca be

Chaper 6. Sochasc Regressors / 6-3 used o calculae esmaors of he varace compoes ha wll be used he followg esmao procedure. For he frs sage we eed o calculae fed values of he resposes usg ol he exogeous varables as regressors. Noe ha * * β x x 0 0 β * * I B Γx M 0 O 0 M Xβ. * * βgx 0 0 x β G Thus wh equao 6. we ma express he reduced form as I- B - α + I- B - Γ x + I- B - ε α * + X * β* + ε *. Ths has he same form as he seemgl urelaed regresso wh error compoes model equao 6.9. Thus oe ca use equao 6.0 o ge fed regresso coeffces ad hus fed values. Aleravel we have see ha ordar leas squares provdes ubased ad cosse esmaes for hs model. Thus hs echque would also serve for compug he frs sage fed values. For he secod sage wre he model equao 6. he same fasho as equao 6.6 o ge g g B g + x β g + α g + ε g. Recall ha B g s a G- vecor of parameers ad g s wh he gh row omed. Sackg over T elds B g g Y g B g + X β g + α g T + ε g Y g X + αgt + ε. g β g Le Var α g T + ε g σ α J T + σ ε I T. Replacg Y g b Y ˆ g elds he wo-sage leas squares esmaors Bˆ g Yˆ g Yˆ + g Yˆ g σ T T T + T g ˆ α J σ ε I σ α J σ I g G. ε β g X X X Seco 6.5 Smulaeous equao models wh lae varables Smulaeous equao models wh lae varables comprse a broad framework for hadlg complex ssems of equaos as well as logudal daa models. Ths framework ha orgaed he work of Jöreskog Keeslg ad Wle see Bolle 989EP page 6 s wdel appled socolog pscholog ad educaoal sceces. Lke he roducos o smulaeous equaos Secos 6.4.3 ad 6.4.4 he ssems of equaos ma be used for boh dffere pes of resposes mulvarae as well as dffere mes of resposes logudal or boh. The esmao echques preferred pscholog ad educao are kow as covarace srucure aalss. To keep our reame self-coaed we frs oule he framework Seco 6.5. he cross-secoal coex. Seco 6.5. he descrbes logudal daa applcaos.

6-4 / Chaper 6. Sochasc Regressors Seco 6.5. Cross-secoal models We beg b assumg ha x are decall ad depedel dsrbued observable draws from a populao. We rea x as he exogeous vecor ad s he edogeous vecor. The x-measureme equao s x τ x + Λ x ξ + δ. 6. Here δ s he dsurbace erm ξ s a vecor of lae uobserved varables ad Λ x s a marx of regresso coeffces ha relaes ξ o x. I a measureme error coex we mgh use Λ x I ad erpre τ x + ξ o be he rue values of he observables ha are corruped b he error δ. I oher applcaos deas from classc facor aalss drve he model se-up. Tha s here ma be ma observable exogeous measuremes ha are drve b relavel few uderlg lae varables. Thus he dmeso of x ma be large relave o he dmeso of ξ. I hs case we ca reduce he dmeso of he problem b focusg o he lae varables. Moreover he lae varables more closel correspod o socal scece heor ha do he observables. To complee he specfcao of hs measureme model we assume ha E ξ µ ξ ad E δ 0 so ha E x τ x + Λ x µ ξ. Furher we use Var δ Θ δ ad Var ξ Φ so ha Var x Λ x Φ Λ x + Θ δ. Tha s ξ ad δ are muuall ucorrelaed. Specfcao of he -measureme equao s smlar. Defe τ + Λ η + ε. 6.3 Here ε s he dsurbace erm η s a vecor of lae varables ad Λ s a marx of regresso coeffces ha relaes η o. We use Θ ε Var ε ad assume ha η ad ε are muuall ucorrelaed. Noe ha equao 6.3 s o a usual mulple lear regresso equao because regressor η s uobserved. To lk he exogeous ad edogeous lae varables we have he srucural equao η τ η + Bη + Γξ + ς. 6.4 Here τ η s a fxed ercep B ad Γ are marces of regresso parameers ad ς s a mea zero dsurbace erm wh secod mome Var ς Ψ. A drawback of a srucural equao model wh lae varables summarzed equaos 6.-6.4 s ha s overparameerzed ad oo uweld o use whou furher resrcos. The correspodg advaage s ha hs model formulao capures a umber of dffere models uder a sgle srucure hus makg a desrable framework for aalss. Before descrbg mpora applcaos of he model we frs summarze he mea ad varace parameers ha are useful hs model formulao. Mea Parameers Wh E ξ µ ξ from equao 6.4 we have E η τ η + B E η + E Γξ + ς τ η + B E η + Γ µ ξ. Thus E η I - B - τ η + Γ µ ξ assumg ha I - B s verble. Summarzg we have E x τ x + Λ xµ ξ. 6.5.4 E τ + Λ I B τη + Γµ ξ Covarace Parameers From equao 6.4 we have η I - B - τ η + Γξ + ς ad Var η Var I B Γξ + ζ I B Var Γξ + ζ I B I B ΓΦΓ + Ψ I B.

Chaper 6. Sochasc Regressors / 6-5 Thus wh equao 6.3 we have Var Λ +. Var η Λ + Θε Λ I B ΓΦΓ + Ψ I B Λ Θε Wh equao 6. we have Cov x Cov Λ η + ελ ξ + δ Λ Cov ηξ Λ Summarzg we have Var Cov x Cov x Var x Λ I B x I B Γξξ Λ Λ I B ΓΦΛ x Λ Cov. ΓΦΓ + Ψ I B Λ + Θε Λ I B ΓΦΛ x + Λ xφγ I B Λ Λ xφλ x Θδ x. 6.5 Idefcao ssues Wh he radom samplg assumpo oe ca cossel esmae he meas ad covaraces of he observables specfcall he lef-had sdes of equao 6.5.4 ad 6.5. The model parameers are gve erms of he rgh-had sdes of hese equaos. There are geerall more parameers ha ca be uquel defed b he daa. Idefcao s demosraed b showg ha he ukow parameers are fucos ol of he meas ad covaraces ad ha hese fucos lead o uque soluos. I hs case we sa ha he ukow parameers are defed. Oherwse he are sad o be uderdefed. There are ma approaches avalable for hs process. We wll llusrae a few cojuco wh some specal cases descrbed below. Dealed broad reames of hs opc are avalable ma sources; see for example Bolle 989EP. Specal cases As oed above he model summarzed equaos 6.-6.4 s overparameerzed ad oo uweld o use alhough does ecompass ma specal cases ha are drecl releva for applcaos. To provde focus ad uo we ow summarze a few of hese specal cases. Cosder ol he x-measureme equao. Ths s he classc facor aalss model see for example Johso ad Wcher 999G. Assume ha boh x ad are used drecl he srucural equao model whou a addoal lae varables. Tha s assume x ξ ad η. The equao 6.4 represes a srucural equao model based o observables roduced Seco 6.4.3. Moreover assumg he B 0 he srucural equao model wh observables reduces o he mulvarae regresso model. Assume ha s used drecl he srucural equao model bu ha x s measured wh error so ha x τ x + ξ + δ. Assumg o feedback effecs for so ha B 0 he equao 6.4 represes he classc errors varables model. Ma oher specal cases appear he leraure. Our focus s o logudal specal cases Seco 6.5..

6-6 / Chaper 6. Sochasc Regressors Pah dagrams The popular of srucural equao models wh lae varables educao ad pscholog s due par o pah dagrams. Pah dagrams due o Sewall Wrgh 98B are pcoral represeaos of ssem of equaos. These dagrams show he relaos amog all varables cludg dsurbaces ad errors. These graphcal relaos allow ma users o readl udersad he cosequeces of modelg relaoshps. Moreover sascal sofware roues have bee developed o allow aalss o specf he model graphcall whou resorg o algebrac represeaos. Table 6. summarzes he prmar smbols used o make pah dagrams. Table 6. Prmar smbols used pah dagrams Recagular or square box sgfes a observed varable x η ς. ε η ξ ξ Crcle or ellpse sgfes a lae varable Ueclosed varable sgfes a dsurbace erm error eher he srucural equao or measureme equao Sragh arrow sgfes a assumpo ha he varable a he base of arrow causes varable a head of arrow Curved wo-headed arrow sgfes a assocao bewee wo varables η η Source: Bolle 989EP Two sragh sgle-headed arrows coecg wo varables sgfes a feedback relao or recprocal causao Esmao echques Esmao s pcall doe usg maxmum lkelhood assumg ormal; somemes usg srumeal varable esmao for al values. Descrpos of alerave echques cludg geeralzed leas squares ad uweghed leas squares ca be foud Bolle 989EP. Ieresgl he lkelhood esmao s cusomar doe b maxmzg he lkelhood over all he observables. Specfcall assumg ha x are jol mulvarae ormal wh momes gve equaos 6.5.4 ad 6.5 oe maxmzes he lkelhood over he ere sample. I coras mos of he maxmum lkelhood esmao preseed hs ex has bee for he lkelhood of he respose or edogeous varables codoal o he exogeous observables. Specfcall suppose ha observables coss of exogeous varables x ad edogeous varables. Le θ be a vecor of parameers ha dexes he codoal dsrbuo

Chaper 6. Sochasc Regressors / 6-7 of he edogeous varables gve he exogeous varables sa p x θ. Assume ha here s aoher se of parameers θ ha are urelaed o θ ad ha dexes he dsrbuo of he exogeous varables sa p x θ. Wh hs se-up he complee lkelhood s gve b p x θ p x θ. If our eres s ol he parameers ha fluece he relaoshp bewee x ad we ca be coe wh maxmzg he lkelhood wh respec o θ. Thus he dsrbuo of he exogeous varables p x θ s o releva o he eres a had ad ma be gored. Because of hs phlosoph our pror examples hs ex we dd o cocer ourselves wh he samplg dsrbuo of he x s. See Egle e al. 983E. Seco 7.4. wll dscuss hs furher. Alhough he requreme ha he wo ses of parameers be urelaed s a resrcve assumpo ha s geerall o esed provdes he aals some mpora freedoms. Wh hs assumpo he samplg dsrbuo of he exogeous varables does o provde formao abou he codoal relaoshp uder vesgao. Thus we eed o make resrcve assumpos abou he shape of hs samplg dsrbuo of he exogeous varables. As a cosequece we eed o model he exogeous varables as mulvarae ormal or eve requre ha he be couous. To llusrae a major dsco bewee he mulple lear regresso model ad he geeral lear model s ha he laer formulao easl hadles caegorcal regressors. The geeral lear model s abou he codoal relaoshp bewee he respose ad he regressors mposg few resrcos o he behavor of he regressors. For he srucural equao model wh lae varables he parameers assocaed wh he dsrbuo of exogeous varables are θ {τ x µ ξ Λ x Θ δ Φ}. Assumg for example mulvarae ormal oe ca use equaos 6.5.4 ad 6.5 o compue he codoal lkelhood of x Appedx B. However s dffcul o wre dow a se of parameers θ ha are a subse of he full model parameers ha are o relaed o θ. Thus maxmum lkelhood for he srucural equaos model wh lae varables requres maxmzg he full lkelhood over all observables. Ths s coras o our frs look a srucural equaos models Seco 6.5.3 whou measureme equaos 6.5. ad 6.5. where we could solae he codoal model parameers from he samplg dsrbuo of he exogeous varables. To summarze maxmum lkelhood esmao for srucural equao models wh lae varables cusomarl emplos maxmum lkelhood esmao where he lkelhood s wh respec o all he observables. A full model of boh he exogeous ad edogeous varables s specfed; maxmum lkelhood esmaors are well kow o ejo ma opmal properes. A cosequece of hs specfcao of he exogeous varables s ha s dffcul o hadle caegorcal varables; mulvarae dsrbuos for dscree varables are much less well udersood ha he couous varable case. Seco 6.5. Logudal daa applcaos The srucural equao model wh lae varables provdes a broad framework for modelg logudal daa. Ths seco provdes a seres of examples o demosrae he breadh of hs framework. Specal case - Auoregressve model Suppose ha represes he readg abl of he h chld; assume ha we observe he se of chldre over 4 me perods. We mgh represe a chld s readg abl as β 0 + β - + ς for 3 4 6.6 usg as he basele measureme. We summarzes hese relaos usg

6-8 / Chaper 6. Sochasc Regressors β0 0 0 0 0 ς β0 β 0 0 0 ς η + + τ + Bη + ς η. 3 β0 0 β 0 0 3 ς 3 4 β0 0 0 β 0 4 ς 4 Thus hs model s a specal case of equaos 6.-6.4 b choosg η ad Γ 0. Graphcall we ca express hs model as: 3 4 ς ς ς 3 ς 4 Fgure 6. Pah dagram for he model equao 6.6 Ths basc model could be exeded several was. For example oe ma wsh o cosder evaluao a uequall spaced me pos such as 4h 6h 8h ad h grades. Ths suggess usg slope coeffces ha deped o me. I addo oe could also use more ha oe lag predcor varable. For some oher exesos see he followg couao of hs basc example. Specal case - Auoregressve model coued. Auoregressve model wh lae varables ad mulple dcaors Suppose ow ha readg abl s cosdered a lae varable deoed b η ad ha we have wo varables ad ha measure hs abl kow as dcaors. The dcaors follow a measureme error model j λ 0 + λ η + ε j 6.7a ad he lae readg varable follows a auoregressve model η β 0 + β η - + ς. 6.7b Wh he oao 3 3 4 4 ε defed smlarl ad η η η η η η 3 η 3 η 4 η 4 we ca express equao 6.7a as equao 6.3. Here τ s λ 0 mes a vecor of oes ad Λ s λ mes a de marx. Equao 6.7b ca be expressed as he srucural equao 6.4 usg oao smlar o equao 6.6b. Graphcall we ca express equaos 6.7a ad 6.7b as:

Chaper 6. Sochasc Regressors / 6-9 ε ε ε ε ε 3 ε 3 ε 4 ε 4 3 3 4 4 η η η 3 η 4 ς ς ς 3 ς 4 Fgure 6. Pah dagram for he model equaos 6.7a ad 6.7b Specal case - Auoregressve model - coued. Auoregressve model wh lae varables mulple dcaors ad predcor varables. Coug wh hs example we mgh also wsh o corporae exogeous predcor varables such as x he umber of hours readg ad x a rag of emooal suppor gve a chld s home. Oe wa of dog hs s o smpl add hs as a predcor he srucural equao replacg equao 6.5.7b b η β 0 + β η - + γ x + γ x + ς. Jöreskog ad Goldberger 975S roduced a model wh mulple dcaors of a lae varable ha could be explaed b mulple causes; hs s ow wdel kow as a MIMIC model. Growh curve models Iegrag srucural equao modelg wh lae varables ad logudal daa modelg has bee he subjec of exesve research rece ears; see for example Duca e al. 999EP. Oe wdel adoped approach cocers growh curve modelg. Example 5. Deal daa coued. I he Seco 5. Deal daa example we used o represe he deal measureme of he h chld measures a four mes correspodg o ages 8 0 ad 4 ears of age. Usg srucural equao modelg oao we could represe he -measureme equao as 8 ε 0 β 0 ε + Λ η + ε. 3 β ε 3 4 4 ε 4 Noe ha hs represeao assumes ha all chldre are evaluaed a he same ages ad ha hese ages correspod o a kow parameer marx Λ. Aleravel oe could form groups so ha all chldre wh a group are measured a he same se of ages ad le Λ var b group. Takg τ η 0 B 0 ad usg he x measureme drecl whou error we ca wre he srucural equao as

6-30 / Chaper 6. Sochasc Regressors β β β ς 0 00 0 0 η + Γx + β β0 β GENDER ς Thus hs model serves o express erceps ad growh raes assocaed wh each chld β 0 ad β as a fuco of geder. ς. Wlle ad Saer 994EP roduced growh curve modelg he coex of srucural equaos wh lae varables. There are several advaages ad dsadvaages whe usg srucural equaos o model growh curves whe compared o our Chaper 5 mullevel models. The ma advaage of srucural equao models s ha s sraghforward o corporae mulvarae resposes. To llusrae our deal example here ma be more ha oe deal measureme of eres or ma be of eres o model deal ad vsual acu measuremes smulaeousl. The ma dsadvaage of srucural equao models also relaes o s mulvarae respose aure; s dffcul o hadle ubalaced srucure wh hs approach. If chldre came o he clc for measuremes a dffere ages hs would complcae he desg cosderabl. Moreover f o all observaos were o avalable ssues of mssg daa are more dffcul o deal wh hs coex. Fall we have see ha srucural equaos wh lae varables mplcl assumes couous daa ha ca be approxmaed b mulvarae ormal; f he predcor varables are caegorcal such as geder hs poses addoal problems. Furher readg Oher roducos o he cocep of exogee ca be foud mos graduae ecoomercs exs; see for example Greee 00E ad Haash 000E. The ex b Wooldrdge 00E gves a roduco wh a specal emphass o pael daa. Arellao ad Hooré 00E provde a more sophscaed overvew of pael daa exogee. The colleco of chapers Máás ad Sevesre 996E provde aoher perspecve as well as a roduco o srucural equaos wh error compoes. For oher mehods for hadlg edogeous regressors wh heerogee erms we refer o Arellao 003E Balag 00E ad Wooldrdge 00E. There are ma sources ha roduce srucural equaos wh lae varables. Bolle 989EP s a wdel ced source ha has bee avalable for ma ears.

Chaper 6. Sochasc Regressors / 6-3 Appedx 6A. Lear Projecos Suppose ha {x } s a radom vecor wh fe secod momes. The he lear projeco of oo x s x E x x - E x provded ha E x x s verble. To ease oao we defe β E x x - E x ad deoe hs projeco as L x x β. Ohers such as Goldberger 99E use E* x o deoe hs projeco. B he lear of expecaos s eas o check ha L. x s a lear operaor. As a cosequece f we defe ε hrough he equao x β + ε he E* ε x 0. Noe ha hs resul s a cosequece of he defo o a model assumpo ha requres checkg. Suppose ha we ow assume ha he model s of he form x β + ε ad ha E x x s verble. The he codo E ε x 0 s equvale o checkg he codo ha Lε x 0. Tha s checkg for a correlao bewee he dsurbace erm ad he predcors s equvale o checkg ha he lear projeco of he dsurbace erm o he predcors s zero. Lear projecos ca be jusfed as mmum mea square predcors. Tha s he choce of β ha mmzes E - x β s β E x x - E x. As a example suppose ha we have a daa se of he form {x } ad defe he expecao operaor o be ha probabl dsrbuo ha assgs / probabl o each oucome he emprcal dsrbuo. The we are aempg o mmze E x β xβ. The soluo s E x x E x x x x he famlar ordar leas squares esmaor.

Chaper 7. Modelg Issues / 7- Chaper 7. Modelg Issues 003 b Edward W. Frees. All rghs reserved Absrac. As roduced Chaper logudal ad pael daa are ofe heerogeeous ad ma suffer from problems of aro. Ths chaper descrbes models for hadlg hese edeces as well as models desged o hadle omed varable bas. Heerogee ma be duced b fxed effecs radom effecs or 3 wh subjec covaraces. Dsgushg amog hese mechasms ca be praccall dffcul alhough as he chaper pos ou s o alwas ecessar. Moreover he chaper descrbes he well-kow Hausma es for dsgushg bewee esmaors based o fxed versus radom effecs. As poed ou b Mudlak 978aE he Hausma es provdes a es of he sgfcace of me-cosa omed varables. Ths chaper emphaszes ha a mpora feaure of logudal ad pael daa s he abl o hadle cera pes of omed varables. Ths abl s oe of he mpora beefs of usg logudal ad pael daa; coras aro s oe of he ma drawbacks. The chaper revews mehods for deecg bases arsg due o aro ad roduces models ha provde correcos for aro dffcules. 7. Heerogee Heerogee s a commo feaure of ma logudal ad pael daa ses. Whe we hk of logudal daa we hk of repeaed measuremes o subjecs. Ths ex emphaszes repeaed observaos over me alhough oher pes of cluserg are of eres. For example oe could model he faml u as a subjec ad have dvdual measuremes of faml members as he repeaed observaos. Smlarl oe could have a geographc area such as a sae as he subjec ad have dvdual measuremes of ows as he repeaed observaos. Regardless of he aure of he repeo he commo heme s ha dffere observaos from he same subjec or observaoal u ed o be relaed o oe aoher. I coras he word heerogee refers o hgs ha are ulke or dssmlar. Thus whe dscussg heerogee he coex of logudal daa aalss we mea ha observaos from dffere subjecs ed o be dssmlar whereas observaos from he same subjec ed o be smlar. We refer o models whou heerogee compoes as homogeous. Two approaches o modelg heerogee I mulvarae aalss here are ma mehods for quafg relaoshps amog radom varables. The goal of each mehod s o udersad he jo dsrbuo fuco of radom varables; dsrbuo fucos provde all he deals o he possble oucomes of radom varables boh solao of oe aoher ad as he occur jol. There are several mehods for cosrucg mulvarae dsrbuos see Hougaard 987G ad Huchso ad La 990G for dealed revews. For appled logudal daa aalss we focus o wo dffere mehods of geerag jol depede dsrbuo fucos hrough commo varables ad covaraces. Wh he varables--commo echque for geerag mulvarae dsrbuos a commo eleme serves o duce depedeces amog several radom varables. We have

7- / Chaper 7. Modelg Issues alread used hs modelg echque exesvel begg Chapers ad 3. Here we used he vecor of parameers α o deoe me-cosa characerscs of a subjec. I Chaper he fxed parameers duced smlares amog dffere observaos from he same subjec hrough he mea fuco. I Chaper 3 he radom vecors α duced smlares hrough he covarace fuco. I each case α s commo o all observaos wh a subjec ad hus duces a depedec amog hese observaos. Alhough subjec-specfc commo varables are wdel used for modelg heerogee he do o cover all he mpora logudal daa applcaos. We have alread dscussed me-specfc varables Chaper deoed as λ ad wll exed hs dscusso o crossclassfed daa Chaper 8 ha s corporag boh subjec-specfc ad me-specfc commo varables. Aoher mpora area of applcaos volves clusered daa descrbed Chaper 5. We ca also accou for heerogee b drecl modelg he covarace amog observaos wh a subjec. To llusrae Seco 3. o error compoes we saw ha a commo radom α duced a posve covarace amog observaos wh a subjec. We also saw ha we could model hs feaure usg he compoud smmer correlao marx. The advaage of he compoud smmer covarace approach s ha also allows for models of egave depedece. Thus modelg he jo relao amog observaos drecl usg covaraces ca be smpler ha a approach usg commo varables ad ma also cover addoal dsrbuos. Furher for seral correlaos modelg covaraces drecl s much smpler ha alerave approaches. We kow ha for ormall dsrbued daa modelg he covarace fuco ogeher wh he mea s suffce o specf he ere jo dsrbuo fuco. Alhough hs s o rue geeral correcl specfg he frs wo momes suffces for much appled work. We ake up hs ssue furher Chapers 9 ad 0 whe dscussg he geeralzed esmag equaos approach. Praccal defcao of heerogee ma be dffcul For ma logudal daa ses a aals could cosder ma alerave sraeges for modelg he heerogee. Oe could use subjec-specfc erceps ha ma be fxed or radom. Oe could use subjec-specfc slopes ha aga ma be fxed or radom. Aleravel oe could use covarace specfcaos o model he edec for observaos from he same subjec o be relaed. As he followg llusrao from Joes 993S shows ma be dffcul o dsgush amog hese alerave models whe ol usg he daa o ad model specfcao. Fgure 7. shows paels of mes seres plos for 3 subjecs. The daa are geeraed wh o seral correlao over me bu wh hree dffere subjec-specfc parameers α 0 α ad α 3 -. Wh perfec kowledge of he subjec-specfc parameers oe would correcl use a scalar mes he de marx for he covarace srucure. However f hese subjecspecfc varables are gored a correlao aalss shows a srog posve seral correlao. Tha s from he frs pael Fgure 7. we see ha observaos ed o oscllae abou he overall mea of zero a radom fasho. However he secod pael shows ha all observaos are above zero ad he hrd pael dcaes ha almos all observaos are below zero. Thus a aalss whou subjec-specfc erms would dcae srog posve auocorrelao. Alhough o he correc formulao a me seres model such as he AR model would serve o capure he heerogee he daa.

Chaper 7. Modelg Issues / 7-3 Subjec α 0 6 4 0 - -4 5 5 5 Subjec α 6 4 0 - -4 5 5 5 Subjec 3 α 3 6 4 0 - -4 5 5 5 Tme Fgure 7.. Dffere subjec-specfc parameers ca duce posve seral correlao. Theorecal defcao wh heerogee ma be mpossble Thus defg he correc model formulao o represe heerogee ca be dffcul. Moreover he presece of heerogee defg all he model compoes ma be mpossble. For example our rag lear model heor has led us o beleve ha wh N observaos ad p lear parameers we requre ol ha N>p order o def varace compoes. Alhough hs s rue cross-secoal regresso he more complex logudal daa seg requres addoal assumpos. The followg example due o Nema ad Sco 948E llusraes some of hese complexes. Example Nema-Sco o defcao of varace compoes Cosder he fxed effecs model α + ε where Var ε σ ad Cov ε ε σ ρ. The ordar leas squares esmaor of α s + /. Thus he resduals are e - / ad e - / - e. Because of hese relaos urs ou ha ρ cao be esmaed despe havg - degrees of freedom avalable for esmag he varace compoes. Esmao of regresso coeffces whou complee defcao s possble Foruael complee model defcao s o requred for all ferece purposes. To llusrae f our ma goal s o esmae or es hpoheses abou he regresso coeffces he we do o requre kowledge of all aspecs of he model. For example cosder he oe-wa fxed effecs model α + X β + ε.

7-4 / Chaper 7. Modelg Issues For he balaced case T T Kefer 980E showed how o cossel esmae all compoes of Var ε R ha are eeded for ferece abou β. Tha s appl he commo rasformao marx Q I T - J o each equao o ge * Q Q X β + Q ε X * β + ε * because Q 0. Noe ha Var ε * Q R Q R*. Wh hs rasformed equao he populao parameers β ca be cossel ad roo- esmaed. Furher elemes of R* ca be cossel esmaed ad used o ge feasble geeralzed leas squares esmaors. Example - Nema-Sco - coued 0 - Here we have T Q - ad 0 - σ σ ρ ρ R σ. Thus σ ρ σ ρ σ ρ σ - ρ R* Q R Q 4 ρ. Usg mome based esmaors we ca esmae he qua σ - ρ e cao separae he erms σ ad ρ. Ths example shows ha he balaced basc fxed effecs model feasble geeralzed leas squares esmao s possble eve whou complee defcao of all varace compoes. More geerall cosder he case where we have ubalaced daa ad varable slopes represeed wh he model Z α + X β + ε where Var ε R. For hs model Seco.5.3 we roduced he rasformao Q I Z Z Z Z. Applg hs rasform o he model elds * Q Q X β + Q ε X * β + ε * because Q Z 0. Noe ha Var ε * Q R Q R *. For hs model we see ha a ordar leas squares esmaor of β s ubased ad roo- cosse. Whou kowledge of he varace compoes R * oe ca sll use robus sadard errors o ge asmpocall correc cofdece ervals ad ess of hpoheses. The case for feasble geeralzed leas squares s more complex. Now f R σ I he R * s kow up o a cosa; hs case he usual geeralzed leas squares esmaor gve equao.6 s applcable. For oher suaos Kug 996O provded suffce codos for he defcao ad esmao of a feasble geeralzed leas squares esmaor. 7. Comparg fxed ad radom effecs esmaors I Seco 3. we roduced he samplg ad fereal bases for choosg a radom effecs model. However here are ma saces whe hese bases do o provde suffce gudace o dcae whch pe of esmaor fxed or radom effecs he aals should emplo. I Seco 3. we saw ha he radom effecs esmaor s derved usg geeralzed leas squares ad hus has mmum varace amog all ubased lear esmaors. However Seco 6. we

Chaper 7. Modelg Issues / 7-5 saw ha fxed effecs esmaors do o rel o assumpo SEC6 zero correlao bewee he me-cosa heerogee varables ad he regressor varables. Ofemes aalss look o feaures of he daa o provde addoal gudace. Ths seco roduces he well-kow Hausma es for decdg wheher o use a fxed or radom effecs esmaor. The Hausma 978E es s based o a erpreao due o Mudlak 978aE ha he fxed effecs esmaor s robus o cera omed varable model specfcaos. Throughou hs seco we maa assumpo SEC he src exogee of he regressor varables codoal o he uobserved effecs. To roduce he Hausma es we frs reur o a verso of our Seco 3. error compoes model α + x β + ε + u. 3.* Here as Seco 3. α s a radom varable ha s ucorrelaed wh he dsurbace erm ε ad he explaaor varables x see Chaper 6. However we have also added u a erm for uobserved omed varables. Ulke α he cocer s ha he u qua ma represe a fxed effec or a radom effec ha s correlaed wh eher he dsurbace erms or explaaor varables. If u s prese he he heerogee erm α * α + u does o sasf he usual assumpos requred for ubased ad cosse regresso coeffce esmaors. We do however resrc he omed varables o be me-cosa; hs assumpo allows us o derve ubased ad cosse esmaors eve he presece of omed varables. Takg averages over me equao 3.* we have α + x β + ε + u. Subracg hs from equao 3.* elds x β + ε ε x. Based o hese devaos we have removed he effecs of he uobserved varable u. Thus he equao.6 fxed effecs esmaor T T b FE x x x x x x s o corruped b u ; urs ou o be ubased ad cosse eve he presece of omed varables. For oaoal purposes we have added he subscrp FE o sugges he movao of hs esmaor eve hough here ma o be a fxed effecs α equao 3.*. The Hausma es sasc compares he robus esmaor b FE o he geeralzed leas squares esmaor b EC. Uder he ull hpohess of o omed varables b EC s more effce because s a geeralzed leas squares esmaor. Uder he alerave hpohess of mecosa omed varables b FE s sll ubased ad cosse. Hausma 978E showed ha he sasc χ b b Var b Var b b b FE FE GLS FE has a asmpoc ch-square dsrbuo wh K degrees of freedom uder he ull hpohess. I s a wdel used sasc for deecg omed varables. GLS FE GLS

7-6 / Chaper 7. Modelg Issues Example: Icome ax pames To llusrae he performace of he fxed effecs esmaors ad omed varable ess hs seco exames daa o deermas of come ax pames roduced Seco 3.. Specfcall we beg wh he error compoes model wh K 8 coeffces esmaed usg geeralzed leas squares. The parameer esmaes as well as he correspodg -sascs appear Table 7.. Also Table 7. are he correspodg fxed effecs esmaors. The fxed effecs esmaors are robus o me-cosa omed varables. The sadard errors ad correspodg - sascs are compued a sraghforward fasho usg he procedures descrbed Chaper. Comparg he fxed ad radom effecs esmaors Table 7. we see ha he coeffces are qualavel smlar. For each varable he esmaors have he same sg ad smlar orders of magude. The also dcae he same order of sascal sgfcace. To llusrae he wo measures of come LNTPI ad MR are boh srogl sascall sgfca uder boh esmao procedures. The oe excepo s he EMP varable; hs s srogl sascall sgfcal egave usg he radom effecs esmao bu s o sascall sgfca usg he fxed effecs esmao. To assess he overall dffereces amog he coeffce esmaes we ma use he omed varable es sasc due o Hausma 978E. As dcaed Table 7. hs es sasc urs χ FE ou o be 6.0. Comparg hs es sasc o a ch-square dsrbuo wh K 8 degrees of freedom he p-value assocaed wh hs es sasc s Pr ob χ > 6.0 0. 6448. Ths does o provde eough evdece o dcae a serous problem wh omed varables. Thus he radom effecs esmaor s preferred. Table 7.. Comparso of Radom Effecs Esmaors o Robus Aleraves. Based o he Seco 3. Example. Model wh Varable Ierceps bu o Varable Slopes Error Compoes Robus Fxed Effecs Esmao Radom Effecs Esmao Varable Parameer Esmaes -sasc Parameer Esmaes -sasc LNTPI 0.77 9.30 0.760 0.9 MR 0. 3.55 0.5 5.83 MS 0.07 0.8 0.037 0. HH -0.707 -.7-0.689 -.98 AGE 0.00 0.0 0.0 0.0 EMP -0.44-0.99-0.505-3.0 PREP -0.030-0.8-0.0-0.9 DEPEND -0.069-0.83-0.3 -.9 χ FE 6.0 Because of he complexes ad he wdespread usage of hs es he ecoomercs leraure we spl he remader of he dscusso o wo pars. The frs par Seco 7.. roduces some mpora addoal deas he coex of a specal case. Here we show a relaoshp bewee fxed ad radom effecs esmaors roduce Mudlak s alerave hpohess derve he fxed effecs esmaor uder hs hpohess ad dscuss Hausma s es of omed varables. The secod par Seco 7.. exeds he dscusso o corporae ubalaced daa ma varables 3 varable slopes as well as 4 poeal seral correlao ad heeroscedasc. Seco 7.3 wll dscuss alerave samplg bases.

Chaper 7. Modelg Issues / 7-7 7.. A specal case Cosder he Seco 3. error compoes model wh K so ha α + β 0 + β x + ε 7. where boh {α } ad {ε } are..d. as well as depede of oe aoher. For smplc we also assume balaced daa so ha T T. Igorg he varabl {α } he usual ordar leas squares esmaor of β s b T x x T x x HOM. Because hs esmaor excludes he heerogee compoe {α } we label usg he subscrp HOM for homogeeous. I coras from Exercse 7.5 a expresso for he geeralzed leas squares esmaor s b T * * * * x x T * * x x EC / / * where σ * x x x ad σ Tσ α + σ. As descrbed Tσ α + σ Seco 3. boh esmaors are ubased cosse ad asmpocall ormal. Because b EC s a geeralzed leas squares esmaor has a smaller varace ha b HOM. Tha s Var b EC Var b HOM where * * T x x Var b EC σ. 7. Also oe ha b EC ad b HOM are approxmael equvale whe he heerogee varace s small. Formall because x * x ad * as σ α 0 we have ha b EC b HOM as σ α 0. Ths seco also cosders a alerave esmaor T FE x x T x x b. 7.3 Ths esmaor could be derved from he model equao 7. b assumg ha he erms {α } are fxed o radom compoes. I he oao of Chaper we ma assume ha α * α + β 0 are he fxed compoes ha are o ceered abou zero. A mpora po of hs seco s ha he esmaor defed equao 7.3 s ubased cosse ad asmpocall ormal uder he model ha cludes radom effecs equao 7.. Furher sraghforward calculaos show ha Var T b FE σ x x. 7.4 We oe ha b EC ad b FE are approxmael equvale whe he heerogee varace s large. Formall because x * x -x ad * - as σ α we have ha b EC b FE as σ α. To relae he radom ad fxed effecs esmaors we defe he so-called bewee groups esmaor

7-8 / Chaper 7. Modelg Issues x x b. 7.5 B x x Ths esmaor ca be movaed b averagg all observaos from a subjec ad he compug a ordar leas squares esmaor usg he daa { x }. As wh he oher esmaors hs esmaor s ubased cosse ad asmpocall ormal uder he equao 7. model. Furher sraghforward calculaos show ha T + σ T x x Var b B σ α. 7.6 To erpre he relaos amog b EC b FE ad b B we ce he followg decomposo due o Maddala 97E b EC - b FE + b B. 7.7 Here he erm Var b EC measures he relave precso of he wo esmaors of β. Var b B Because b EC s he geeralzed leas squares esmaor we have ha Var b EC Var b B so ha 0. Omed varables model of correlaed effecs Thus assumg he radom effecs model equao 7. s a adequae represeao we expec each of he four esmaors of β o be close o oe aoher. However ma daa ses hese esmaors ca dffer dramacall. To expla hese dffereces Mudlak 978aE proposed wha we wll call a model of correlaed effecs. Here we erpre α o represe me-cosa or permae characerscs of ha are uobserved ad hece omed. Mudlak roduced he possbl ha {α } are correlaed wh he observed varables x. Tha s he lae varables α fal he exogee assumpo SE6 descrbed Seco 6... To express he relaoshp bewee α ad x we cosder he fuco E [α x ]. Specfcall for our specal case Mudlak assumed ha α η + γ x where {η } s a..d. sequece ha s depede of {ε }. Thus he model of correlaed effecs s η + β 0 + β x + γ x + ε. 7.8 Uder hs model oe ca show ha he geeralzed leas squares esmaor of β s b FE. Furher he esmaor b FE s ubased cosse ad asmpocall ormal. I coras he esmaors b HOM b B ad b EC are based ad cosse. To compare he model of correlaed effecs equao 7.8 wh he basele model equao 7. we eed ol exame he ull hpohess H 0 : γ 0. Ths s cusomarl doe usg he Hausma 978E es sasc b EC b FE χ FE. 7.9 Varb FE - Varb EC Uder he ull hpohess of he model equao 7. hs es sasc has a asmpoc as ch-square dsrbuo wh degree of freedom. Ths provdes he bass for comparg he wo models. Moreover we see ha he es sasc wll be large whe here s a large dfferece bewee he fxed ad radom effecs esmaors. I addo s sraghforward o cosruc he es sasc based o a f of he radom effecs model equao 7. o ge b EC ad Var b EC ad a f of he correspodg fxed effecs model o ge b FE ad Var b FE. Thus oe eed o cosruc he augmeed varable x equao 7.8.

Chaper 7. Modelg Issues / 7-9 7.. Geeral case Exeso o he geeral case follows drecl. To corporae erceps we use a modfcao of he lear mxed effecs model equao 3.5. Thus we assume ha E α α ad re-wre he model as Z α + X β + ε * 7.0 where ε * ε + Z α - α ad Var ε * Z D Z + R V. Sraghforward calculaos Exercse 7.3 show ha he geeralzed leas squares esmaor of β s GLS GLS Z V Z Z V Z X V X V C b wh GLS X Z V Z Z V Z X V X X V C. From equao.6 we have ha he correspodg fxed effecs esmaor s Z FE FE / / R Q R X C b where Z FE / / X R Q R X C ad / / Z R Z Z R Z Z R I Q. From Exercse 7.5 he bewee-groups esmaor s B B V Z Z Z V Z V X V Z Z V Z Z V X C b where B X V Z Z V Z Z V X X V Z Z V Z Z V X C. To relae hese hree esmaors he exeso of Maddala s 97E resul s B FE GLS b b I b + where Var Var B b GLS b Var B B C b ad Var GLS GLS C b see Exercse 7.5. Aga he marx s a wegh marx ha quafes he relave precso of he wo esmaors b GLS ad b B.

7-0 / Chaper 7. Modelg Issues Correlaed effecs model For a model of correlaed effecs ha descrbes he correlao bewee {α } ad {X } le x vecx a KT vecor bul b sackg vecors {x x T }. For oao we deoe he observed depede varables b o z x a q+k vecor of observed effecs ad le o be he assocaed colum vecor ha s o o o T. We assume ha he relaoshp bewee α ad X ca be descrbed hrough he codoal momes E[ α o ] Σ Σ x Ex Var α o D 7. x x α ad [ ] * where Σ αx Covα x E α x x T ad Σ x Var x. Equao 7. ca be movaed b jo ormal of α ad o bu s also useful whe some compoes of o are caegorcal. Wh dspla 7. we have E [ o ] E E [ α o ] o Z Σα xσ x x Ex + Xβ 7. ad Var o E Var α o o + Var E α o o R + Z D * Z. 7.3 [ ] [ ] [ ] The correlaed effecs alers he form of he regresso fuco equao 7. bu o he codoal varace equao 7.3. Now uder he model of correlaed effecs summarzed equaos 7. ad 7.3 s eas o see ha he radom effecs esmaor s geerall based. I coras he fxed effecs esmaor s ubased ad has varace / / Var b FE X R Q R X whch s he same as uder he fxed effecs model formulao; see Seco.5.3. Aga a exeso of he Hausma 978E es allows us o compare he basele model ad he model of correlaed effecs. The es sasc s χ. 7.4 FE b b Var b Var b b b FE GLS FE Uder he ull hpohess of he model equao 7.0 hs es sasc has a asmpoc as ch-square dsrbuo wh K degrees of freedom. As Seco 7.. hs es sasc s uvel pleasg ha large dffereces bewee he fxed ad radom effecs esmaors allow us o rejec he ull hpohess of o correlaed effecs. To summarze he fxed effecs esmaor s eas o compue ad s robus o omed varable bas. The esmaor has desrable properes uder a varao of he radom effecs model ha we call a model of correlaed effecs. Uder hs model of correlaed effecs formulao he ma subjec-specfc fxed parameers geerall assocaed wh fxed effecs models eed o be compued. I addo o he esmaor self sadard errors assocaed wh hs esmaor are eas o compue. Furher he equao 7.4 es sasc provdes a smple mehod for assessg he adequac of he radom effecs model; hs could lead o furher followup vesgaos ha ma ur lead o a mproved model specfcao. GLS FE GLS

Chaper 7. Modelg Issues / 7- Example: Icome ax pames - coued The aalss above s based o he error compoes model. Ma addoal feaures could be f o he daa. Afer addoal model explorao we reaed he varable erceps ad also used subjec-specfc varable slopes for he come varables LNTPI ad MR. I addo o accommodae seral paers ax lables we specfed a AR compoe for he errors. Par of he raoale comes from he aure of ax lables. Tha s we hpohesze ha he ax labl creases wh come. Moreover because dvduals have dffere audes owards ax-shelerg programs lve dffere saes ha have her ow ax programs ha affec he amou of he federal ax ad so o we expec coeffces assocaed wh come o dffer amog dvduals. A smlar argume ca be made for MR because hs s smpl a olear fuco of come. The radom effecs esmaed geeralzed leas squares ad robus fxed effecs esmaors are gve Table 7.. We ow see some mpora dffereces he wo esmao mehodologes. To llusrae examg he coeffces assocaed wh he MS ad EMP varables he radom effecs esmaor dcaes ha each varable s srogl sascall egavel sgfca whereas he robus esmaors do o sgal sascal sgfcace. Ths s also rue bu o a lesser exe of he coeffce assocaed wh AGE. To compare he vecor of radom versus fxed effecs esmaors he omed varable es FE sasc urs ou o be χ 3.68. Usg K 6 degrees of freedom he p-value assocaed wh hs es sasc s Prob χ >3.680.034. I coras o he error compoes model hs provdes evdece o dcae a serous problem wh omed varables. The resul of he hpohess es suggess usg he robus esmaors. Ieresgl boh radom ad fxed effecs esmao dcae ha use of a ax preparer PREP sgfcal lowers he ax labl of a axpaer whe corollg for come ad demographc characerscs. Ths was o a fdg he error compoes model. B roducg wo varable slopes he umber of esmaor comparsos dropped from egh o sx. Examg equao 7.0 we see ha he varables cluded he radom effecs formulao are o loger cluded he X β poro. Thus equao 7.0 he umber of rows of β K refers o he umber of varables o assocaed wh he radom effecs poro; we ca hk of hese varables as assocaed wh ol fxed effecs. Ths mples amog oher hgs ha he Seco 7..3 omed varable es s o avalable for he radom coeffces model where here are o varables assocaed wh ol fxed effecs. Thus Seco 7.3 roduces a es ha wll allow us o cosder he radom coeffces model. Table 7.. Comparso of Radom Effecs Esmaors o Robus Aleraves. Based o he Seco 3. Example. Model wh Varable Ierceps ad wo Varable Slopes Robus Fxed Effecs Esmao Radom Effecs Esmao Varable Parameer Esmaes -sasc Parameer Esmaes -sasc LNTPI MR MS -0.97-0.46-0.603-3.86 HH -.870-4.4-0.79-3.75 AGE -0.464 -.8-0.359 -.5 EMP -0.98-0.68-0.66-5.05 PREP -0.474 -.5-0.300-3. DEPEND -0.304 -.56-0.38 -.84 AR 0.454 3.76 0.53 3.38 χ FE 3.68

7- / Chaper 7. Modelg Issues 7.3 Omed varables Parcularl he socal sceces where observaoal leu of expermeal daa are predoma problems of omed varables aboud. The possbl of uobserved omed varables ha affec boh he respose ad explaaor varables ecourage aalss o dsgush bewee cause ad effec. We have alread see oe approach for hadlg cause ad effec aalss hrough mulple ssems of equaos Secos 6.4 ad 6.5. Furher he srucure of logudal daa allows us o cosruc esmaors ha are less suscepble o bas arsg from omed varables ha commo aleraves. For example Seco 7. we saw ha fxed effecs esmaors are robus o cera pes of me-cosa omed varables. Ths seco roduces esmaors ha are robus o oher pes of omed varables. These omed varable robus esmaors do o provde proeco from all pes of omed varables; he are sesve o he aure of he varables beg omed. Thus as a maer of pracce aalss should alwas aemp o collec as much formao as possble regardg he aure of he omed varables. Specfcall Seco 7. showed how a fxed effecs esmaor s robus o assumpo SEC6 zero correlao bewee he me-cosa heerogee varables ad he regressor varables. Uforuael he fxed effecs rasform sweeps ou me-cosa varables ad hese varables ma be he focus of a sud. To remed hs hs seco shows how o use paral formao abou he relaoshp bewee he uobserved heerogee varables ad he regressor varables. Ths dea of paral formao s due o Hausma ad Talor 98E who developed a srumeal varable esmao procedure. We cosder a broader class of omed varables models ha uder cera crcumsaces also allow for me-varg omed varables. Here we wll see ha he fxed effecs esmaor s a specal case of a class ha we call augmeed regresso esmaors. Ths class o ol provdes exesos bu also gves a bass for provdg heeroscedasc-cosse sadard errors of he esmaors. To se he sage for addoal aalses we frs reur o a verso of he Seco 3. error compoes model x + ε + u 3.** α + β + x β We have ow spl up he x β o wo poros oe for me-varg explaaor varables x ad β ad oe for me-cosa explaaor varables x ad β. As before he u erm preses uobserved omed varables ha ma be a fxed effec or a radom effec ha s correlaed wh eher he dsurbace erms or explaaor varables. As poed ou Seco.3 f a explaaor varable s me-cosa he he fxed effecs esmaor s o loger esmable. Thus he echques roduced Seco 7. o loger mmedael appl. However b examg devaos from he mea x β + ε ε x we see ha we ca sll derve ubased ad cosse esmaors of β eve he presece of omed varables. To llusrae oe such esmaor s T T x x x x x b FE x. Moreover wh he addoal assumpo ha u s o correlaed wh x we wll be able o provde cosse esmaors of β. Ths s a srog assumpo; sll he eresg aspec s ha wh logudal/pael daa we ca derve esmaors wh desrable properes eve he presece of omed varables.

Chaper 7. Modelg Issues / 7-3 Whe hpoheszg relaoshps amog varables he breakdow bewee me-varg ad me-cosa varables ca be arfcal. Thus our dscusso below we refer o explaaor varables ha eher are or are o relaed o omed varables. 7.3. Models of omed varables I s helpful o revew he samplg bass for he model order o descrbe how varaos samplg ma duce omed varable bas. The samplg ca be descrbed wo sages see for example Seco 3. ad Ware 985S. Specfcall we have: Sage. Draw a radom sample of subjecs from a populao. The vecor of subjec-specfc parameers α s assocaed wh he h subjec. Sage. Codoal o α draw realzaos of { z x } for T for he h subjec. Summarze hese draws as { Z X }. We follow oao roduced Seco 7. ad deoe he observed depede varables b o z x a q+k vecor of observed effecs ad le o be he assocaed colum vecor ha s o o o T. We ow use a uobserved varable model also cosdered b Pala ad Yao 99B Pala Yao ad Velu 994B ad ohers; Frees 00S provdes a summar. I he bologcal sceces omed varables are kow as umeasured cofouders. Here we assume ha he lae vecor α s depede of he observed varables o e we have omed mpora possbl me-varg varables he secod samplg sage. Specfcall assume a he secod sage we have Sage *. Codoal o α draw realzaos { Z X U } for he h subjec. Here o {Z X } represes observed depede varables whereas U represes he uobserved depede varables. Momes of he depede varables are specfed as E [ α o U ] Z α + X β + U γ 7.5 ad Var [ α o U ] R. 7.6 Thus γ s a g vecor of parameers ha sgals he presece of omed varables U. Usg he same oao coveo as o le u be he colum vecor assocaed wh U. To esmae he model parameers smplfg assumpos are requred. Oe roue s o make full dsrbuoal assumpos such as mulvarae ormal ha perms esmao usg maxmum lkelhood mehods. I he followg we sead use assumpos o he samplg desg because hs perms esmao procedures ha lk back o Seco 7. whou he ecess of assumg ha a dsrbuo follows a parcular paramerc model. To specf he model ha s used for ferece furher use Σ uo Covu o o capure he correlao bewee uobserved ad observed effecs. For smplc we drop he subscrp o he T g T q+k marx Σ uo. Now we ol eed be cocered wh hose observed varables ha are relaed o he uobservables. Specfcall we ma re-arrage he observables o wo peces o o o so ha Σ uo Cov u o Cov u o Σ uo 0. Tha s he frs pece of o s correlaed o he uobservables whereas he secod pece s o. To preve a drec relao bewee u ad o we also assume ha o ad o have zero covarace. Wh hese coveos we assume ha E u α o Σ Var o o Eo. 7.7 [ ] uo

7-4 / Chaper 7. Modelg Issues A suffce codo for 7.7 s jo ormal of o ad u codoal o α. A advaage of assumg equao 7.7 drecl s ha also allows us o hadle caegorcal varables wh o. A mplcao of 7.7 s ha Covu o 0; however o mplc dsrbuoal assumpos for o are requred. For he samplg desg of he observables we assume ha he explaaor varables are geeraed from a error compoes model. Specfcall we assume ha Σ o + Σ o for s Cov o s o so ha Var o I Σ o + J Σ o. Σ o for s We furher assume ha he covarace bewee u s ad o s me-cosa. Thus Cov u s o Σ uo for all s. Ths elds Cov u o J Σ uo. The from Frees 00S we have E uo T o o * β + z α * + x β + z γ + x * 7.8 [ α o ] z α + x β + γσ Σ + Σ o E o 0 γ * where β0 γ Σuo Σo + Σo E o T. I Equao 7.8 we have lsed ol hose explaaor varables z ad x ha we hpohesze wll be correlaed wh he uobserved varables. Ths s a esable hpohess. Furher equao 7.8 suggess ha b corporag he erms z ad x as regresso erms he aalss we ma avod omed varable bas. Ths s he opc of Seco 7.3.. Specal case: Error compoes model wh me-cosa explaaor varables Wh he error compoes model we have q ad z so ha equao 7.8 reduces o * E α x β + α + x β + x γ. [ ] * 0 To esmae he coeffces of hs codoal regresso equao we requre ha he explaaor varables o be lear combaos of oe aoher. I parcular f here are a me-cosa varables x he mus o be cluded x. I oher words we requre ha me-cosa varables be ucorrelaed wh omed varables. 7.3. Augmeed regresso esmao Ths subseco cosders a augmeed regresso model of he form E [ o ] X β + G γ 7.9 ad Var [ o ] R + Z D Z V. 7.0 Here γ s a g vecor of coeffces ad G s a kow fuco of depede varables {o } such as equao 7.8. Furher D s a posve defe varace-covarace marx o be esmaed. Oe ma smpl use weghed leas squares o deerme esmaes of β ad γ. Explcl we use mmzers of he weghed sum of squares WSS β γ Xβ + G γ W Xβ + G γ 7.

Chaper 7. Modelg Issues / 7-5 o defe esmaors of β ad γ. We deoe hese esmaors as b AR ad γˆ AR respecvel. Here he AR subscrp deoes arfcal regresso as Davdso ad MacKo 990E or augmeed regresso as Arellao 993E. The po s ha o specalzed sofware s requred for he omed varable esmaor γˆ AR or omed varable bas correced regresso coeffce esmaor b AR. Dffere choces of he G perm us o accommodae dffere daa feaures. To llusrae s eas o see usg W V ad omg G ha b AR reduces o b GLS. Furher Frees 00S shows ha b AR reduces o b FE whe W V ad G Z Z R Z ZR X. As aoher example cosder he radom coeffces desg. Here we assume ha q K ad x z. Thus from equao 7.8 we have * E α o β + x α + β + x γ. [ ] * 0 Hpoheszg ha all varables are poeall relaed o omed varables we requre g K ad G x. A advaage of he augmeed regresso formulao compared o he Seco 7.. omed varable es s ha perms a drec assessme of he hpohess of omed varables H 0 : γ 0. Ths ca be doe drecl usg a Wald es sasc of he form χ AR γˆ AR Var γˆ AR γˆ AR. Here Var γˆ AR ca be esmaed based o he varace specfcao equao 7.6 or usg a robus alerave as roduced Seco 3.4. Example - Icome ax pames - Coued We coue wh he example from Seco 7..3 b frs cosderg he model wh varable erceps wo varable slopes ad a AR seral correlao coeffce for he errors. The usual radom effecs geeralzed leas squares esmaors are preseed Table 7.3. These coeffces also appear Tables 7. ad 7. where we leared ha hs model suffered from omed varable bas. Table 7.3 preses he fs from he augmeed regresso model where we have augmeed he regresso usg he averages from all he explaaor varables x. Table 7.3 shows ha he averages of boh of he come varables LNTPI ad MR are sascall sgfcal dffere from zero. Furher from he overall es of H 0 : γ 0 he es sasc s χ AR 57.5. Usg a ch-square wh 8 degrees of freedom he p-value assocaed wh hs es s less ha 0.000. As Seco 7..3 hs dcaes serous poeal problems from omed varables ad suggess usg he robus bas-correced esmaors Table 7.3. Compared o he fxed effecs esmaors Table 7. we see ha he resuls are qualavel smlar. The varables DEPEND PREP ad HH are sascall sgfcal egave for boh models whereas EMP ad AGE are o sascall sgfca for eher esmao procedure. The esmao procedures gve dffere resuls for he maral saus MS varable wh he augmeed regresso procedure. The advaage of he augmeed regresso procedure s ha perms esmao of a mea for he LNTPI ad MR varables somehg ha was o possble usg he fxed effecs robus esmaors. The augmeed regresso procedures also allow us o gve robus esmaors for he radom coeffces model. Table 7.3 summarzes esmao of he radom coeffces model wh a AR seral correlao coeffce for he errors. Table 7.3 also shows he robus esmaors calculaed usg augmeao wh he averages from all he explaaor varables x. Aga he es sasc of χ AR 56.69 dcaes serous poeal problems from omed varables. The bas-correced esmaors are qualavel smlar o hose of he augmeed regresso resuls for he varable ercep ad wo varable slope model.

7-6 / Chaper 7. Modelg Issues Table 7.3. Comparso of Radom Effecs Esmaors o Robus Aleraves. Based o he Seco 3. Example. Model wh Varable Ierceps ad wo Varable Slopes Model wh Varable Ierceps ad egh Varable Slopes Radom Coeffces Radom Effecs Esmao Augmeed Regresso Esmao Radom Effecs Esmao Augmeed Regresso Esmao Varable Parameer -sasc Parameer -sasc Parameer -sasc Parameer -sasc Esmaes Esmaes Esmaes Esmaes LNTPI.70 3.40.35.94.30.05.407 3.06 MR 0.005 0.46 0.03.77 0.009 0.9 0.09.85 MS -0.603-3.86-0.563-3. -0.567-3.53-0.670 -.96 HH -0.79-3.75 -.056 -.67-0.79 -.97 -.3-3.4 AGE -0.359 -.5-0.386 -.08-0.9 -.54-0.485 -.76 EMP -0.66-5.05-0.35 -.0-0.557 -.58-0.334 -. PREP -0.300-3. -0.96 -.5-0.94-3.4-0.3 -.3 DEPEND -0.38 -.84-0.77 -.9-0.7-3.56-0.70 -.44 LNTPIAVG 0.964 6.84 0.68 4.9 MRAVG -0.09-7.08-0.09-6.66 MSAVG -0.65 -.07-0.057-0. HHAVG 0.398 0.95 0.565.40 AGEAVG -0.053-0.3 0.56 0.79 EMPAVG -0.489 -. -0.35 -. DEPAVG 0.039 0. 0.089 0.53 PREPAVG -0.007-0.07-0.038-0.43 AR 0.53 3.38 0.8.70 0.006 0.3-0.043-0.96 χ AR 57.5 56.69

Chaper 7. Modelg Issues / 7-7 Seco 7.4 Samplg selecv bas ad aro 7.4. Icomplee ad roag paels Hsorcall he prmar approaches o pael/logudal daa aalss he ecoomerc ad bosascs leraures assumed balaced daa of he followg form. : : T T Ths form suggesed usg echques from mulvarae aalss as descrbed b for example Rao 987B ad Chamberla 98E. However here are ma was whch a complee balaced se of observaos ma o be avalable due o delaed er earl ex ad erme orespose. Ths seco begs b cosderg ubalaced suaos where he lack of balaced s plaed or desged b he aals. Seco 7.4. he dscusses ubalaced suaos ha are o plaed. Whe uplaed we call he oresposes mssg daa. We wll be parcularl cocered wh suaos whch he mechasms for mssgess are relaed o he respose dscussed Seco 7.4.3. To accommodae mssg daa for subjec we use he oao T for he umber of observaos ad j o deoe he me of he jh observao. Thus he mes for he avalable se of observaos are { T }. For hs chaper we assume ha hese mes are dscree ad a subse of { T}. Chaper 6 descrbed he couous me case. Defe M o be he T T desg marx ha has a he h j colum ad j h row ad s zero oherwse j T. Specfcall wh hs desg marx we have M M. 7. M T T Usg hs desg marx mos of he formulas Chapers hrough 6 carr hrough for plaed mssg daa. The case of me-varg parameers urs ou o be more complex; Appedx 8A. wll llusrae hs po he coex of he wo-wa model. For smplc hs ex we use he oao { T } o deoe a ubalaced observao se. Pael surves of people provde a good llusrao of plaed mssg daa; hese sudes are ofe desged o have complee observaos. Ths s because we kow ha people become red of respodg o surves o a regular bass. Thus pael surves are pcall desged o replesh a cera proporo of he sample a each ervew me. Specfcall for a rollg or roag pael a fxed proporo of dvduals eer ad ex he pael durg each ervew me. Fgure 7.4. llusraes hree waves of a roag pael. A me he frs 3 dvduals ha comprse Wave are ervewed. A me dvduals from Wave are dropped ad ew dvduals ha comprse Wave are added. A me 3 dvduals from Wave are dropped ad ew dvduals ha comprse Wave 3 are added. Ths paer s coued wh dvduals dropped ad added a each ervew me. Thus a each me 3 dvduals are ervewed. Each perso sas he surve for a mos 3 me perods.

7-8 / Chaper 7. Modelg Issues Fgure 7.4.. Three Waves of a Roag Pael. The sold les dcae he dvduals ha are ervewed a each of fve ervew mes. Wave Wave Wave Wave + + 3 3+ 4 Wave 3 4+ 5 Iervew me 3 4 5 Number of subjecs ervewed 3 3 3 7.4. Uplaed orespose Few dffcules arse whe he daa are o balaced due o a plaed desg. However whe he daa are ubalaced due o uforesee eves hs lack of balace represes a poeal source of bas. To provde some specfc examples we aga reur o pael surves of people. Verbeek ad Njma Chaper 8 of Máás ad P. Sevesre 996 provde he followg ls of pes of uplaed orespose. Tpes of pael surve orespose Ial orespose. A subjec coaced cao or wll o parcpae. Because of lmed formao hs poeal problem s ofe gored he aalss. U orespose. A subjec coaced cao or wll o parcpae eve afer repeaed aemps subseque waves o clude he subjec. Wave orespose. A subjec does o respod for oe or more me perods bu does respod he precedg ad subseque mes for example he subjec ma be o vacao. Aro. A subjec leaves he pael afer parcpag a leas oe surve. I s also of cocer o deal wh surve em orespose where he ems are reaed as covaraes he aalss. Here formao o oe or more varables s mssg. For example dvduals ma o wsh o repor come age ad so o. However we resrc cosderao o mssg resposes. To udersad he mechasms ha lead o uplaed orespose we model sochascall. Le r j be a dcaor varable for he jh observao wh a oe dcag ha hs respose s observed ad a zero dcag ha he respose s mssg. Le r r r T r r T summarze he daa avalabl for all subjecs. The eres s wheher or o he resposes fluece he mssg daa mechasm. For oao we use Y T T o be he colleco of all poeall observed resposes.

Chaper 7. Modelg Issues / 7-9 Mssg daa models I he case where Y does o affec he dsrbuo of r we follow Rub 976G ad call hs case mssg compleel a radom MCAR. Specfcall he mssg daa are MCAR f fr Y fr where f. s a geerc probabl mass fuco. A exeso of hs dea s Lle 995G where he adjecve covarae depede s added whe Y does o affec he dsrbuo of r codoal o he covaraes. If he covaraes are summarzed as {X Z} he he codo correspods o he relao fr Y X Z fr X Z. To llusrae hs po cosder a example of Lle ad Rub 987G where X correspods o age ad Y correspods o come of all poeal observaos. If he probabl of beg mssg does o deped o come he he mssg daa are MCAR. If he probabl of beg mssg vares b age bu does o b come over observaos wh a age group he he mssg daa are covarae depede MCAR. Uder he laer specfcao s possble for he mssg daa o var b come. For example ouger people ma be less lkel o respod o a surve. Ths shows ha he mssg a radom feaure depeds o he purpose of he aalss. Specfcall s possble ha a aalss of he jo effecs of age ad come ma ecouer serous paers of mssg daa whereas a aalss of come corolled for age suffers o serous bas paers. Whe he daa are MCAR Lle ad Rub 987G Chaper 3 descrbe several approaches for hadlg parall mssg daa. Oe opo s o rea he avalable daa as f oresposes were plaed ad use ubalaced esmao echques. Aoher opo s o ulze ol subjecs wh a complee se of observaos b dscardg observaos from subjecs wh mssg resposes. A hrd opo s o mpue values for mssg resposes. Lle ad Rub oe ha each opo s geerall eas o carr ou ad ma be sasfacor wh small amous of mssg daa. However he secod ad hrd opos ma o be effce. Furher each opo mplcl reles heavl o he MCAR assumpo. Lle ad Rub 987G advocae modelg he mssg daa mechasms; he call hs he model-based approach. To llusrae cosder a lkelhood approach usg a seleco model for he mssg daa mechasm. Now paro Y o observed ad mssg compoes usg he oao Y Y obs Y mss. Wh he lkelhood approach we base ferece o he observed radom varables. Thus we use a lkelhood proporoal o he jo fuco fr Y obs. We also specf a seleco model b specfg he codoal mass fuco fr Y. Suppose ha he observed resposes ad he seleco model dsrbuos are characerzed b a vecors of parameers θ ad ψ respecvel. The wh he relao fr Y obs θ ψ fy obs θ fr Y obs ψ we ma express he log lkelhood of he observed radom varables as Lθψ log fr Y obs θψ log fy obs θ + log fr Y obs ψ. I he case where he daa are MCAR he fr Y obs ψ fr ψ does o deped o Y obs. Lle ad Rub 987G also cosder he case where he seleco mechasm model dsrbuo does o deped o Y mss bu ma deped o Y obs. I hs case he call he daa mssg a radom MAR. I boh he MAR ad MCAR cases we see ha he lkelhood ma be maxmzed over he parameers separael for each case. I parcular f oe s ol eresed he maxmum lkelhood esmaor of θ he he seleco model mechasm ma be gored. Hece boh suaos are ofe referred o as he gorable case.

7-0 / Chaper 7. Modelg Issues Example Icome ax pames Le represe ax labl ad x represe come. Cosder he followg fve seleco mechasms. The axpaer s o seleced mssg wh probabl ψ whou regard o he level of ax labl. I hs case he seleco mechasm s MCAR. The axpaer s o seleced f he ax labl s less ha $00. I hs case he seleco mechasm depeds o he observed ad mssg respose. The seleco mechasm cao be gored. The axpaer s o seleced f he come s less ha $0000. I hs case he seleco mechasm s MCAR covarae depede. Tha s assumg ha he purpose of he aalss s o udersad ax lables codoal o kowledge of come srafg based o come does o serousl bas he aalss. The probabl of a axpaer beg seleced decreases wh ax labl. For example suppose he probabl of beg seleced s log -ψ. I hs case he seleco mechasm depeds o he observed ad mssg respose. The seleco mechasm cao be gored. The axpaer s followed over T perods. I he secod perod a axpaer s o seleced f he frs perod ax s less ha $00. I hs case he seleco mechasm s MAR. Tha s he seleco mechasm s based o a observed respose. The secod ad fourh seleco mechasms represe suaos where he seleco mechasm mus be explcl modeled; hese are kow as o-gorable cases. I hese suaos whou explc adjusmes procedures ha gore he seleco effec ma produce serousl based resuls. To llusrae a correco for seleco bas a smple case we oule a example due o Lle ad Rub 987G. Seco 7.4.3 descrbes addoal mechasms. Example- Hsorcal heghs Lle ad Rub 987G dscuss daa due o Wacher ad Trusell 98G o he hegh of me recrued o serve he mlar. The sample s subjec o cesorg ha mmum hegh sadards were mposed for admsso o he mlar. Thus he seleco mechasm s f > c r 0 oherwse where c s he kow mmum hegh sadard mposed a he me of recrume. The seleco mechasm s o-gorable because depeds o he dvdual s hegh. For hs example addoal formao s avalable o provde relable model ferece. Specfcall based o oher sudes of male heghs we ma assume ha he populao of heghs s ormall dsrbued. Thus he lkelhood of he observables ca be wre dow ad ferece ma proceed drecl. To llusrae suppose ha c c s cosa. Le µ ad σ deoe he mea ad sadard devao of. Furher suppose ha we have a radom sample of + m me whch m me fall below he mmum sadard hegh c ad we observe Y obs. The jo dsrbuo for observables s fr Y obs µ σ fy obs µ σ fr Y obs { f > c Prob > c } Prob c m Now le φ ad Φ represe he des ad dsrbuo fuco for he sadard ormal dsrbuo. Thus he log-lkelhood s.

Chaper 7. Modelg Issues / 7- Lµ σ log fr Y obs µ σ µ c µ log φ + m log Φ. σ σ σ Ths s eas o maxmze µ ad σ. If oe gored he cesorg mechasms he oe would derve esmaes of he observed daa from he log lkelhood µ log φ σ σ eldg dffere ad based resuls. 7.4.3 No-gorable mssg daa For o-gorable mssg daa Lle 995G recommeds: Avod mssg resposes wheever possble b usg approprae follow-up procedures. Collec covaraes ha are useful for predcg mssg values. Collec as much formao as possble regardg he aure of he mssg daa mechasm. For he hrd po f lle s kow abou he mssg daa mechasm he s dffcul o emplo a robus sascal procedure o correc for he seleco bas. There are ma models of mssg daa mechasms. A geeral overvew appears Lle ad Rub 987G. Verbeek ad Njma Chaper 8 of Máás ad P. Sevesre 996E surve more rece ecoomerc pael daa leraure. Lle 995G surves he problem of aro. Raher ha surve hs developg leraure we gve a few models of o-gorable mssg daa. Heckma wo-sage procedure Heckma 976E developed hs procedure he coex of cross-secoal daa. Because reles o correlaos of uobserved varables s also applcable o fxed effecs pael daa models. Thus assume ha he respose model follows a oe-wa fxed effecs. As roduced Chaper hs model ca be expressed as α + x β + ε. Furher assume ha he samplg respose mechasm s govered b he lae uobserved varable r * where r * w γ+ η. * The varables w ma or ma o clude he varables x. We observe f r 0 ha s f * * f r 0 r crosses he hreshold 0. Thus we observe r. To complee he specfcao 0 oherwse we assume ha {ε η } are decall ad depedel dsrbued ad ha he jo dsrbuo of ε η s bvarae ormal wh meas zero varaces σ ad σ η ad correlao ρ. Noe ha f he correlao parameer ρ equals zero he he respose ad seleco models are depede. I hs case he daa are MCAR ad he usual esmao procedures are ubased ad asmpocall effce. Uder hese assumpos basc mulvarae ormal calculaos show ha E r * 0 α + x β + β λ λw γ

7- / Chaper 7. Modelg Issues where β λ ρσ ad a λ a φ. Here λ s he verse of he so-called Mlls rao. Ths Φ a calculao suggess he followg wo-sep procedure for esmag he parameers of eres. Heckma s wo-sage procedure. Use he daa { r w } ad a prob regresso model o esmae γ. Call hs esmaor g H.. Use he esmaor from sage o creae a ew explaaor varable x K+ λw g H. Ru a oe-wa fxed effecs model usg he K explaaor varables x as well as he addoal explaaor varable x K+. Use b H ad b λh o deoe he esmaors of β ad β λ respecvel. Seco 9.. wll roduce prob regressos. We also oe ha he wo-sep mehod does o work absece of covaraes o predc he respose ad for praccal purposes requres varables w ha are o x see Lle ad Rub 987. To es for seleco bas we ma es he ull hpohess H 0 : β λ 0 he secod sage due o he relao β λ ρσ. Whe coducg hs es oe should use heeroscedasc correced sadard errors. Ths s because he codoal varace Var r * 0 depeds o he observao. Specfcall Var r * 0 σ - ρ δ where δ λ λ + w γ ad λ φw γ/φw γ. Ths procedure assumes ormal for he seleco lae varables o form he augmeed varables. Oher dsrbuo forms are avalable he leraure cludg he logsc ad uform dsrbuos. A deeper crcsm rased b Lle 985G s ha he procedure reles heavl o assumpos ha cao be esed usg he daa avalable. Ths crcsm s aalogous o he hsorcal heghs example where we reled heavl o he ormal curve o fer he dsrbuo of heghs below he cesorg po. Despe hese crcsms Heckma s procedure s wdel used he socal sceces. Hausma ad Wse procedure To see how o exed he Heckma procedure o error compoe pael daa models we ow descrbe a procedure orgall due o Hausma ad Wse 979E; see also he developme Verbeek ad Njma Chaper 8 of Máás ad P. Sevesre 996E. For smplc we work wh he error compoes model descrbed Seco 3. α + x β + ε. We also assume ha he samplg respose mechasm s govered b he lae varable of he form r * ξ + w γ + η. Ths s also a error compoes model. The varables w ma or ma o clude he varables * f r 0 x. As before r dcaes wheher s observed ad r. The radom 0 oherwse varables α ε ξ ad η are assumed o be jol ormal each wh mea zero ad varace α σ α 0 σ αξ 0 ε 0 σ ε 0 σ εη Var. ξ σ αξ 0 σ ξ 0 η 0 σ εη 0 σ η If σ αξ σ εη 0 he he seleco process s depede of he observao process.

Chaper 7. Modelg Issues / 7-3 I s eas o check ha b EC s ubased ad cosse f E r x β. Uder codoal ormal oe ca check ha σ T αξ σ εη σ + ξ E + r xβ g g + g s Tσ ξ σ η σ η Tσ ξ + σ η s where g E ξ + η r. The calculao of g volves he mulvarae ormal dsrbuo ha requres umercal egrao. Ths calculao s sraghforward alhough compuaoall esve. Followg hs calculao esg for gorabl ad producg bas-correced esmaors proceeds as he Heckma case. Chaper 9 wll dscuss he fg of bar depede resposes o error compoe models. For oher addoal deals we refer he reader o Verbeek ad Njma Chaper 8 of Máás ad P. Sevesre 996. EM algorhm Seco 7.4 has focused o roducg specfc models of o-gorable orespose. Geeral robus models of orespose are o avalable. Raher a more approprae sraeg s o focus o a specfc suao collec as much formao as possble regardg he aure of he seleco problem ad he develop a model for hs specfc seleco problem. The EM algorhm s a compuaoal devce for compug model parameers. Alhough specfc o each model has foud applcaos a wde vare of models volvg mssg daa. Compuaoall he algorhm eraes bewee he E for codoal expecao ad M for maxmzao seps. The E sep fds he codoal expecao of he mssg daa gve he observed daa ad curre values of he esmaed parameers. Ths s aalogous o he me-hoored rado of mpug mssg daa. A ke ovao of he EM algorhm s ha oe mpues suffce sascs for mssg values o he dvdual daa pos. For he M sep oe updaes parameer esmaes b maxmzg a observed log lkelhood. Boh he suffce sascs ad he log lkelhood deped o he model specfcao. Ma roducos of he EM algorhm are avalable he leraure. Lle ad Rub 987G provde a dealed reame.

7-4 / Chaper 7. Modelg Issues 7. Exercses ad Exesos Seco 7. 7. Fuller-Baese Trasform Cosder he Seco 3. error compoes model wh α + X β + ε ad Q I Z Z Z Z Var V σ I + σ α J. Recall from Seco.5.3 ha. / σ a. Defe ψ ad P I Q. Show ha V / σ P + ψ Q. Tσ α + σ * X P Q X * * * * Show ha he geeralzed leas squares esmaor of β s b EC X X X. * b. Trasform he error compoes model usg P + ψ Q ad + ψ c. Now cosder he specal case of he error compoes model equao 7.. Show ha he geeralzed leas squares esmaor of β s b EC T * * * * x ψ T ψ T ψ T x T * * T ψ x ψ T x * * where x x x ψ ad ψ. d. Now cosder he balaced daa case so ha T T for each. Show ha b T * * * * x x T * * x x EC. e. For he model cosdered par d show ha varace of b EC s as gve equao 7.. 7. Maddala s decomposo Cosder he specal case of he error compoes model equao 7.. a. Show equao 7.6. Tha s show Var b B Tσ + σ T x x α ε 7.5. b. Show equao 7.4. Tha s show x x where b B as equao x x Var T x x b FE σ ε x x where b FE as T x x equao 7.3. c. Use pars a ad b ad he expressos for b EC ad Var b EC Seco 7.. o show +. Var b EC Var b FE Var b B d. Show equao 7.7. Tha s wh pars a-c ad he expressos for b EC ad Var b EC Var b EC Seco 7.. show b EC - b FE + b B where. Var b T B.

Chaper 7. Modelg Issues / 7-5 7.3. Mxed lear model esmao wh erceps Cosder he lear mxed effecs model descrbed Seco 3.3 where α are reaed as radom wh mea E α α ad varace-covarace marx Var α D depede of he error erm. The we ma re-wre he model as Z α + X β + ε * where ε * ε + Z α - α ad Var ε * Z D Z + R V a posve defe T T marx. a. Show ha we ca express he geeralzed leas squares esmaor of β as GLS GLS Z V Z Z V Z X V X V C b wh GLS X Z V Z Z V Z X V X X V C. b. Show ha Var RE GLS C b. c. Now cosder he error compoes model so ha q D σ α z ad Z. Use par a o show ha { } + + w w w w EC T T x x X Q x x x x X Q X b ζ ζ where T J I Q w ζ ζ x x ad w ζ ζ. d. Cosder par c ad assume addo ha K. Show ha + + w T w w T EC x x T x x x x T x x b ζ ζ e. As par a show ha he geeralzed leas squares esmaor of α s GLS GLS b X Z V Z V Z Z V a. f. Show ha for he case cosdered par c wh q D σ α z ad Z ha EC w EC w a b x where b EC s gve par c. 7.4. Robus esmao Cosder he lear mxed effecs model descrbed Problem 7.3. Le Z FE / / X R Q R X C where / / Z R Z Z R Z Z R I Q. Recall ha Z FE FE / / R Q R X C b. a. Show ha β b FE E.

/ Chaper 7. Modelg Issues 7-6 b. Show ha Var FE FE C b. 7.5. Decomposg he radom effecs esmaor Cosder he lear mxed effecs model descrbed Problem 7.3. A alerave esmaor of β s he so-called bewee-groups esmaor gve as B B V Z Z Z V Z V X V Z Z V Z Z V X C b where B X V Z Z V Z Z V X X V Z Z V Z Z V X C. a. Show ha Var B B C b. b. Now cosder he error compoes model so ha q D σ α z ad Z. Use par a o show ha w w w w B T T x x x x x x b ζ ζ. c. Show ha a alerave form for b B s w w w w B x x x x x x b ζ ζ. d. Use equao A.4 of Appedx A.5 o esablsh + Z R Z D Z V Z. e. Use par d o esablsh R Z Z R Z Z R R V Z Z V Z Z V V. f Use Problem 7.3a 7.4 ad pars a e o show ha C B + C FE C GLS ha s show ha Var Var Var + GLS FE B b b b. g. Prove Maddala s decomposo: B FE GLS b b I b + where Var Var B b GLS b. 7.6. Omed varable es Cosder he lear mxed effecs model descrbed Problems 7.3 7.4 ad 7.5. a. Show ha GLS FE GLS b b b Var Cov. b. Use par a o show ha GLS FE GLS FE b b b b Var Var Var. c. Show ha GLS FE GLS FE GLS FE FE b b b b b b Var Var χ has a asmpoc as ch-square dsrbuo wh K degrees of freedom.

Chaper 8. Damc Models / 8- Chaper 8. Damc Models 003 b Edward W. Frees. All rghs reserved Absrac. Ths chaper cosders models of logudal daa ses wh loger me dmesos ha were cosdered earler chapers. Wh ma observaos per subjec aalss have several opos for roducg more complex damc model feaures ha address quesos of eres or ha represe mpora edeces of he daa or boh. Oe opo s based o he seral correlao srucure; hs chaper exeds he basc srucures ha were roduced Chaper. Aoher damc opo s o allow parameers o var over me. Moreover for a daa se wh a log me dmeso relave o he umber of subjecs we have a opporu o model he cross-secoal correlao a mpora ssue ma sudes. The chaper also cosders he Kalma fler approach ha allows he aals o corporae ma of hese feaures smulaeousl. Throughou he assumpo of exogee of he explaaor varables s maaed. Chaper 6 cosdered lagged depede varables as explaaor varables aoher wa of roducg damc feaures o he model. 8. Iroduco Because logudal daa var over me as well as he cross-seco we have opporues o model he damc or emporal paers he daa. For he daa aals whe s mpora o cosder damc aspecs of a problem? Par of he aswer o hs queso ress o he purpose of he aalss. If he ma fereal ask s forecasg of fuure observaos as roduced Chaper 4 he he damc aspec s crcal. I hs sace ever opporu for udersadg damc aspecs should be explored. I coras oher problems he focus s o udersadg relaos amog varables. Here he damc aspecs ma be less crcal. Ths s because ma models sll provde he bass for cosrucg ubased esmaors ad relable esg procedures whe damc aspecs are gored a he prce of effcec. To llusrae for problems wh large sample szes he cross-seco effcec ma o be a mpora ssue. Noeheless udersadg he damc correlao srucure s mpora for achevg effce parameer esmaors; hs aspec ca be val especall for daa ses wh ma observaos over me. The mporace of damcs s flueced b he sze of he daa se boh hrough he choce of he sascal model ad he pe of approxmaos used o esablsh properes of parameer esmaors. For ma logudal daa ses he umber of subjecs s large relave o he umber of observaos per subjec T. Ths suggess he use of regresso aalss echques; hese mehods are desged o udersad relaoshps amog varables observed ad uobserved ad o accou for subjec-level heerogee. I coras for oher problems T s large relave o. Ths suggess borrowg from oher sascal mehodologes such as mulvarae me seres. Here alhough relaoshps amog varables are mpora udersadg emporal paers s he focus of hs mehodolog. We remark ha he modelg echques preseed Chapers -5

8- / Chaper 8. Damc Models are based o he lear model. I coras Seco 8.5 preses a modelg echque from he mulvarae me seres leraure he Kalma fler. The sample sze also flueces he properes of our esmaors. For logudal daa ses where s large compared o T hs suggess he use of asmpoc approxmaos where T s bouded ad eds o f. However for oher daa ses we ma acheve more relable approxmaos b cosderg saces where ad T approach f ogeher or where s bouded ad T eds o f. For ma models hs dsco s o a mpora oe for applcaos. However for some models such as he fxed effecs lagged depede varable model Seco 6.3 he dfferece s crcal. There he approach where T s bouded ad eds o f leads o based parameer esmaors. Ths chaper deals wh problems where he damc aspec s mpora eher because of he fereal purposes uderlg he problem or he aure of he daa se. We ow oule several approaches ha are avalable for corporag damc aspecs o a logudal daa model. Perhaps he eases wa for hadlg damcs s o le oe of he explaaor varables be a prox for me. For example we mgh use x j for a lear red me model. Aoher echque s o use me dumm varables ha s bar varables ha dcae he presece or absece of a perod effec. To llusrae Chaper we roduced he wo-wa model α + λ + x β + ε. 8. Here he parameers {λ }are me-specfc quaes ha do o deped o subjecs. Chaper cosdered he case where {λ } were fxed parameers. I Chaper 3 we allowed {λ } o be radom. Seco 8.3 exeds hs dea b allowg several parameers he logudal daa model o var wh me. To llusrae oe example ha we wll cosder s x β + ε ha s where regresso parameers β var over me. Ulke cross-secoal daa wh logudal daa we also have he abl o accommodae emporal reds b lookg a chages eher he respose or he explaaor varables. Ths echque s sraghforward ad aural some areas of applcao. To llusrae whe examg sock prces because of facal ecoomcs heor we exame proporoal chages prces whch are smpl reurs. As aoher example we ma wsh o aalze he model α + x β + ε 8. where - - s he chage or dfferece. I geeral oe mus be war of hs approach because ou lose al observaos whe dfferecg. Re-wrg equao 8. we have α + - + x β + ε. A geeralzao of hs s α + γ - + x β + ε 8.3 where γ s a parameer o be esmaed. If γ he he model equao 8.3 reduces o he model equao 8.. If γ 0 he he model equao 8.3 reduces o our usual oe-wa model. Thus he parameer γ s a measure of he relaoshp bewee ad -. Because measures he regresso of - o s called a auoregressve parameer. The model equao 8.3 s a example of a lagged depede varable model ha was roduced Seco 6.3.

Chaper 8. Damc Models / 8-3 Aoher wa o formulae a auoregressve model s α + x β + ε 8.4 where ε ρ ε - + η. Here he auoregresso s o he dsurbace erm o he respose. The models equaos 8.3 ad 8.4 are smlar e he dffer some mpora aspecs. To see hs use equao 8.4 wce o ge - ρ - α + x β + ε ρ α + x - β + ε - α * + x - ρ x - β + η where α * α -ρ. Thus equao 8.4 s smlar o equao 8.3 wh γ ρ; he dfferece les he varable assocaed wh β. Seco 8. explores furher he modelg sraeg of assumg seral correlao drecl o he dsurbace erms leu of he respose. There Seco 8. oes ha because of he assumpo of bouded T oe eed o assume saoar of errors. Ths sraeg was used mplcl Chapers -5 for hadlg he damcs of logudal daa. Fall Seco 8.5 shows how o adap he Kalma fler echque o logudal daa aalss. The Kalma fler approach s a flexble echque ha allows aalss o corporae me-varg parameers ad broad paers of seral correlao srucures o he model. Furher we wll show how o use hs echque o smulaeousl model cross-secoal heerogee emporal aspecs as well as spaal paers. 8. Seral correlao models Oe approach for hadlg he damcs s hrough he specfcao of he covarace srucure of he dsurbace erm ε. Ths seco exames saoar ad o-saoar specfcaos of he correlao srucure for equall spaced daa ad he roduces opos for daa ha ma o be equall spaced. 8.. Covarace srucures Recall from Seco.5. ha R Var ε s a T T emporal varace-covarace marx. Here he eleme he rh row ad sh colum s deoed b R rs. For he h subjec we defe Var ε R τ a T T submarx of R ha ca be deermed b removg he rows ad colums of R ha correspod o resposes o observed. We deoe hs depedece of R o parameers usg Rτ. Here τ s he vecor of ukow parameers called varace compoes. Seco.5. roduced four specfcaos of R: o correlao compoud smmer auoregressve of order oe ad v usrucured. The auoregressve model of order oe s a sadard represeao used me seres aalss. Ths feld of sud also suggess alerave correlao srucures. For example oe could eera auoregressve models of hgher order. Furher movg average models sugges he Toeplz specfcao of R: R rs σ r-s. Ths defes elemes of a Toeplz marx. R rs σ r-s for r-s < bad ad R rs 0 for r-s bad. Ths s he baded Toeplz marx. Whe he bad s q + hs Toeplz specfcao correspods o a movg average model of order q also kow as a MAq srucure. More complex auoregressve movg average models ma be hadled a smlar fasho see for example Jerch ad Schlucer 986B. The Toeplz specfcao suggess a geeral lear varace srucure of he form R τ R +τ R + + τ dmτ R dmτ

8-4 / Chaper 8. Damc Models where dmτ s he dmeso of τ ad R R R dmτ are kow marces. As poed ou Seco 3.5.3 o MIVQUE esmao hs geeral srucure accommodaes ma alhough o all such as auoregressve covarace srucures. Aoher broad covarace srucure suggesed b he mulvarae aalss leraure s he facor-aalc srucure of he form R ΛΛ + Ψ where Λ s a marx of ukow facor loadgs ad Ψ s a ukow dagoal marx. A mpora advaage of he facor aalc specfcao s ha easl allows he daa aals o esure ha he esmaed varace marx wll be posve or o-egave defe whch ca be mpora some applcaos. The covarace srucures were descrbed he coex of specfcao of R alhough he also appl o specfcao of Var α D. 8.. Nosaoar srucures Wh large ad a bouded T we eed o resrc ourselves o saoar models. For example we have alread cosdered he usrucured model for R. Makg hs specfcao mposes o addoal resrcos o R cludg saoar. The prmar advaage of saoar models s ha he provde parsmoous represeaos for he correlao srucure. However parsmoous osaoar models are also possble. To llusrae suppose ha he subjec-level damcs are specfed hrough a radom walk model ε ε - + η. Here {η } s a..d. sequece wh Var η σ η whch s depede of {ε 0 }. Wh he oao Var ε 0 σ 0 we have Var ε σ 0 + σ η ad Cov ε r ε s Var ε r σ 0 + r σ η for r < s. Ths elds R σ 0 J + σ η R RW where L L R RW 3 L 3 M M M O M 3 L T Noe ha R s a fuco of ol wo ukow parameers. Furher hs represeao allows us o specf a osaoar model whou dfferecg he daa ad hus whou losg he al se of observaos. As show Exercse 4.6 hs marx has a smple verse ha ca speed compuaos whe T s large. More geerall cosder ε ρ ε - + η whch s smlar o he AR specfcao excep ha we o loger requre saoar so ha we ma have ρ. To specf covaraces we frs defe he fuco f ρ S ρ + ρ + + ρ - ρ. f ρ ρ Pleasa calculaos show ha Var ε σ 0 + σ η S ρ ad Cov ε r ε s ρ s-r Var ε r for r < s. Ths elds R σ 0 R AR ρ + σ η R RW ρ where T ρ ρ L ρ ad R AR ρ ρ ρ M T ρ ρ ρ M T ρ ρ M T 3 L L O L ρ ρ T T 3 M

Chaper 8. Damc Models / 8-5 S S S S S S S S S S S S S S S S 3 3 3 3 3 ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ T T T T T T T RW L M O M M M L L L R. A smpler expresso assumes ha ε 0 s a cosa eher kow or a parameer o be esmaed Seco 8.5 wll dscuss a esmao mehod usg he Kalma fler. I hs case we have σ 0 0 ad ρ σ η RW R R. I s eas o see ha he Choleks square roo of ρ RW R s 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 / ρ ρ ρ ρ L L M M O M M M L L L RW R. Ths suggess usg he rasformao / * * * T T T RW T ρ ρ ρ M M M R. whch s he same as he Pras-Wso rasform excep for he frs row. The Pras-Wso rasform s he usual oe for a saoar specfcao. The po of hs example s ha we do o requre ρ < ad hus do o requre saoar. 8..3 Couous me correlao models Whe daa are o equall spaced me a aural formulao s o sll cosder subjecs draw from a populao e wh resposes as realzaos of a couous-me sochasc process. The couous-me sochasc process seg s aural he coex of bomedcal applcaos where for example we ca evso paes arrvg a a clc for esg a rregularl spaced ervals. Specfcall for each subjec we deoe he se of resposes as { for ε R}. Here deoes ha me of he observao ha we allow o exed over he ere real le R. I hs coex s covee o use he subscrp j for he order of observaos wh a subjec whle sll usg for he subjec. Observaos of he h subjec are ake a me j so ha j j deoes he jh respose of he h subjec. Smlarl le x j ad z j deoe ses of explaaor varables a me j. We ma he model he dsurbace erms ε j j z j α + x j β as realzaos from a couous me sochasc process ε. I ma applcaos hs s assumed o be a mea zero Gaussa process whch perms a wde choce of correlao srucures. Parcularl for uequall spaced daa a paramerc formulao for he correlao srucure s useful. I hs seg we have R rs Cov ε r ε s σ ρ r s where ρ s he correlao fuco of {ε }. Two wdel used choces clude he expoeal correlao model ρu exp φ u for φ > 0 8.5 ad he Gaussa correlao model ρu exp φ u for φ > 0. 8.6

8-6 / Chaper 8. Damc Models Dggle Heager Lag ad Zeger 00S provde addoal deals regardg he couousme model. Aoher advaage of couous me sochasc process models s ha he easl perm dexg b ordergs oher ha me. B far he mos eresg orderg oher ha me s a spaal orderg. Spaal ordergs are of eres whe we wsh o model pheomea where resposes ha are close o oe aoher geographcall ed o be relaed o oe aoher. For some applcaos s sraghforward o corporae spaal correlaos o our models. Ths s doe b allowg d j o be some measure of spaal or geographcal locao of he jh observao of he h subjec. The usg a measure such as Eucldea dsace we erpre d j d k o be he dsace bewee he jh ad kh observaos of he h subjec. Oe could use he correlao srucure eher of he equaos 8.5 or 8.6. Aoher sraghforward approach ha hadles oher applcaos s o reverse he role of ad j allowg o represe he me perod or replcao ad j o represe he subjec. To llusrae suppose ha we cosder observg purchases of surace each of he ff saes he US over e ears. Suppose ha mos of he heerogee s due o he perod effecs ha s chages surace purchases are flueced b chages he cour-wde ecoom. Because surace s regulaed a he sae level we expec each sae o have dffere expereces due o local regulao. Furher we ma be cocered ha saes close o oe aoher share smlar ecoomc evromes ad hus wll be more relaed o oe aoher ha saes ha are geographcall dsa. Wh hs reversal of oao he vecor represes all he subjecs he h me perod ad he erm α represes emporal heerogee. However he basc lear mxed effecs model hs approach esseall gores cross-secoal heerogee ad reas he model as successve depede cross-secos. More deals o hs approach are Seco 8.. To see how o allow for cross-secoal heerogee emporal damcs ad spaal correlaos smulaeousl cosder a basc wo-wa model α + λ + x β + ε where for smplc we assume balaced daa. Sackg over we have α x ε α + + + λ x β ε. M M M M α x ε where s a vecor of oes. We re-wre hs as α + λ + X β + ε. 8.7 Defe H Var ε o be he spaal varace marx whch we assume does o var over me. Specfcall he jh eleme of H s H j Cov ε ε j σ ρ d d j where d s a measure of geographc locao. Assumg ha {λ } s..d. wh varace σ λ we have Var Var α + σ λ + Var ε σ α I + σ λ J + H σ α I + V H. Sackg over we ma express equao 8.7 as a specal case of he mxed lear model wh T. Because Cov r s σ α I for r s we have V Var σ α I J T + V H I T. I s eas o verf ha V - σ α I + T V H - T V H - J T + V H I T. Thus s sraghforward o compue he regresso coeffce esmaor ad he lkelhood as equao 3.0. For example wh X X X T he geeralzed leas squares esmaor of β s

Chaper 8. Damc Models / 8-7 b GLS X V - X - X V - T T T X r σ α I + T VH T VH X s + r s T T X X V H X T α I + T VH T VH s + X VH σ. r r s Reurg o he smpler case of o subjec heerogee suppose ha σ α 0. I hs case we have b GLS T T X V X X V H H. 8.3 Cross-secoal correlaos ad me-seres crossseco models Cross-secoal correlaos are parcularl mpora sudes of govermeal us such as saes or aos. I some felds such as polcal scece whe T s large relave o he daa are referred o as me-seres cross-seco daa. Ths omeclaure dsgushes hs se-up from he pael coex where s large relave o T. To llusrae accordg o Beck ad Kaz 995O me-seres cross-seco daa would pcall rage from 0 o 00 subjecs wh each subjec observed over a log perod perhaps 0 o 50 ears; ma cross-aoal sudes have raos of o T ha are close o. Such sudes volve ecoomc socal or polcal comparsos of coures or saes; because of he lkages amog govermeal us he eres s models ha perm subsaal coemporaeous correlao. Followg he polcal scece leraure we cosder a me-seres cross-seco TSCS model of he form X β + ε 8.8 ha summarzes T resposes over me. Ulke pror chapers we allow for correlao across dffere subjecs hrough he oao Covε ε j V j. Because s o large relave o T fxed effecs heerogee erms could easl be corporaed o he regresso coeffces β b usg bar dcaor dumm varables. Icorporao of radom effec heerogee erms would volve a exeso of he curre dscusso; we follow he leraure ad gore hs aspec for ow. Smso 985O surves a rage of models of eres polcal scece. To complee he specfcao of he TSCS model we eed o make a assumpo abou he form of V j. Four basc specfcaos of cross-secoal covaraces are: σ I j Vj. Ths s he radoal model se-up whch ordar leas squares s 0 j effce. σ I j Vj. Ths specfcao perms heerogee across subjecs. 0 j σ j s Cov ε ε js. Ths specfcao perms cross-secoal correlaos across 0 s subjecs. However observaos from dffere me pos are ucorrelaed.

8-8 / Chaper 8. Damc Models Covε ε js σ j for s ad ε ρ ε - + η. Ths specfcao perms coemporaeous cross-correlaos as well as ra-subjec seral correlao hrough a AR model. Moreover wh some mld addoal assumpos he model has a eas o erpre cross-lag correlao fuco of he form Cov ε ε σ ρ for s <. s js j j The TSCS model s esmaed usg feasble geeralzed leas squares procedures. A he frs sage ordar leas square resduals are used o esmae he varace parameers. Oe ca hk of he model as separae regresso equaos ad use seemgl urelaed regresso echques descrbed Seco 6.4. o compue esmaors. I was he coex of seemgl urelaed regressos ha Parks 967S proposed he coemporaeous cross-correlao wh ra-subjec seral AR correlao model. Geeralzed leas square GLS esmao a regresso coex has drawbacks ha are well documeed see for example Carroll ad Rupper 988G. Tha s GLS esmaors are more effce ha ordar leas squares OLS esmaors whe he varace parameers are kow. However because varace parameers are rarel kow oe mus use sead feasble GLS esmaors. Asmpocall feasble GLS are jus as effce as GLS esmaors. However fe samples feasble GLS ma be more or less effce ha OLS esmaors depedg o he regresso desg ad dsrbuo of dsurbaces. For he TSCS model ha allows for crosssecoal covaraces here are +/ varace parameers. Moreover for he Parks model here are addoal seral correlao parameers. As documeed b Beck ad Kaz 995O he TSCS coex havg hs ma varace parameers meas ha feasble GLS esmaors are effce regresso desgs ha are pcall of eres polcal scece applcaos. Thus Beck ad Kaz 995O recommed usg OLS esmaors of regresso coeffces. To accou for he cross-secoal correlaos he recommed usg sadard errors ha are robus o he presece of cross-secoal correlaos ha he call pael-correced sadard errors. I our oao hs s equvale o he robus sadard errors roduced Seco.5.3 whou he subjec-specfc fxed effecs e reversg he roles of ad. Tha s for he asmpoc heor we ow requre depedece over me e allow for cross-secoal correlao across subjecs. Specfcall for balaced daa oe compues pael-correced sadard errors as: Procedure for compug pael-correced sadard errors. Calculae OLS esmaors of β b OLS ad he correspodg resduals e x b OLS. T. Defe he esmaor of he jh cross-secoal covarace o be ˆ σ j T ee j. 3. Esmae he varace of b OLS usg X ˆ X σ jxx j XX. j For ubalaced daa seps ad 3 eed o be modfed o alg daa from he same me perods. Beck ad Kaz 995O provde smulao sudes ha esablsh ha he robus - sascs resulg from he use of pael-correced sadard errors are preferable o he ordar - sascs usg eher OLS or feasble GLS. The also argue ha hs procedure ca be used wh seral AR correlao b frs applg a Pras-Wso rasformao o he daa o duce depedece over me. Usg smulao he demosrae ha hs procedure s superor o he feasble GLS esmaor usg he Parks model. For geeral applcaos we cauo he reader

Chaper 8. Damc Models / 8-9 ha b reversg he roles of ad oe ow reles heavl o he depedece over me sead of subjecs. The presece of eve mld seral correlao meas ha he usual same asmpoc approxmaos are o loger vald. Thus alhough pael-correced sadard errors are deed robus o presece of cross-secoal correlaos o use hese procedures oe mus be especall careful abou modelg he damc aspecs of he daa. 8.4 Tme-varg coeffces 8.4. The model Begg wh he basc wo-wa model equao 8. more geerall we use subjec-varg erms z α α + + z αq α q z α α ad me-varg erms z λ λ + + z λr λ r z λ λ. Wh hese erms we defe he logudal daa mxed model wh me-varg coeffces as z α α + z λ λ + x β + ε T. 8.9 Here α α α q s a q vecor of subjec-specfc erms ad z α z α z αq s he correspodg vecor of covaraes. Smlarl λ λ λ r s a r vecor of mespecfc erms ad z λ z λ z λr s he correspodg vecor of covaraes. We use he oao T o dcae he ubalaced aure of he daa. A more compac form of equao 8.9 ca be gve b sackg over. Ths elds a marx form of he logudal daa mxed model Z α α + Z λ λ + X β + ε.. 8.0 X x x K xt of dmeso T K Z α z α z α... z α T of dmeso T q marx ad z λ 0 L 0 0 z 0 Z λ L λ : 0 0 M 0 M z L O M λ T of dmeso T rt where 0 s a T rt-t zero marx. Fall λ λ λ T s he rt vecor of me-specfc coeffces. We assume ha sources of varabl {ε } {α } ad {λ } are muuall depede ad mea zero. The o-zero meas are accoued for he β parameers. The dsurbaces are depede bewee subjecs e we allow for seral correlao ad heeroscedasc hrough he oao Var ε σ R. Furher we assume ha he subjec-specfc effecs {α } are radom wh varace-covarace marx σ D a q q posve defe marx. Tme-specfc effecs λ have varace-covarace marx σ Σ λ a rt rt posve defe marx. For each varace compoe we separae ou he scale parameer σ o smplf he esmao calculaos descrbed Appedx 8A.. Wh hs oao we ma express he varace of each subjec as Var σ V α + Z λ Σ λ Z λ where V α Z α D Z α + R. To see how he model equao 8.0 s a specal case of he mxed lear model ake Ths expresso uses marces of covaraes K ε ε ε ε K α α α α K

8-0 / Chaper 8. Damc Models X Z λ Zα 0 0 L 0 X Z λ 0 Zα 0 L 0 X X Z M 3 λ Z λ3 ad Zα 0 0 Zα3 L 0. M M M M O M X Z λ 0 0 0 L Zα Wh hese choces we ca express he model equao 8.0 as a mxed lear model gve b Z α α + Z λ λ + X β + ε. 8.4. Esmao B wrg equao 8.0 as a mxed lear model we ma appeal o he ma esmao resuls for hs laer class of models. To llusrae for kow varace parameers drec calculaos show ha he geeralzed leas squares GLS esmaor of β s b GLS X V X X V where Var σ V. Furher here s a vare of was of calculag esmaors of varace compoes. These clude maxmum lkelhood resrced maxmum lkelhood ad several ubased esmao echques. For smaller daa ses oe ma use mxed lear model sofware drecl. For larger daa ses drec appeal o such sofware ma be compuaoall burdesome. Ths s because he me-varg radom varables λ are commo o all subjecs oblerag he depedece amog subjecs. However compuaoal shorcus are avalable ad are descrbed deal Appedx 8A.. Example Forecasg Wscos loer sales coued Table 8. repors he esmao resuls from fg he wo-wa error compoes model equao 8. wh ad whou a AR erm. For comparso purposes he fed coeffces for he oe-wa model wh a AR erm are also preseed hs able. As Table 4.3 we see ha he goodess of f sasc AIC dcaes ha he more complex wo-wa models provde a mproved f compared o he oe-wa models. As wh he oe-wa models he auocorrelao coeffce s sascall sgfca eve wh he me-varg parameer λ. I each of he hree models Table 8. ol he populao sze POP ad educao levels MEDSCHYR have a sgfca effec o loer sales.

Chaper 8. Damc Models / 8- Table 8. Loer Model Coeffce Esmaes Based o -sample daa of 50 ZIP codes ad T 35 weeks. The respose s aural logarhmc sales. Oe-wa Error compoes model wh AR erm Two-wa Error compoes model Two-wa Error compoes model wh AR erm Varable Parameer -sasc Parameer -sasc Parameer -sasc esmae esmae esmae Iercep 3.8.8 6.477.39 5.897.3 PERPERHH -.085 -.36 -.0 -.43 -.80 -.40 MEDSCHYR -0.8 -.53-0.98 -.79-0.948 -.70 MEDHVL 0.04 0.8 0.00 0.7 0.00 0.75 PRCRENT 0.03.53 0.08.44 0.09.49 PRC55P -0.070 -.0-0.07 -.00-0.07 -.0 HHMEDAGE 0.8.09 0.8.06 0.0.08 MEDINC 0.043.58 0.004.59 0.004.59 POPULATN 0.057.73 0.00 5.45 0.00 4.6 NRETAIL 0.0 0.0-0.009 -.07-0.003-0.6 Var α σ α 0.58 0.564 0.554 Var ε σ 0.79 0.0 0.04 Var λ σ λ 0.4 0.4 AR corr ρ 0.555 5.88 0.58 5.54 AIC 70.97-09.6-574.0 8.4.3 Forecasg For forecasg we wsh o predc T + L z α T + L α + z λ T + L λ T + L + x T + L β + ε T + L 8. for L lead me us he fuure. We use Chaper 4 resuls for bes lear ubased predco BLUP. To calculae hese predcors we use he sum of squares S ZZ Z λ Vα Z λ. The deals of he dervao of BLUPs are Appedx 8A.3. As ermedae resuls s useful o provde he BLUPs for λ ε ad α. The BLUP of λ urs ou o be λ BLUP S + Σ λ ZZ Z λ Vα e GLS 8. where we use he vecor of resduals e GLS - X b GLS. The BLUP of ε urs ou o be e BLUP R Vα e GLS Z λ λ BLUP 8.3 ad he BLUP of α s DZ V e Z λ. 8.4 a BLUP α α GLS λ We remark ha he BLUP of ε ca also be expressed as e Z a + Z λ BLUP X b BLUP α BLUP λ BLUP + GLS.

8- / Chaper 8. Damc Models ˆ Wh hese quaes he BLUP forecas of T + L s T + L x T + LbGLS + z α T + Lα BLUP + z λ T + L Cov λ T + L λ σ Σλ λ BLUP T L + ε R e BLUP + Cov ε σ. 8.5 A expresso for he varace of he forecas error Var ˆ T + L T + L s gve Appedx 8A.3 equao 8A.0. Equaos 8. - 8.5 provde suffce srucure o calculae forecass for a wde vare of models. Addoal compuaoal deals appear Appedx 8A.3. Sll s srucve o erpre he BLUP forecas a umber of specal cases. We frs cosder he case of depedel dsrbued me-specfc compoes {λ }. Example 8.4. Idepede me-specfc compoes We cosder he specal case where {λ } are depede ad assume ha T + L > T so ha Cov λ T + L λ 0. Thus from equao 8.4 we have he BLUP forecas of T + L s ˆ T + L x T + L b GLS + z α T + La BLUP + Cov T L ε σ R e BLUP + ε. 8.6 Ths s smlar appearace o he forecas formula Chaper 4 equao 4.4. However oe ha eve whe {λ } are depede he me-specfc compoes appear b GLS e BLUP ad a BLUP. Thus he presece of {λ } flueces he forecass. Example 8.4. Tme-varg coeffces Suppose ha he model s x β + ε where {β } are..d. We ca re-wre hs as: z λ λ + x β + ε where E β β λ β - β ad z λ x. Wh hs oao ad equao 8.6 he forecas of T + L s ˆ b. T + L x T + L Example 8.4.3 Two-wa error compoes model Cosder he basc wo-wa model gve equao 8. wh {λ } beg depede ad decall dsrbued. Here we have ha q r ad D σ α /σ z α T +L Z α. Furher Z λ I : 0 where 0 s a T T-T marx of zeroes ad Σ λ σ λ /σ I T. Thus from equao 8.6 we have ha he BLUP forecas of T + L s ˆ a + b. Here from equao 8. we have λ BLUP Z where Z λ s gve above GLS T + L BLUP x T + L GLS σ λ Vα Z λ + IT Z λ Vα e GLS σ λ ζ V α I T J ad α Tσ ζ. Furher equao 8.6 elds σ + T σ α

Chaper 8. Damc Models / 8-3 ζ. a BLUP xb GLS Z λ λ BLUP T For addoal erpreao we assume balaced daa so ha T T; see Balag 995E Tσ α page 38. To ease oao recall ζ. Here we have σ + Tσ α ζ σ ˆ λ T + L x T + L b GLS + ζ xb GLS x b GLS. σ + ζ σ λ Example 8.4.4 Radom walk model Through mor modfcaos oher emporal paers of commo e uobserved compoes ca be easl cluded. For hs example we assume ha r {λ } are decall ad depedel dsrbued so ha he paral sum process {λ +λ + +λ } s a radom walk process. Thus he model s z α α + s λs + x β + ε T. 8.7 Sackg over hs ca be expressed marx form as equao 8.0 where he T T marx Z λ s a lower ragular marx of s for he frs T rows ad zero elsewhere. Tha s 0 0 L 0 0 L 0 0 L 0 0 L 0 Z 0 0 0 λ L L. M M M O 0 0 L 0 0 0 L L T Thus ca be show ha ˆ b + λ + z α + Cov ε ε σ R e. T + L x T + L GLS s s BLUP α T + L BLUP T L BLUP + 8.5 Kalma fler approach The Kalma fler approach orgaed me seres aalss. I s a echque for esmag parameers from complex recursvel specfed ssems. The esseal dea s o use echques from mulvarae ormal dsrbuos o express he lkelhood recursvel a easl compuable fasho. The parameers ma be derved usg maxmum or resrced maxmum lkelhood. If hs s our frs exposure o Kalma flers please skp ahead o he example Seco 8.6 ad he roduco of he basc algorhm Appedx D. We ow cosder a class of models kow as sae space models. These models are well kow he me seres leraure for her flexbl descrbg broad caegores of damc srucures. As we wll see he ca be readl f usg he Kalma f algorhm. These models have bee explored he logudal daa aalss leraure exesvel b Joes 993S. We use rece modfcaos roduced b Tsmkas ad Ledoler 998S of hs srucure for lear mxed effecs models. Specfcall we cosder equao 8.9 whch he me seres leraure s called he observao equao. The me-specfc quaes of equao 8.9 are λ λ... λ r ; hs vecor s our prmar mechasm for specfg he damcs. I s updaed recursvel hrough he raso equao λ Φ λ - + η. 8.8 Here {η } are decall ad depedel dsrbued mea zero radom vecors. Wh sae space models s also possble o corporae a damc error srucure such as a ARp model. The auoregressve of order p ARp model for he dsurbaces {ε } has he form

8-4 / Chaper 8. Damc Models ε φ ε - +φ ε - + + φ p ε -p + ζ 8.9 where {ζ } are all assumed o be decall ad depedel dsrbued mea zero radom varables. Harve 989S llusraes he wde rage of choces of damc error srucures. Furher we shall see ha sae space models readl accommodae spaal correlaos amog he resposes. Boh he lear mxed effecs models ad he sae space models are useful for forecasg. Because here s a uderlg couous sochasc process for he dsurbaces boh allow for uequall spaced me observaos. Furhermore boh accommodae mssg daa. Boh classes of models ca be represeed as specal cases of he lear mxed model. For sae space models he relaoshp o lear mxed models has bee emphaszed b Tsmkas ad Ledoler 994S 997S 998S. Perhaps because of her loger hsor we fd ha he lear mxed effecs models are easer o mpleme. These models are ceral adequae for daa ses wh shorer me dmesos. However for loger me dmesos he addoal flexbl provded b he ewer sae space models leads o mprove model fg ad forecasg. We frs express he logudal daa model equaos 8.9 ad 8.0 as a specal case of a more geeral sae space model. To hs ed hs seco cosders he raso equaos he se of observaos avalable ad he measureme equao. I he descrbes how o calculae he lkelhood assocaed wh hs geeral sae space model. The Kalma fler algorhm s a mehod for effcel calculag he lkelhood of complex me seres usg codog argumes. Appedx D roduces he geeral dea of he algorhm as well as exesos o clude fxed ad radom effecs. Ths seco preses ol he compuaoal aspecs of he algorhm. 8.5. Traso equaos We frs collec he wo sources of damc behavor ε ad λ o a sgle raso equao. Equao 8.8 specfes he behavor of he lae me-varg varable λ. For he measureme error ε we assume ha s govered b a Markova srucure. Specfcall we use a ARp srucure as specfed equao 8.9. To hs ed defe he p vecor ξ ε ε - ε -p+ so ha we ma wre φ φ L φ p φ p ζ 0 L 0 0 0 ξ 0 L 0 0 ξ 0 + Φ ξ + η. M M O M M M 0 0 L 0 0 The frs row s he ARp model equao 8.9. Sackg hs over elds ξ Φ ξ η ξ M M + M I Φ ξ + η. ξ Φ ξ η Here ξ s a p vecor I s a de marx ad s a Kroecker drec produc see Appedx A.6. We assume ha {ζ } are decall dsrbued wh mea zero ad varace σ. The spaal correlao marx s defed as H Varζ ζ / σ for all. We assume o cross-emporal spaal correlao so ha Covζ s ζ j 0 for s. Thus

Chaper 8. Damc Models / 8-5 Var p 0 0 0 H η σ where 0 p- s a p- p- zero marx. To alze he recurso we use ζ 0 0. We ma ow collec he wo sources of damc behavor ε ad λ o a sgle raso equao. From equaos 8.8 ad 8.9 we have η Tδ η ξ λ Φ I 0 0 Φ η η ξ Φ I λ Φ ξ λ δ + + +. 8.0 We assume ha {λ } ad {ξ } are depede sochasc processes ad express he varace usg * Var Var Var p Q 0 0 0 H 0 0 Q η 0 0 η η Q σ σ. 8. Fall o alze he recurso we assume ha δ 0 s a vecor of parameers o be esmaed. 8.5. Observao se To allow for ubalaced daa we use oao aalogous o ha roduced Seco 5.4.. Specfcall le { } deoe he se of subjecs ha are avalable a me where { } { }. Furher defe M o be he desg marx ha has a he j h colum ad zero oherwse j. Wh hs desg marx we have M M M. Wh hs oao we have q α α α I M α α α M M. 8. Smlarl wh ε 0 0 ξ we have 0 0 0 0 0 0 0 0 ξ ξ ξ I ξ ξ ξ M L L M L L M ε ε ε p ξ M ξ ξ ξ I M I 0 0 0 0 L M L. 8.3 8.5.3 Measureme equaos Wh hs observao se for he h me perod ad equao 8.9 we ma wre

/ Chaper 8. Damc Models 8-6 + + + ε ε ε M M M L M O M M L L M M λ z z z α α α z 0 0 0 z 0 0 0 z β x x x λ λ λ α α α 8.4 Wh equaos 8. ad 8.3 we have ξ W λ Z α Z X β λ α + + + δ W α Z β X + + 8.5 where x x x X M q I M z 0 0 0 z 0 0 0 z Z Z α α α α L M O M M L L α α α α M λ λ λ λ z z z Z M ad W 0 0 L M. Thus W [Z λ : W ] so ha [ ] ξ W λ Z ξ λ W Z δ W λ λ +. Equao 8.5 collecs he me observaos o a sgle expresso. To complee he model specfcao we assume ha {α } are decall ad depedel dsrbuo wh mea zero ad D σ - Var α. Thus Var α σ I D. Here we wre he varace of α as a marx mes a cosa σ so ha we ma cocerae ou he cosa he lkelhood equaos. 8.5.4 Ial codos We frs re-wre he measureme ad observao equaos so ha he al uobserved sae vecor δ 0 s zero. To hs ed defe 0 * δ T δ δ r r. Wh hese ew varables we ma express equaos 8.5 ad 8.0 as r r ε W δ Z α δ T W X β + + + + * 0 8.6 ad δ * T δ - * + η. 8.7 where δ 0 * 0. Wh equao 8.6 we ma cosder he al sae varable δ 0 o be eher fxed radom or a combao of he wo. Wh our assumpos of ζ 0 0 ad λ 0 as fxed we ma rewre equao 8.6 as: * 0 : r r δ W α Z λ β Φ W X + +. 8.8

Chaper 8. Damc Models / 8-7 Hece he ew vecor of fxed parameers o be esmaed s β λ 0 wh correspodg K+r marx of covaraes X : W Φr. Thus wh hs reparameerzao we r heceforh cosder he sae space model wh he assumpo ha δ 0 0. 8.5.5 The Kalma fler algorhm We defe he rasformed varables * X* ad Z* as follows. Recursvel calculae: ad d +/ T + d /- + K - W d /- d +/ X T + d /- X+ K X - W d /- X d +/ Z T + d /- Z+ K Z - W d /- Z P +/ T + P /- - P /- W F - W P /- T + + Q + F + W + P +/ W + K + T + P +/ W + F - +. 8.9a 8.9b 8.9c 8.30a 8.30b 8.30c We beg he recursos equaos 8.9a-c wh d /0 0 d /0 X 0 ad d /0 Z 0. Also for equao 8.30a use P /0 Q. The h compoe of each rasformed varable s * - W d /- X * X - W d /- X 8.3a 8.3b Z * Z - W d /- Z. 8.3c From equaos 8.9-8.3 oe ha he calculao of he rasformed varables are uaffeced b scale chages {Q }. Thus usg he sequece {Q *} defed equao 8. he Kalma fler algorhm elds he same rasformed varables ad rescaled codoal varaces F * σ - F. Lkelhood equaos To calculae parameer esmaors ad he lkelhood we use he followg sums of T T T * * * * * * * * * squares: S XXF X F X S XZF X F Z S ZZF Z F Z S XF T T T * * * * * * X F S ZF Z F ad S F geeralzed leas square esmaor of β s: * * * F. Wh hs oao he b GLS { S XXF - S XZF S ZZF + I D - - S ZXF } - { S XF - S ZF S ZZF + I D - - S ZXF }. 8.3 Le τ deoe he vecor of he oher varace compoes so ha σ τ represe all varace compoes. We ma express he coceraed logarhmc lkelhood as: Lσ τ - { N l π + N l σ + σ - Error SS T + * l de F + l de D + l des ZZF + I D - }. 8.33

8-8 / Chaper 8. Damc Models where Error SS S F - S ZF S ZZF + I D - - S ZF S XF - S ZF S ZZF + I D - - S ZXF b GLS. 8.34 The resrced logarhmc lkelhood s: L REML σ τ - {l de SXXF - S XZF S ZZF + I D - - S ZXF - K l σ }+ Lσ τ 8.35 up o a addve cosa. Esmaes of he varace compoes σ ad τ ma be deermed eher b maxmzg 8.33 or 8.35. Ths ex uses 8.35 whch eld he REML esmaors. The resrced maxmum lkelhood esmaor of σ s: s REML Error SS / N-K. 8.36 Wh equao 8.35 he coceraed resrced log lkelhood s: L REML τ - {l de SXXF - S XZF S ZZF + I D - - Maxmzg L REML τ over τ elds he REML esmaor of τ. S ZXF - K l s REML }+ L s REML τ.8.37 8.6 Example: Capal asse prcg model The capal asse prcg model CAPM s a represeao ha s wdel used facal ecoomcs. A uvel appealg dea ad oe of he basc characerscs of he CAPM model s ha here should be a relaoshp bewee he performace of a secur ad he performace of he marke. Oe raoale s smpl ha f ecoomc forces are such ha he marke mproves he hose same forces should ac upo a dvdual sock suggesg ha also mprove. We measure performace of a secur hrough he reur. To measure performace of he marke several marke dces exs for each exchage. As a llusrao below we use he reur from he value weghed dex of he marke creaed b he Ceer for Research Secures Prces CRSP. The value weghed dex s defed b assumg a porfolo s creaed whe vesg a amou of moe proporo o he marke value a a cera dae of frms lsed o he New York Sock Exchage he Amerca Sock Exchage ad he Nasdaq Sock Marke. Aoher raoale for a relaoshp bewee secur ad marke reurs comes from facal ecoomcs heor. Ths s he CAPM heor arbued o Sharpe 964O ad Ler 965O ad based o he porfolo dversfcao deas of Markowz 95O. Oher hgs equal vesors would lke o selec a reur wh a hgh expeced value ad low sadard devao he laer beg a measure of rskess. Oe of he desrable properes abou usg sadard devaos as a measure of rskess s ha s sraghforward o calculae he sadard devao of a porfolo a combao of secures. Oe ol eeds o kow he sadard devao of each secur ad he correlaos amog secures. A oable secur s a rsk-free oe ha s a secur ha heorecall has a zero sadard devao. Ivesors ofe use a 30- da U.S. Treasur bll as a approxmao of a rsk-free secur argug ha he probabl of defaul of he U.S. goverme wh 30 das s eglgble. Posg he exsece of a rsk-free asse ad some oher mld codos uder he CAPM heor here exss a effce froer called he secures marke le. Ths froer specfes he mmum expeced reur ha vesors should demad for a specfed level of rsk. To esmae hs le we use he equao β 0 + β x m + ε 8.38 where s he secur reur excess of he rsk-free rae x m s he marke reur excess of he rsk-free rae. We erpre β as a measure of he amou of he h secur s reur ha s arbued o he behavor of he marke. Accordg o he CAPM heor he ercep β 0 s zero bu we clude o sud he robusess of he model.

Chaper 8. Damc Models / 8-9 To assess he emprcal performace of he CAPM model we sud secur reurs from CRSP. We cosder 90 frms from he surace carrers ha were lsed o he CRSP fles as a December 3 999. The surace carrers cosss of hose frms wh sadard dusral classfcao SIC codes ragg from 630 hrough 633 clusve. For each frm we used sx mohs of daa ragg from Jauar 995 hrough December 999. Table 8. summarzes he performace of he marke hrough he reur from he value weghed dex VWRETD ad rsk free srume RISKFREE. We also cosder he dfferece bewee he wo VWFREE ad erpre hs o be he reur from he marke excess of he rsk-free rae. Table 8.. Summar Sascs for Marke Idex ad Rsk Free Secur Based o sx mohl observaos Jauar 995 o December 999. Varable Mea Meda Mmum Maxmum Sadard devao VWRETD Value weghed dex.09.946-5.677 8.305 4.33 RISKFREE Rsk free 0.408 0.45 0.96 0.483 0.035 VWFREE Value weghed.684.57-6.068 7.880 4.34 excess of rsk free Source: Ceer for Research Secures Prces Table 8.3 summarzes he performace of dvdual secures hrough he mohl reur RET. These summar sascs are based o 5400 mohl observaos ake from 90 frms. The dfferece bewee he reur ad he correspodg rsk free srume s RETFREE. Table 8.3. Summar Sascs for Idvdual Secur Reurs Based o 5400 mohl observaos Jauar 995 o December 999 ake from 90 frms. Varable Mea Meda Mmum Maxmum Sadard devao RET Idvdual secur reur.05 0.745-66.97 0.500 0.038 RETFREE Idvdual secur reur excess of rsk free 0.645 0.340-66.579 0.085 0.036 To exame he relaoshp bewee marke ad dvdual frm reurs a rells plo s gve Fgure 8.. Here ol a subse of 8 radoml seleced frms s preseed; he subse allows oe o see mpora paers. Each pael he fgure represes a frm s experece; hus he marke reurs o he horzoal axs are commo o all frms. I parcular oe he flueal po o he lef-had sde of each pael correspodg o a Augus 998 mohl reur of 5.7%. So ha hs po would o domae a oparamerc le was f for each pael. The les supermposed show a posve relaoshp bewee he marke ad dvdual frm reurs alhough he ose abou each le s subsaal.

8-0 / Chaper 8. Damc Models -5-0 -5 0 5-5 -0-5 0 5-5 -0-5 0 5 SIGI TMK TREN TRH UICI UNM 50 0-50 EQ HSB KCLI NWLI OCAS PLFE RET 50 0-50 AET CIA CINF CSH CSLI EFS 50 0-50 -5-0 -5 0 5-5 -0-5 0 5 VWRETD -5-0 -5 0 5 Fgure 8.. Trells Plo of Reurs versus Marke Reur. A radom sample of 8 frms are ploed each pael represes a frm. Wh each pael frm reurs versus marke reurs are ploed. A oparamerc le s supermposed o provde a vsual mpresso of he relaoshp bewee he marke reur ad dvdual frm s reur. Several fxed effecs models were f usg equao 8.38 as a framework. Table 8.4 summarzes he f of each model. Based o hese fs we wll use he varable slopes wh a AR error erm model as he basele for vesgag me varg coeffces. Summar measure Homogeeous model Table 8.4. Fxed Effecs Models Varable Varable erceps slopes model model Varable erceps ad slopes model Varable slopes model wh AR erm Resdual sd 9.59 9.6 9.53 9.54 9.53 devao s - l Lkelhood 3975. 39488.6 39646.5 39350.6 3960.9 AIC 39753. 39490.6 39648.5 3935.6 3964.9 AR corr ρ -0.084 -sasc for ρ -5.98 For me-varg coeffces we vesgae models of he form: β 0 + β x m + ε 8.39 where ad ε ρ ε ε - + η 8.40 β - β ρ β β - - β + η. 8.4

Chaper 8. Damc Models / 8- Here {η } are..d. ose erms. These are depede of {η } ha are muuall depede ad decal for each frm. For equaos 8.40 ad 8.4 we assume ha {ε } ad {β } are saoar AR processes. The slope coeffce β s allowed o var b boh frm ad me. We assume ha each frm has s ow saoar mea β ad varace Var β. I s possble o vesgae he model equao 8.39-8.4 for each frm. However b cosderg all frms smulaeousl we allow for effce esmao of commo parameers β 0 ρ ε ρ β ad σ Var ε. To express hs model formulao he oao of Seco 8.3 frs defe j o be a vecor wh a oe he h row ad zeroes elsewhere. Furher defe β0 β β β β M x z λ j xm ad λ M. j xm β β β Thus wh hs oao we have β 0 + β x m + ε z λ λ + x β + ε. Ths expresses he model as a specal case of equao 8.8 gorg he me-vara radom effecs poro ad usg r me-varg coeffces. A mpora compoe of model esmao roues s Var λ σ Σ λ. Sraghforward calculaos show ha hs marx ma be expressed as Var λ R AR ρ β Σ β where Σ β σ β I ad R AR s defed Seco 8... Thus hs marx s hghl srucured ad easl verble. However has dmeso T T whch s large. Specal roues mus ake advaage of he srucure o make he esmao compuaoall feasble. The esmao procedure Appedx 8A. assumes ha r he umber of me-varg coeffces s small. See for example equao 8A.5. Thus we look o he Kalma fler algorhm for hs applcao. To appl he Kalma fler algorhm we use he followg coveos. For he updag marx for me-varg coeffces equao 8.8 we use Φ I ρ β. For he error srucure equao 8.9 we use a AR srucure so ha p ad Φ ρ ε. Thus we have β β M λ β β Iρβ 0 ρβ 0 δ ad T I ξ ε 0 I ρε 0 ρε M ε for he vecor of me-varg parameers ad updag marx. As Seco 8.5. we assume ha {λ } ad {ξ } are depede sochasc processes ad express he varace usg Var η 0 ρ β σ β I 0 Q. 0 Var η 0 ρε σ ε I To reduce he complex we assume ha he al vecor s zero so ha δ 0 0. For he measureme equaos we have

/ Chaper 8. Damc Models 8- m m m m x x x x M j j j z z z Z λ λ λ λ M M ad m m m m x x x x M j j j x x x X M M M. Furher we have W M ad hus W M x m : M. For parameer esmao we have o specfed a me-vara radom effecs. Thus we eed ol use pars a ad b of equaos 8.9 ad 8.3 as well as all of equao 8.30. To calculae parameer esmaors ad he lkelhood we use he followg sums of squares: S XXF T * * * X F X S XF T * * * F X ad S F T * * * F. Wh hs oao he geeralzed leas square esmaor of β s b GLS S XXF - S XF. We ma express he coceraed logarhmc lkelhood as: Lσ τ - { N l π + N l σ + σ - Error SS + T * de l F } where Error SS S F S XF b GLS. The resrced logarhmc lkelhood s L REML σ τ - {l de SXXF - K l σ }+ Lσ τ up o a addve cosa. For predco we ma aga use bes lear ubased predcors BLUPs roduced Chaper 4 ad exeded Seco 8.3.3. Pleasa calculaos show ha he BLUP of β s m GLS GLS m T T m GLS BLUP b b x x b b x 0 Var + β β β ρ ρ σ L 8.4 where m T m m x x L x Var ε ε β β ρ σ ρ σ AR m AR m R X R X + ad m T m m x x dag L X. Table 8.5 summarzes he f of he me-varg CAPM model based o equaos 8.39-8.4 ad he CRSP daa. Whe fg he model wh boh auoregressve processes equaos 8.40 ad 8.4 ca be dffcul o separae he damc sources hus flaeg ou he lkelhood surface. Whe he lkelhood surface s fla s dffcul o oba covergece of he lkelhood maxmzao roue. Fgure 8. shows ha he lkelhood fuco s less resposve o chages he ρ β parameer compared o he ρ ε parameer.

Chaper 8. Damc Models / 8-3 Logarhmc Lkelhood -9850 Logarhmc Lkelhood -9867-9868 -9869-9950 -9870-987 -0050-0. -0. 0.0 0. 0. Correlao Parameer - Epslo -987-9873 -9874-0. -0. 0.0 0. 0. Correlao Parameer - Bea Fgure 8.. Logarhmc lkelhood as a fuco of he correlao parameers. The lef-had pael shows he log lkelhood as a fuco of ρ ε holdg he oher parameers fxed a her maxmum lkelhood values. The rgh-had pael show log lkelhood as a fuco of ρ β. The lkelhood surface s flaer he dreco of ρ β ha ρ ε. Table 8.5 Tme-Varg CAPM Models Parameer σ ρ ε ρ β σ β Model f wh ρ ε parameer Esmae 9.57-0.084-0.86 0.864 Sadard Error 0.4 0.09 0.40 0.069 Model f whou ρ ε parameer Esmae 9.57-0.65 0.903 Sadard Error 0.4 0.6 0.068 Because of he eres he chages of he slope parameer he model was he re-f whou he correlao parameer for he ose process ρ ε. The lkelhood surface was much seeper for hs reduced model ad resulg sadard errors are much sharper as see Table 8.5. A alerave model would be o cosder ρ β equal o zero e rea ρ ε. We leave hs as a exercse for he reader. Wh he fed model parameer esmaes Table 8.5 bea predco ad forecasg s possble. For llusrao purposes we calculaed he predcos of he slope for each me po usg equao 8.4. Fgure 8. summarzes hese calculaos for he Lcol Naoal Corporao. For referece urs ou ha he geeralzed leas square esmaor of β LINCOLN for hs me perod s b LINCOLN 0.599. The upper pael of Fgure 8.3 shows he me seres of he me-varg effce predcors of he slope. The lower pael of Fgure 8.3 shows he me seres of Lcol reurs over he same me perod. Here we see he fluece of he frm s reurs o he effce predcor of β LINCOLN. For example we see ha he large drop Lcol s reur for Sepember of 999 leads o a correspodg drop he predcor of he slope.

8-4 / Chaper 8. Damc Models BLUP 0.7 0.6 0.5 0.4 995 996 997 Year 998 999 000 relc 0 0 0-0 -0 995 996 997 Year 998 999 000 Fgure 8.3. Tme seres plo of BLUP predcors of he slope assocaed wh he marke reurs ad reurs for he Lcol Naoal Corporao. The upper pael shows he BLUP predcors of he slopes. The lower pael shows he mohl reurs.

Chaper 8. Damc Models / 8-5 Appedx 8A. Iferece for he Tme-varg Coeffce Model Appedx 8A. The Model To allow for ubalaced daa recall he desg marx M specfed equao 7.. To allow for he observao se descrbed Seco 7.4. we ma use he marx form of he lear mxed effecs model equao 8.9 wh oe excepo. Tha excepo s o expad he defo Z λ a T Tr marx of explaaor varables. Wh he oao equao 7. we have λ λ λ λ M I r. 8A. M M λ T λt Thus o complee he specfcao of equao 8.9 we wre z λ 0 L 0 0 z λ L 0 Zλ M I r block dag z λ L z λ T M I r. 8A. M M O M 0 0 z L λ T To express he model more compacl we use he mxed lear model specfcao. Furher we also use he oao Var ε σ R σ blockdagr R ad oe ha Var α σ I D. Wh hs oao we ma express he varace-covarace marx of as Var σ V where V Z α I D Z α + Z λ Σ λ Z λ + R. 8A.3 Appedx 8A. Esmao For kow varaces he usual geeralzed leas squares GLS esmaor of β s b GLS X V - X - X V - he scale parameer σ drops ou. To smplf calculaos we oe ha boh R ad Z α I D Z α are block dagoal marces ad hus have readl compuable verses. Thus we defe V α R + Z α I D Z α blockdagv α V α where V α s defed Seco 8.3.. Wh hs oao we use equao A.4 of Appedx A.5 o wre V - V α + Z λ Σ λ Z λ - V α - - V α - Z λ Z λ V α - Z λ + Σ λ - - Z λ V α -. 8A.4 I equao 8A.4 ol he block dagoal marx V α ad he rt rt marx Z λ V - - α Z λ +Σ λ requre verso o he N N marx V. Defe he followg sums of squares: S XX X V α X S XZ X V α Z λ S ZZ Z λ Vα Z λ S Z Z λ Vα ad S X X V α. Wh hs oao ad equao 8A.4 we ma express he GLS esmaor of β as

8-6 / Chaper 8. Damc Models b GLS S XX - S XZ S ZZ + Σ λ - - S XZ - S X - S XZ S ZZ + Σ λ - - S Z. 8A.5 Lkelhood equaos We use he oao τ o deoe he remag parameers so ha {σ τ} represe he varace compoes. From sadard ormal heor see Appedx B he logarhmc lkelhood s Lβσ τ - {N l π + N l σ + σ - - Xβ V - - Xβ + l de V }. 8A.6 The correspodg resrced log lkelhood s L R βσ τ - {l de X V - X K l σ } + Lβσ τ + cosa. 8A.7 Eher 8A.6 or 8A.7 ca be maxmzed o deerme a esmaor of β. The resul s also he geeralzed leas squares esmaor b GLS gve equao 8A.5. Usg b GLS for β equaos 8A.6 ad 8A.7 elds coceraed lkelhoods. To deerme he REML esmaor of σ we maxmze L R b GLS σ τ holdg τ fxed o ge s REML N-K - X b GLS V - - X b GLS. 8A.8 Thus he log-lkelhood evaluaed a hese parameers s Lb GLS REML s τ - {N l π + N l s + N-K + l de V }. 8A.9 The correspodg resrced log-lkelhood s REML L R - {l de X V - X K l s REML } + Lb GLS s REML τ + cosa. 8A.0 The lkelhood expressos equaos 8A.9 ad 8A.0 are uvel sraghforward. However because of he umber of dmesos he ca be dffcul o compue. We ow provde alerave expressos ha alhough more complex appearace are smpler o compue. Usg equao 8A.4 we ma express Error SS N-K s REML V - - V - X b GLS S S Z S ZZ + Σ - λ - S Z - S X b GLS + S Z S ZZ + Σ - λ - S XZ b GLS. 8A. From equao A.5 of Appedx A.5 we have l de V l de V α + l de Σ λ + l dez λ V α - Z λ + Σ λ -. 8A. Thus he logarhmc lkelhood evaluaed a hese parameers s Lb GLS REML s τ - {N l π + N l s + N-K + S V + l de Σ λ + l des ZZ + Σ - λ } where S V l de V α REML. The correspodg resrced log-lkelhood s L R τ - {l de SXX - S XZ S ZZ + Σ - λ - S XZ K l s REML } + Lb GLS Maxmzg L R τ over τ elds he REML esmaor of τ sa τ REML. 8A.3 s REML τ + cosa. 8A.4

Chaper 8. Damc Models / 8-7 Appedx 8A.3 Predco To derve he BLUP predcor of λ we le c λ be a arbrar vecor of cosas ad se w c λ λ. Wh hs choce we have E w 0. Usg equao 4.7 we have c λ λ BLUP σ - c λ Cov λ V - - X b GLS σ - c λ Cov λ Z λ λ V - - X b GLS. Wh Wald s devce hs elds λ BLUP Σ λ Z λ V - - X b GLS. Furher usg equao 8A.4 we have Σ λ Z λ V - Σ λ Z λ V α - - V α - Z λ Z λ V α - Z λ + Σ λ - - Z λ V α - Σ λ I - S ZZ S ZZ + Σ λ - - Z λ V α - Σ λ S ZZ + Σ λ - - S ZZ S ZZ + Σ λ - - Z λ V α - S ZZ + Σ λ - - Z λ V α - Thus λ BLUP S ZZ + Σ λ - - Z λ V α - - X b GLS S ZZ + Σ λ - - S Z S ZX b GLS. 8A.5 To smplf hs expresso we recall he vecor of resduals e GLS - X b GLS. Ths elds λ BLUP S + Σ λ Z λ Vα Xb GLS S ZZ + Σ λ ZZ Z λ Vα e GLS as equao 8.. We ow cosder predcg a lear combao of resduals w c ε ε where c ε s a vecor of cosas. Wh hs choce we have E w 0. Sraghforward calculaos show ha c ε σ R for j Cov w j. 0 for j Usg hs equaos 4.7 8A.4 ad 8A.5 eld c ε e BLUP σ - Covc ε ε V - - X b GLS σ - Covc ε ε V α - - X b GLS - Covc ε ε V α - Z λ S ZZ + Σ λ - - S Z S ZX b GLS c + ε R Vα X b GLS Z λ S ZZ Σ λ S Z S ZXb GLS c R V e Z λ ε α GLS λ BLUP. Ths elds equao 8.3. Smlarl we derve he BLUP predcor of α. Le c α be a arbrar vecor of cosas ad se w c α α. For hs choce of w we have E w 0. Furher we have σ c DZ Cov c α j α α α 0 Usg hs equao 8A.4 ad 4.7 elds c α α BLUP σ - Covc α α V - - X b GLS for for j. j σ - Covc α α V α - - X b GLS - Covc α α V α - Z λ S ZZ + Σ λ - - S Z S ZX b GLS

8-8 / Chaper 8. Damc Models c DZ V e Z λ α α α GLS λ BLUP Usg Wald s devce we have he BLUP of α gve equao 8.4. Forecasg Frs oe from he calculao of BLUPs equao 4.7 ha he BLUP projeco s lear. Tha s cosder esmag he sum of wo radom varables w + w. The s mmedae ha BLUPw + w BLUPw + BLUPw. Wh hs ad equao 8.9 we have BLUP BLUP z α + BLUP z λ + BLUP x β BLUP ε ˆ T + L T + L α T + L λ T + L T + L T + L + T + L z α T + LaBLUP + z λ T + LBLUP λ T + L + x T + LbGLS + BLUP ε T L. + From equao 4.7 ad he expresso of λ BLUP we have BLUP λ σ Cov λ V Xb σ Cov λ λ Z λv Xb T + L T + L GLS σ Cov λ λ Σ λ. From equao 4.7 ad he calculao of e BLUP we have BLUP ε σ Cov ε V Xb T + L T + L σ Covε T + L ε V α T + L GLS λ BLUP. T + L + XbGLS Z λ SZZ Σλ SZ SZXbGLS e Z λ Cov ε ε R e σ Cov ε T + L ε Vα GLS λ BLUP T + L BLUP Thus he BLUP forecas of T + L s σ. ˆ T + L x T + LbGLS + σ z λ T + L Cov λ T + L λ Σλ λ BLUP + z α T + Lα BLUP GLS T + L ε R e BLUP + σ Cov ε as equao 8.6. For forecasg we wsh o predc w T + L gve equao 8.. I s eas o see ha Var σ z Dz + z Var λ z Var ε. 8A.6 Nex we have Cov T + L + + + α T L α T + L λ T + L T + L λ T + L T L To calculae he varace of he forecas error we use equao 4.9. Frs oe ha X V - X S XX - S XZ S ZZ + Σ λ - - S XZ. 8A.7 V - X Cov T + L j T + L j j Vα X - j j Cov Vα Z T + L j j λ j SZZ + Σλ SZX. 8A.8

Chaper 8. Damc Models / 8-9 Smlarl we have Cov L T + V - Cov L T + + + j j L T j j L T Cov Cov V α - + + + j j j j L T j j j j L T Cov Cov λ α λ ZZ λ α Z V Σ S Z V. 8A.9 Thus usg equao 4.8 he varace of he forecas error s L T L T + + ˆ Var + + + + X V x X X V X V x Cov Cov L T L T L T L T Cov Cov + + V L T L T + Var L T + 8A.0 where Cov L T + V - X s specfed equao 8A.8 X V - X s specfed equao 8A.7 Var L T + s specfed equao 8A.5 ad Cov Cov + + V L T L T s specfed equao 8A.9.

Chaper 9. Bar Depede Varables / 9-003 b Edward W. Frees. All rghs reserved Chaper 9. Bar Depede Varables Absrac. Ths chaper cosders suaos where he respose of eres akes o values 0 or a bar depede varable. To llusrae oe could use o dcae wheher or o a subjec possesses a arbue or o dcae a choce made; for example wheher or o a axpaer emplos a professoal ax preparer o fle come ax reurs. Regresso models ha descrbe he behavor of bar depede varables are more complex ha lear regresso models. Thus Seco 9. revews basc modelg ad fereal echques whou he heerogee compoes. Recall ha we refer o models whou heerogee compoes as homogeeous models. Secos 9. ad 9.3 clude heerogee compoes b descrbg radom ad fxed effecs models. Seco 9.4 roduces a broader class of models kow as margal models ha ca be esmaed usg a mome-based procedure kow as geeralzed esmag equaos. 9. Homogeeous models To roduce some of he complexes ecouered wh bar depede varables deoe he probabl ha he respose equals b p Prob. The we ma erpre he mea respose o be he probabl ha he respose equals ha s E 0 Prob 0 + Prob p. Furher sraghforward calculaos show ha he varace s relaed o he mea hrough he expresso Var p - p. Lear probabl models Whou heerogee erms we beg b cosderg a lear model of he form x β + ε 9. kow as a lear probabl model. Assumg E ε 0 we have ha E p x β ad Var x β -x β. Lear probabl models are wdel appled because of he ease of parameer erpreaos. For large daa ses he compuaoal smplc of ordar leas squares esmaors s aracve whe compared o some complex alerave olear models roduced below. Furher ordar leas squares esmaors for β have desrable properes. I s sraghforward o check ha he are cosse ad asmpocall ormal uder mld codos o he explaaor varables {x }. However lear probabl models have several drawbacks ha are serous for ma applcaos. These drawbacks clude: The expeced respose s a probabl ad hus mus var bewee 0 ad. However he lear combao x β ca var bewee egave ad posve f. Ths msmach mples for example ha fed values ma be ureasoable. Lear models assume homoscedasc cosa varace e he varace of he respose depeds o he mea ha vares over observaos. The problem of varg varabl s kow as heeroscedasc. The respose mus be eher a 0 or alhough he regresso models pcall regards dsrbuo of he error erm as couous. Ths msmach mples for example ha he usual resdual aalss regresso modelg s meagless.

9- / Chaper 9. Bar Depede Varables To hadle he heeroscedasc problem a wo-sage weghed leas squares procedure s possble. Tha s he frs sage oe uses ordar leas squares o compue esmaes of β. Wh hs esmae a esmaed varace for each subjec ca be compued usg he relao Var x β -x β. A he secod sage a weghed leas squares s performed usg he verse of he esmaed varaces as weghs o arrve a ew esmaes of β. I s possble o erae hs procedure alhough sudes have show ha here are few advaages dog so see Carroll ad Ruper 988. Aleravel oe ca use ordar leas square esmaors of β wh sadard errors ha are robus o heeroscedasc. 9.. Logsc ad prob regresso models Usg olear fucos of explaaor varables To crcumve he drawbacks of lear probabl models we cosder alerave models whch we express he expecao of he respose as a fuco of explaaor varables p πx β Prob x. We focus o wo specal cases of he fuco π.: z e π z he log case ad z z + e e + πz Φz he prob case. Here Φ. s he sadard ormal dsrbuo fuco. Noe ha he choce of he de fuco a specal kd of lear fuco πz z elds he lear probabl model. Thus we focus o olear choces of π. The verse of he fuco π - specfes he form of he probabl ha s lear he explaaor varables ha s π - p x β. I Chaper 0 we wll refer o hs verse as he lk fuco. These wo fucos are smlar ha he are almos learl relaed over he erval 0. π 0.9 see McCullagh ad Nelder 989 page 09. Ths smlar meas ha wll be dffcul o dsgush bewee he wo specfcaos wh mos daa ses. Thus o a large exe he fuco choce s depede o he prefereces of he aals. Threshold erpreao Boh he log ad prob cases ca be jusfed b appealg o he followg hreshold * erpreao of he model. To hs ed suppose ha here exss a uderlg lear model x β + ε *. *. Here we do o observe he respose e erpre o be he propes o possess a characersc. For example we mgh hk abou he speed of a horse as a measure of s propes o w a race. Uder he hreshold erpreao we do o observe he propes bu we do observe whe he propes crosses a hreshold. I s cusomar o assume ha hs * 0 0 hreshold s 0 for smplc. Thus we observe. * > 0 To see how he log case ca be derved from he hreshold model we assume a log dsrbuo fuco for he dsurbaces so ha Prob ε * a. Because he log + exp a * * dsrbuo s smmerc abou zero we have ha Prob ε a Prob ε a. Thus * * p Prob Prob > 0 Prob ε x β πx β. + exp x β Ths esablshes he hreshold erpreao for he log case. The developme for he prob case s smlar ad s omed.

Chaper 9. Bar Depede Varables / 9-3 Radom ul erpreao Boh he log ad prob cases ca also be jusfed b appealg o he followg radom ul erpreao of he model. I ecoomc applcaos we hk of a dvdual as selecg bewee wo choces. To llusrae Seco 9..3 we wll cosder wheher or o a axpaer chooses o emplo a professoal ax preparer o asss flg a come ax reur. Here prefereces amog choces are dexed b a uobserved ul fuco; dvduals selec he choce ha provdes he greaer ul. For he h dvdual a he h me perod we use he oao u for hs ul. We model ul as a fuco of a uderlg value plus radom ose ha s U j u V j + ε j where j ma be 0 or correspodg o he choce. To llusrae we assume ha he dvdual chooses he caegor correspodg o j f U > U 0 ad deoe hs choce as. Assumg ha u s a srcl creasg fuco we have Prob Prob U 0 < U Prob u V0 + ε 0 < u V + ε Prob ε 0 ε < V V0. To parameerze he problem assume ha he value fuco s a ukow lear combao of explaaor varables. Specfcall we ake V 0 0 ad V x β. We ma ake he dfferece he errors ε 0 - ε o be ormal or logsc correspodg o he prob ad log cases respecvel. I Seco. we wll show ha he logsc dsrbuo s sasfed f he errors are assumed o have a exreme-value or Gumbel dsrbuo. I Seco 9..3 lear combaos of axpaer characerscs wll allow us o model he choce of usg a professoal ax preparer. The aalss allows for axpaer prefereces o var b subjec ad over me. Example 9. Job secur Vellea 999E suded declg job secur usg he PSID Pael Surve of Icome Damcs daabase see Appedx F. We cosder here oe of he regressos preseed b Valea based o a sample of male household heads ha cosss of N 468 observaos over he ears 976-99 clusve. The PSID surve records reasos wh me lef her mos rece emplome cludg pla closures qu ad chaged jobs for oher reasos. However Valea focused o dsmssals lad off or fred because voluar separaos are assocaed wh job secur. Chaper wll expad hs dscusso o cosder he oher sources of job urover. Table 9. preses a prob regresso model ru b Valea 999E usg dsmssals as he depede varable. I addo o he explaaor varables lsed Table 9. oher varables corolled for cossed of educao maral saus umber of chldre race ears of full-me work experece ad s square uo membershp goverme emplome logarhmc wage he U.S. emplome rae ad locao as measured hrough he Meropola Sascal Area resdece. I Table 9. eure s ears emploed a he curre frm. Furher secor emplome was measured b examg CPS Cosumer Prce Surve emplome 387 secors of he ecoom based o 43 dusr caegores ad e regos of he cour. O he oe had he eure coeffce reveals ha more expereced workers are less lkel o be dsmssed. O he oher had he coeffce assocaed wh he eraco bewee eure ad me red reveals a creasg dsmssal rae for expereced workers. The erpreao of he secor emplome coeffces s also of eres. Wh a average eure of abou 7.8 ears he sample we see he low eure me are relavel uaffeced b chages secor emplome. However for more expereced me here s a creasg probabl of dsmssal assocaed wh secors of he ecoom where growh decles. Valea also f a radom effecs model ha wll be descrbed Seco 9.; he resuls were qualavel smlar o hose preseed here.

9-4 / Chaper 9. Bar Depede Varables Table 9. Dsmssal Prob Regresso Esmaes Varable Parameer esmae Sadard error Teure -0.084 0.00 Tme Tred -0.00 0.005 Teure*Tme Tred 0.003 0.00 Chage Logarhmc Secor Emplome 0.094 0.057 Teure* Chage Logarhmc Secor Emplome -0.00 0.009 - Log Lkelhood 707.8 Pseudo-R 0.097 Logsc regresso A advaage of he log case s ha perms closed-form expressos ulke he ormal dsrbuo fuco. Logsc regresso s aoher phrase used o descrbe he log case. Usg p πz he verse of π ca be calculaed as z π - p l p/-p. To smplf fuure preseaos we defe logp l p/-p o be he log fuco. Wh logsc regresso model we represe he lear combao of explaaor varables as he log of he success probabl ha s x β log p. Odds rao erpreao Whe he respose s bar kowg ol he probabl of p summarzes he dsrbuo. I some applcaos a smple rasformao of p has a mpora erpreao. The lead example of hs s he odds rao gve b p/-p. For example suppose ha dcaes wheher or o a horse ws a race ha s f he horse ws ad 0 f he horse does o. Ierpre p o be he probabl of he horse wg he race ad as a example suppose ha p 0.5. The he odds of he horse wg he race s 0.5/.00-0.5 0.3333. We mgh sa ha he odds of wg are 0.3333 o or oe o hree. Equvalel we ca sa ha he probabl of o wg s - p 0.75. Thus he odds of he horse o wg s 0.75/ - 0.75 3. We erpre hs o mea he odds agas he horse are hree o oe. Odds have a useful erpreao from a beg sadpo. Suppose ha we are plag a far game ad ha we place a be of $ wh odds of oe o hree. If he horse ws he we ge our $ back plus wgs of $3. If he horse loses he we lose our be of $. I s a far game he sese ha he expeced value of he game s zero because we w $3 wh probabl p 0.5 ad lose $ wh probabl - p 0.75. From a ecoomc sadpo he odds provde he mpora umbers be of $ ad wgs of $3 o he probables. Of course f we kow p he we ca alwas calculae he odds. Smlarl f we kow he odds we ca alwas calculae he probabl p. The log s he logarhmc odds fuco also kow as he log odds. Logsc regresso parameer erpreao To erpre he regresso coeffces he logsc regresso model β β β β K we beg b assumg ha jh explaaor varable x j s eher 0 or. The wh he oao x x L x L x j K β we ma erpre x L L x β x L 0 L x β j K K Prob xj Prob xj 0 l l. Prob xj Prob xj 0

Chaper 9. Bar Depede Varables / 9-5 Thus β e j Prob Prob x x j j / Prob 0 / Prob x x j j. 0 We oe ha he umeraor of hs expresso s he odds whe x j whereas he deomaor s he odds whe x j 0. Thus we ca sa ha he odds whe x j are expβ j mes as large as he odds whe x j 0. To llusrae f β j 0.693 he expβ j. From hs we sa ha he odds for are wce as grea for x j as x j 0. Smlarl assumg ha jh explaaor varable s couous dffereable we have Prob xj / Prob xj Prob xj xj β l j xβ. x j xj Prob xj Prob xj / Prob xj Thus we ma erpre β j as he proporoal chage he odds rao kow as a elasc ecoomcs. 9.. Iferece for logsc ad prob regresso models Parameer esmao The cusomar mehod of esmao for homogeous models s maxmum lkelhood descrbed furher deal Appedx C. To provde uo we oule he deas he coex of bar depede varable regresso models. The lkelhood s he observed value of he des or mass fuco. For a sgle observao he lkelhood s p f 0. The objecve of maxmum lkelhood p f esmao s o fd he parameer values ha produce he larges lkelhood. Fdg he maxmum of he logarhmc fuco ofe elds he same soluo as fdg he maxmum of he correspodg fuco. Because s geerall compuaoall smpler we cosder he logarhmc log- lkelhood wre as l p f 0. More compacl he loglkelhood of a sgle observao s l p f l π x β + l π x β where p πx β. Assumg depedece amog observaos he lkelhood of he daa se s a produc of lkelhoods of each observao. Thus akg logarhms he log-lkelhood of he daa se s he sum of log-lkelhoods of sgle observaos. The log-lkelhood of he daa se s L β { l π x β + l π x β } 9. where he sum rages over { T }. The log lkelhood s vewed as a fuco of he parameers wh he daa held fxed. I coras he jo probabl mass des fuco s vewed as a fuco of he realzed daa wh he parameers held fxed. The mehod of maxmum lkelhood meas fdg he values of β ha maxmze he loglkelhood. The cusomar mehod of fdg he maxmum s akg paral dervaves wh respec o he parameers of eres ad fdg roos of he hese equaos. I hs case akg paral dervaves wh respec o β elds he score equaos

9-6 / Chaper 9. Bar Depede Varables L β β π x β 0. 9. x π x β π x β π x β The soluo of hese equaos sa b MLE s he maxmum lkelhood esmaor. To llusrae for he log fuco he score equaos equao 9. reduce o x π x β 0 9.3 where πz + exp-z -. We oe ha he soluo depeds o he resposes ol hrough he sascs Σ x. Ths proper kow as suffcec wll be mpora Seco 9.3. A alerave expresso for he score equaos equao 9. s E Var E 0 9.4 β where E π x β E x π x β ad Var π x β π x β. The expresso β equao 9.4 s a example of a geeralzed esmag equao ha wll be roduced formall Seco 9.4. A esmaor of he asmpoc varace of β ma be calculaed akg paral dervaves of he score equaos. Specfcall he erm L β s he formao marx β β β b MLE evaluaed a b MLE. To llusrae usg he log fuco sraghforward calculaos show ha he formao marx s x x π x β π x β. The square roo of he jh dagoal eleme of hs marx elds he sadard error for he jh row of b jmle whch we deoe as seb jmle. To assess he overall model f s cusomar o ce lkelhood rao es sascs olear regresso models. To es he overall model adequac H 0 : β 0 we use he sasc LRT Lb MLE L 0 where L 0 s he maxmzed log-lkelhood wh ol a ercep erm. Uder he ull hpohess H 0 hs sasc has a ch-square dsrbuo wh K degrees of freedom. Appedx C.8 descrbes lkelhood rao es sascs greaer deal. As descrbed Appedx C.9 measures of goodess of f are dffcul o erpre olear models. Oe measure s he so-called max-scaled R R defed as Rms where R expl0/ N R expl / b MLE N value of hs log-lkelhood. ad R max 0 / expl N. Here L 0 /N represes he average max

Chaper 9. Bar Depede Varables / 9-7 9..3 Example: Icome ax pames ad ax preparers To llusrae he mehods descrbed hs seco we reur o he Icome ax pames example roduced Seco 3.. For hs chaper we ow wll use he demographc ad ecoomc characerscs of a axpaer descrbed Table 3. o model he choce of usg a professoal ax preparer deoed b PREP. The oe excepo s ha we wll o cosder he varable LNTAX he ax pad logarhmc dollars. Alhough ax pad s clearl relaed o he choce of a professoal ax preparer s o clear ha hs ca serve as a depede explaaor varable. I ecoomerc ermolog hs s cosdered a edogeous varable see Chaper 6. Ma summar sascs of he daa were dscussed Secos 3. ad 7..3. Tables 9. ad 9. show addoal sascs b level of PREP. Table 9. shows ha hose axpaers usg a professoal ax preparer PREP were more lkel o be marred o he head of a household age 65 ad over ad self-emploed. Table 9. shows ha hose axpaers usg a professoal ax preparer had more depedes larger come ad were a hgher ax bracke. TABLE 9. Averages of Idcaor Varables b Level of PREP PREP Number MS HH AGE EMP 0 67 0.54 0.06 0.07 0.09 69 0.709 0.066 0.65 0. Table 9. Summar Sascs for Oher Varables b Level of PREP Varable PREP Mea Meda Mmum Maxmum Sadard devao DEPEND 0.67 0 6.30.585 0 6.358 LNTPI 0 9.73 9.9-0.8.043.089 0.059 0.78-0.09 3..0 MR 0.987 0 50.68 5.88 5 0 50.536 Table 9.3 provdes addoal formao abou he relao bewee EMP ad PREP. To llusrae for hose self-emploed dvduals EMP 67.9% 3/93 of he me he chose o use a ax preparer compared o 44.5% 488/097 for hose o self-emploed. Pu aoher wa he odds of self-emploed usg a preparer are. 0.679/-0.679 compared o 0.80 0.445/-0.445 for hose o self-emploed. Table 9.3 Cous of Taxpaers b Levels of PREP ad EMP EMP 0 Toal PREP 0 609 6 67 488 3 69 Toal 097 93 90 Dspla 9. shows a fed logsc regresso model usg LNTPI MR ad EMP as explaaor varables. The calculaos were doe usg SAS PROC LOGISTIC. To erpre hs oupu we frs oe ha he lkelhood rao es sasc for checkg model adequac s LRT 67.4 Lb MLE L 0 786.3 78.98. Compared o a ch-square wh K3 degrees of freedom hs dcaes ha a leas oe of he varables LNTPI MR ad EMP s a sascall sgfca predcors of PREP. Addoal model f sascs cludg Akake s formao crero AIC ad Schwarz s crero SC are descrbed Appedx C.9.

9-8 / Chaper 9. Bar Depede Varables We erpre he R sasc o mea ha here s subsaal formao regardg PREP ha s o explaed b LNTPI MR ad EMP. I s useful o cofrm he calculao of hs sasc hs beg exp L0/ N exp 786.3/90 R 0.05079. exp L / N exp 78.98/90 b MLE Dspla 9. Seleced SAS Oupu The LOGISTIC Procedure Model Iformao Respose Varable PREP Number of Respose Levels Number of Observaos 90 Lk Fuco Log Opmzao Techque Fsher's scorg Model F Sascs Iercep Iercep ad Crero Ol Covaraes AIC 788.3 76.98 SC 793.385 747.630 - Log L 786.3 78.98 R-Square 0.0508 Max-rescaled R-Square 0.0678 Tesg Global Null Hpohess: BETA0 Tes Ch-Square DF Pr > ChSq Lkelhood Rao 67.4 3 <.000 Score 65.0775 3 <.000 Wald 60.5549 3 <.000 Aalss of Maxmum Lkelhood Esmaes Sadard Parameer DF Esmae Error Ch-Square Pr > ChSq Iercep -.3447 0.7754 9.430 0.005 LNTPI 0.88 0.0940 4.007 0.0455 MR 0.008 0.00884.4964 0. EMP.009 0.693 35.539 <.000 Odds Rao Esmaes Po 95% Wald Effec Esmae Cofdece Lms LNTPI.07.004.45 MR.0 0.994.09 EMP.743.969 3.8

Chaper 9. Bar Depede Varables / 9-9 For parameer erpreao we oe ha he coeffce assocaed wh EMP s b EMP.009. Thus we erpre he odds assocaed wh hs esmaor exp.009.743 o mea ha self-emploed axpaers EMP are.743 mes more lkel o emplo a professoal ax preparer compared o axpaers ha are o self-emploed. 9. Radom effecs models Ths seco roduces models ha use radom effecs o accommodae heerogee. Seco 9.3 follows up wh he correspodg fxed effec formulao. I coras he lear models poro of he ex we frs roduced fxed effecs Chaper followed b radom effecs Chaper 3. The cossec bewee hese seemgl dffere approaches s ha he ex approaches daa modelg from a applcaos oreao. Specfcall for esmao ad ease of explaaos wh users pcall he fxed effecs formulao s smpler ha he radom effecs alerave lear models. Ths s because fxed effecs models are smpl specal cases of aalss of covarace models represeaos ha are famlar from appled regresso aalss. I coras olear cases such as models wh bar depede varables radom effecs models are smpler ha correspodg fxed effecs aleraves. Here hs s par compuaoal because radom effecs summar sascs are easer o calculae. Furher as we wll see Seco 9.3 sadard esmao roues such as maxmum lkelhood eld fxed effecs esmaors ha do o have he usual desrable asmpoc properes. Thus he fxed effecs formulao requres specalzed esmaors ha ca be cumbersome o compue ad expla o users. As Seco 9. we expressed he probabl of a respose equal o oe as a fuco of lear combaos of explaaor varables. To accommodae heerogee we corporae subjec-specfc varables of he form Prob α πα + x β. Here he subjec-specfc effecs accou ol for he erceps ad do o clude oher varables. Chaper 0 wll roduce exesos o varable slope models. We assume ha {α } are radom effecs hs seco. To movae he radom effecs formulao we ma assume he wo-sage samplg scheme roduced Seco 3.. Sage. Draw a radom sample of subjecs from a populao. The subjecspecfc parameer α s assocaed wh he h subjec. Sage. Codoal o α draw realzaos of { x } for T for he h subjec. I he frs sage oe draws subjec-specfc effecs {α } from a populao. I he secod sage for each subjec oe draws a radom sample of T resposes T ad also observes he explaaor varables {x }. Radom effecs lkelhood To develop he lkelhood frs oe ha from he secod samplg sage codoal o α he lkelhood for he h subjec a he h observao s π α + x β f p ; β α. π α + x β f 0 We summarze hs as p ; β α π α + x β π α + x β. Because of he depedece amog resposes for a subjec codoal o α he codoal lkelhood for he h subjec s

9-0 / Chaper 9. Bar Depede Varables T α + x β π α + x p ; β α π β. Takg expecaos over α elds he ucodoal lkelhood. Thus he ucodoal lkelhood for he h subjec s T p ; β τ π a + x β π a + x β d Fα a. 9.5 I equao 9.5 τ s a parameer of he dsrbuo of α F α.. Alhough o ecessar s cusomar o use a ormal dsrbuo for F α.. I hs case τ represes he varace of hs mea zero dsrbuo. Wh a specfcao for F α he log-lkelhood for he daa se s L β τ l p β τ. To deerme maxmum lkelhood esmaors oe maxmzes he log-lkelhood Lβ τ as a fuco of β ad τ. Closed form aalcal soluos for hs maxmzao problem do o exs geeral alhough umercal soluos are feasble wh moder compug equpme. The maxmum lkelhood esmaors ca be deermed b solvg for he roos of he K + score equaos L β τ 0 ad L β τ 0. β τ Furher asmpoc varaces ca be compued b akg he marx of secod dervaves of he log-lkelhood kow as he formao marx. Appedx C provdes addoal deals. There are wo commol used specfcaos of he codoal dsrbuo he radom effecs model. A log model for he codoal dsrbuo of a respose. Tha s Prob α π α + x β. + exp α + x β A prob model for he codoal dsrbuo of a respose. Tha s Prob α Φ α + β where Φ s he sadard ormal dsrbuo fuco. x There are o mpora advaages or dsadvaages whe choosg he codoal probabl π o be eher a log or a prob. The lkelhood volves roughl he same amou of work o evaluae ad maxmze alhough he log fuco s slghl easer o evaluae ha he sadard ormal dsrbuo fuco. The prob model ca be easer o erpre because ucodoal probables ca be expressed erms of he sadard ormal dsrbuo fuco. Tha s assumg ormal for α we have x β Prob E Φ α + x β Φ. + τ Example - Icome ax pames ad ax preparers - Coued To see how a radom effecs depede varable model works wh a daa se we reur o he Seco 9..3 example. Dspla 9. shows a fed model usg LNTPI MR ad EMP as explaaor varables. The calculaos were doe usg he SAS procedure NLMIXED. Ths procedure uses a umercal egrao echque for mxed effec models called adapve Gaussa quadraure see Phero ad Baes 000 for a descrpo. Dspla 9. shows ha hs radom effecs specfcao s o a desrable model for hs daa se. B codog o he radom effecs he parameer esmaes ur ou o be hghl correlaed wh oe aoher.

Chaper 9. Bar Depede Varables / 9- Dspla 9. Seleced SAS Oupu The NLMIXED Procedure Specfcaos Depede Varable PREP Dsrbuo for Depede Varable Bomal Dsrbuo for Radom Effecs Normal Opmzao Techque Dual Quas-Newo Iegrao Mehod Adapve Gaussa Quadraure F Sascs - Log Lkelhood 79.7 AIC smaller s beer 79.7 AICC smaller s beer 79.8 BIC smaller s beer 755.5 Correlao Marx of Parameer Esmaes Row Parameer blntpi bmr bemp sgma.0000-0.9995-0.9900-0.9986-0.9988 blntpi -0.9995.0000 0.985 0.9969 0.997 3 bmr -0.9900 0.985.0000 0.9940 0.9944 4 bemp -0.9986 0.9969 0.9940.0000 0.9997 5 sgma -0.9988 0.997 0.9944 0.9997.0000 Mullevel model exesos We saw ha here were a suffce umber of applcaos o devoe a ere chaper 5 o a mullevel framework for lear models. However exesos o olear models such as bar depede varable models are ol ow comg o regular use he appled sceces. Ths subseco preses he developme of hree-level exesos of prob ad log models due o Gbbos ad Hedeker 997B. We use a se of oao smlar o ha developed Seco 5... Le here be subjecs schools for example j J clusers wh each subjec classrooms for example ad T j observaos wh each cluser over me or sudes wh a classroom. Combg he hree levels Gbbos ad Hedeker cosdered * α + z α + x β + ε. 9.6 j j Here α represes a level-hree heerogee erm α j represes a level-wo vecor of heerogee erms ad ε j represes he level-oe dsurbace erm. All hree erms have zero mea; he meas of each level are alread accoued for x j β. The lef-had varable of equao 9.6 s lae; we acuall observe j ha s a bar varable correspodg o wheher he lae varable * j crosses a hreshold. The codoal probabl of hs eve s Pr ob α α π α + z α + x β j j j where π. s eher log or prob. To complee he model specfcao we assume ha {α } ad {α j } are each..d. ad depede of oe aoher. j j j j j

9- / Chaper 9. Bar Depede Varables The model parameers ca be esmaed va maxmum lkelhood. As poed ou b Gbbos ad Hedeker he ma compuaoal echque s o ake advaage of he depedece ha we pcall assume amog levels mullevel modelg. Tha s defg j j jtj he codoal probabl mass fuco s T j j α + z α + x β π α + z α + x j p ; β α α π β. j j j j Iegrag over he level-wo heerogee effecs we have p j ; β Σ α p j ; β α a d Fα a. 9.7 a Here F α. s he dsrbuo fuco of {α j } ad Σ are he parameers assocaed wh. Followg Gbbos ad Hedeker we assume ha α j s ormall dsrbued wh mea zero ad varace-covarace Σ. Iegrag over he level-hree heerogee effecs we have j p... ; β Σ σ 3 p ; β Σ a d F 3 a. 9.8 J Here F α3. s he dsrbuo fuco of {α } ad σ 3 s he parameer assocaed wh pcall ormal. Wh equaos 9.7 ad 9.8 he log-lkelhood s Σ σ 3 a J j l p... J ; β Σ 3 j L β σ. 9.9 Compuao of he log-lkelhood requres umercal egrao. However he egral equao 9.8 s ol -dmesoal ad he egral equao 9.7 depeds o he dmeso of {α j } sa q ha s pcall ol or. Ths s coras o a more drec approach ha combes he heerogee erms α α α J. The dmeso of hs vecor s +q J ; usg hs drecl equao 9.5 s much more compuaoall ese. j j j α Example 9.. Faml smokg preveo To llusrae her procedures Gbbos ad Hedeker 997B cosdered daa from he Televso School ad Faml Smokg Preveo ad Cessao Projec. I her repor of hs sud Gbbos ad Hedeker cosdered 600 seveh grade sudes from 35 classrooms wh 8 schools. The daase was ubalaced; here were bewee ad 3 classrooms from each school ad bewee ad 8 sudes from each classroom. The schools were radoml assged o oe of four sud codos: a socal ressace classroom whch a school-based currculum was used o promoe obacco use preveo ad cessao a elevso based currculum a combao of socal ressace ad elevso based currcula ad o reames corol. A obacco ad healh scale was used o classf each sude as kowledgeable or o boh before ad afer he erveo. Table 9.4 provdes emprcal probables of sudes kowledge afer he erveos b pe of erveo. Ths able suggess ha socal ressace classroom currcula are effecve promog obacco preveo awareess.

Chaper 9. Bar Depede Varables / 9-3 Table 9.4 Tobacco ad Healh Scale Pos-erveo Performace Emprcal Probables Perce. Socal Televso Cou Kowledgeable ressace based Yes No classroom currculum No No 4 4.6 58.4 No Yes 46 48.3 5.7 Yes No 380 63. 36.8 Yes Yes 383 60.3 39.7 Toal 600 5.9 47. Gbbos ad Hedeker esmaed boh log ad prob models usg pe of erveo ad performace of he pre-erveo es as explaaor varables. The cosdered a model wh radom effecs as well as a model wh classroom as he secod level ad school as he hrd level as well as wo wo-level models for robusess purposes. For boh models he socal ressace classroom currculum was sascall sgfca. However he also foud ha he model whou radom effecs dcaed ha he elevso based erveo was sascall sgfca whereas he hree-level model dd o reveal such a srog effec. 9.3 Fxed effecs models As Seco 9. we express he probabl of he respose beg a oe as a olear fuco of lear combaos of explaaor varables. To accommodae heerogee we corporae subjec-specfc varables of he form p πα + x β. Here he subjec-specfc effecs accou ol for he erceps ad do o clude oher varables. Exesos o varable slope models are possble bu as we wll see eve varable ercep models are dffcul o esmae. We assume ha {α } are fxed effecs hs seco. Maxmum lkelhood esmao Smlar o equao 9. he log-lkelhood of he daa se s L { l π α + x β + l π α + x β }. 9.0 Ths log-lkelhood ca be maxmzed o eld maxmum lkelhood esmaors of α ad β ha we deoe as a MLE ad b MLE. Noe ha here are + K parameers o be esmaed smulaeousl. As Seco 9. we cosder he log specfcao of π so ha p π α + x β. 9. + exp α + x β Because l πx/-πx x we have ha he log-lkelhood equao 9.0 s π α + x β L l π α + x β + l + π α xβ { l π α + x β + α + x β }. 9. Sraghforward calculaos show ha he score equaos are:

9-4 / Chaper 9. Bar Depede Varables L α π α + x β L β α. 9.3 ad x π + x β Fdg he roos of hese equaos eld our maxmum lkelhood esmaors. Example - Icome ax pames ad ax preparers - Coued To see how maxmum lkelhood works wh a daa se we reur o he Seco 9..3 example. For hs daa se we have 58 axpaers ad cosder K3 explaaor varables LNTPI MR ad EMP. Fg hs model elds - log-lkelhood 46.04 ad R 0.6543. Accordg o sadard lkelhood rao ess he addoal ercep erms are hghl sascall sgfca. Tha s he lkelhood rao es sasc for assessg he ull hpohess H 0 : α α 58 s LRT 78.98 46.04 30.957. The ull hpohess s rejeced based o a comparso of hs sasc wh a ch-square dsrbuo wh 58 degrees of freedom. Uforuael he above aalss s based o approxmaos ha are kow o be urelable. The dffcul s ha as he subjec sze eds o f he umber of parameers also eds o f. I urs ou ha our abl o esmae β s corruped b our abl o esmae cossel he subjec-specfc effecs {α }. I coras he lear case maxmum lkelhood esmaors are equvale o he leas squares esmaors ha are cosse. The leas squares procedure sweeps ou ercep esmaes whe producg esmaes of β. Ths s o he case olear regresso models. To ge a beer feel for he pes of hgs ha ca go wrog suppose ha we have o explaaor varables. I hs case from dspla 9.3 he roo of he score equao s L exp α + 0. α + exp α The soluo a MLE s exp a MLE or a MLE log. + exp a MLE Thus f he a MLE ad f 0 he a MLE -. Thus ercep esmaors are urelable hese crcumsaces. A examao of he score fucos dspla 9.3 shows ha smlar pheomea also occurs eve he presece of explaaor varables. To llusrae Example 9. we have 97 axpaers who do o use a professoal ax preparer a of he fve ears uder cosderaos 0 whereas 89 axpaers alwas use a ax preparer. Eve whe he ercep esmaors are fe maxmum lkelhood esmaors of global parameers β are cosse fxed effecs bar depede varable models. See Example 9.. Illusrao 9. - Icossec of maxmum lkelhood esmaes Chamberla 978 Hsao 986. Now as a specal case le T K ad x 0 ad x. Usg he score equaos dspla 9. Appedx 9A shows how o calculae drecl he maxmum lkelhood esmaor of β b MLE for hs specal case. Furher Appedx 9A. argues ha he probabl lm of b MLE s β. Hece s a cosse esmaor of β.

Chaper 9. Bar Depede Varables / 9-5 Codoal maxmum lkelhood esmao To crcumve he problem of he ercep esmaors corrupg he esmaor of β we use he codoal maxmum lkelhood esmaor. Ths esmao echque s due o Chamberla 980E he coex of fxed effecs bar depede varable models. We cosder he log specfcao of π as equao 9.. Wh hs specfcao urs ou ha Σ s a suffce sasc for α. The dea of suffcec s revewed Appedx 0A.. I hs coex meas ha f we codo o Σ he he dsrbuo of he resposes wll o deped o α. Illusrao 9. - Suffcec Coug wh he se-up of Illusrao 9. we ow llusrae how o separae he ercep from he slope effecs. Here we ol assume ha T o ha K. To show ha he codoal dsrbuo of he resposes do o deped o α beg b supposg ha he sum Σ + equals eher 0 or. Cosder hree cases. For he frs case assume ha sum equals 0. The he codoal dsrbuo of he resposes s Prob 0 0 + sum ; hs clearl does o deped o α. For he secod case assume ha sum equals. The he codoal dsrbuo of he resposes s Prob + sum whch also does o deped o α. Saed aoher wa f s eher 0 or he he sasc ha s beg codoed o deermes all he resposes resulg o corbuo o a codoal lkelhood. Now cosder he hrd case where he sum equals. Basc probabl calculaos esablsh ha Prob + Prob 0 Prob + Prob Prob 0 exp α + x β + exp α + x β + exp α + x β + exp α + x β Thus f sum equals he Prob 0 Prob Prob 0 + Prob + exp α + x β exp x β. exp α + x β + exp α + x β exp x β + exp x β Thus he codoal dsrbuo of he resposes does o deped o α. We also oe ha f a explaaor varable x j s me-cosa x j x j he he correspodg parameer β j dsappears from he codoal lkelhood.

9-6 / Chaper 9. Bar Depede Varables Codoal lkelhood esmao To defe he codoal lkelhood le S be he radom varable represeg Σ ad le sum be he realzao of Σ. Wh hs oao he codoal lkelhood of he daa se s T p p. Prob S sum Noe ha he rao wh he curl brackes equals oe whe sum equal 0 or T. Takg he log of he fuco ad he fdg values of β ha maxmze elds b CMLE he codoal maxmum lkelhood esmaor. We remark ha hs ca be compuaoall dffcul. Tha s he dsrbuo of S s mess ad s dffcul o compue for moderae sze daa ses wh T more ha 0. Appedx 9A. provdes deals. Illusrao 9.3 - Codoal maxmum lkelhood esmaor To see ha he codoal maxmum lkelhood esmaor s cosse a case where he maxmum lkelhood s o we coue wh Illusrao 9.. As argued Illusrao 9. we eed ol be cocered wh he case +. The codoal lkelhood s + p p exp xβ exp xβ. 9.4 Prob + + exp xβ exp xβ As Example 9. ake K ad x 0 ad x. The b akg he dervave wh respec o β of he log of he codoal lkelhood ad seg hs equal o zero oe ca deerme explcl he codoal maxmum lkelhood esmaor deoed as b CMLE. Sraghforward lm heor shows hs o be a cosse esmaor of β. Appedx 9A. provdes deals. A oe o ermolog - codoal maxmum lkelhood esmao for he log model dffers from he codoal log model ha we wll roduce Seco.. 9.4 Margal models ad GEE For margal models we requre ol he specfcao of he frs wo momes specfcall he mea ad varace of a respose as well as he covaraces amog resposes. Ths s much less formao ha he ere dsrbuo as requred b he lkelhood based approaches Secos 9. ad 9.3. Of course f he ere dsrbuo s assumed kow he we ca alwas calculae he frs wo momes. Thus he esmao echques applcable o margal models ca also be used whe he ere dsrbuo s specfed. Margal models are esmaed usg a specal pe of mome esmao kow as he geeralzed esmag equaos or GEE approach he bologcal sceces. I he socal sceces hs approach s par of he geeralzed mehod of momes or GMM. For he applcaos ha we have md s mos useful o develop he esmao approach usg he GEE oao. However aalss should keep md ha hs esmaor s reall jus aoher pe of GMM esmaor. To descrbe GEE esmaors oe mus specf a mea varace ad covarace srucure. To llusrae he developme we beg b assumg ha he Seco 9. radom effecs model s vald ad we wsh o esmae parameers of hs dsrbuo. The geeral GEE procedure s descrbed Appedx C.6 ad wll be furher developed Chaper 0.

Chaper 9. Bar Depede Varables / 9-7 GEE esmaors for he radom effecs bar depede varable model From Seco 9. we have ha he codoal frs mome of he respose s E α Prob α πα + x β. Thus he mea ma be expressed as µ µ β τ π a + x β d F a. 9.5 α Recall ha τ s a parameer of he dsrbuo fuco F α.. For example f F α. represes a ormal dsrbuo he τ represes he varace. Occasoall s useful o use he oao µ β τ o remd ourselves ha he mea fuco µ depeds o he parameers β ad τ. Le µ µ µ T deoe he T vecor of meas. For hs model sraghforward calculaos show ha he varace ca be expressed as Var µ µ. Regardg covaraces for r s we have Cov r s E r s - µ r µ s E E r s α - µ r µ s E πα + x r β πα + x s β - µ r µ s a + rβ π a + x sβ d Fα a µ r µ s π x. 9.6 Le V V β τ be he usual T T varace-covarace marx for he h subjec; ha s he h dagoal eleme of V s Var whereas for o-dagoal elemes he rh row ad sh colum of V s gve b Cov r s. For GEE we also requre dervaves of cera momes. For he mea fuco from equao 9.5 we wll use µ x π a + x β d Fα a. β As s cusomar appled daa aalss hs calculao assumes a suffce amou of regular of he dsrbuo fuco F α. so ha we ma erchage he order of dffereao ad egrao. I geeral we wll use he oao µ G β µ T µ τ L β β a K T marx of dervaves. GEE esmao procedure The GEE esmaors are compued accordg o he followg geeral recurso. Beg wh al esmaors of β τ sa b 0EE τ 0EE. Tpcall al esmaors b 0EE are calculaed assumg zero covaraces amog resposes. Ial esmaors τ 0EE are compued usg resduals based o he b 0EE esmae. The a he +s sage recursvel:. Use τ EE ad he soluo of he equao V b τ µ b τ EE EE µ b 0 G τ 9.7 K o deerme a updaed esmaor of β sa b +EE.. Use he resduals { - µ b +EE τ EE } o deerme a updaed esmaor of τ sa τ +EE. 3. Repea seps ad ul covergece. EE

9-8 / Chaper 9. Bar Depede Varables Le b EE ad τ EE deoe he resulg esmaors of β ad τ. Uder broad codos b EE s cosse ad asmpocall ormal wh asmpoc varace G µ b EE τ EE V b EE τ EE G µ b EE τ EE. 9.8 The soluo b EE equao 9.7 ca be compued quckl usg eraed reweghed leas squares a procedure descrbed Appedx C.3. However he specfed esmao procedure s sll edous because reles o he umercal egrao compuaos calculag µ equao 9.5 ad Cov r s equao 9.6. Now Seco 9. we saw ha umercal egrao 9.5 could be avoded b specfg ormal dsrbuos for π ad x F α resulg µ Φ β. However eve wh hs specfcao we would sll requre + τ umercal egrao o calculae Cov r s equao 9.6. A sgle umercal egrao s sraghforward moder-da compug evrome. However evaluao of V would requre T T / umercal egraos for he covarace erms. Thus each evaluao of equao 9.7 would requre Σ {T T /} umercal egraos hs s 58 5 5-/ 580 for he Seco 9..3 example. Ma evaluaos of equao 9.7 would be requred pror o successful covergece of he recursve procedure. I summar hs approach s ofe compuaoall prohbve. To reduce hese compuaoal complexes he focus of margal models s he represeao of he frs wo momes drecl wh or whou referece o uderlg probabl dsrbuos. B focusg drecl o he frs wo momes we ma keep he specfcao smple ad compuaoall feasble. To llusrae we ma choose o specf he mea fuco as µ Φ x β. Ths s ceral plausble uder he radom effecs bar depede varable model. For he varace fuco we cosder Var φ µ - µ. Here φ s a overdsperso parameer ha we ma eher assume o be or o be esmaed from he daa. Fall s cusomar he leraure o specf correlaos leu of covaraces. Use he oao Corr r s o deoe he correlao bewee r ad s. To llusrae s commo o use he exchageable correlao srucure specfed as for r s Corr r s. ρ for r s Here he movao s ha he lae varable α s commo o all observaos wh a subjec hus ducg a commo correlao. For hs llusrao he parameers τ φ ρ cosue he varace compoes. Esmao ma he proceed as descrbed he recurso begg wh equao 9.7. However as wh lear models he secod momes ma be msspecfed. For hs reaso he correlao specfcao s commol kow as a workg correlao. For lear models weghed leas squares provdes esmaors wh desrable properes. Alhough o opmal compared o geeralzed leas squares weghed leas squares esmaors are pcall cosse ad asmpocall ormal. I he same fasho GEE esmaors based o workg correlaos have desrable properes eve whe he correlao srucure s o perfecl specfed. However f he correlao srucure s o vald he he asmpoc sadard errors provded hrough he asmpoc varace equao 9.7 are o vald. Isead emprcal sadard errors ma be calculaed usg he followg esmaor of he asmpoc varace of b EE

Chaper 9. Bar Depede Varables / 9-9 G µ V G µ G µ V µ µ V G µ G µ V G µ. 9.9 Specfcall he sadard error of he jh compoe of b EE seb jee s defed o be he square roo of he jh dagoal eleme of he varace-covarace marx dspla 9.9. Example - Icome ax pames ad ax preparers - Coued To see how a margal model works wh a daa se we reur o he Seco 9..3 example. Table 9.5 shows he f of wo models each usg LNTPI MR ad EMP as explaaor varables. The calculaos were doe usg he SAS procedure GENMOD. For he frs model he exchageable workg correlao srucure was used. Parameer esmaes as well as equao 9.9 emprcal sadard errors ad equao 9.8 model-based sadard errors appear Table 9.5. The esmaed correlao parameer ured ou o be ρˆ 0.7. For he secod model a usrucured workg correlao marx was used. Table.5. provdes a example of a usrucured covarace marx. Table 9.6 provdes hese esmaed correlaos. We ma erpre he rao of he esmae o sadard error as a -sasc ad use hs o assess he sascal sgfcace of a varable. Examg Table 9.5 we see ha LNTPI ad MR are o sascall sgfca usg eher pe of sadard error or correlao srucure. The varable EMP rages from beg srogl sascall sgfca for he case wh model-based sadard errors ad a exchageable workg correlao o beg o sascall sgfca for he case of emprcal sadard errors ad a usrucured workg correlao. Overall he GEE esmaes of he margal model provde dramacall dffere resuls whe compared o eher he Seco 9. homogeeous model or he Seco 9. radom effecs resuls. Table 9.5 Comparso of GEE Esmaors Exchageable Workg Correlao Usrucured Workg Correlao Model-Based Esmae Emprcal Sadard Sadard Error Error Parameer Esmae Emprcal Sadard Error Model-Based Sadard Error Iercep -0.9684 0.700 0.585 0.884 0.6369 0.35 LNTPI 0.0764 0.080 0.0594-0.05 0.0754 0.0395 MR 0.004 0.0083 0.0066 0.0099 0.0076 0.005 EMP 0.5096 0.676 0.74 0.797 0.90 0.49 Table 9.6 Esmae of Usrucured Correlao Marx Tme Tme Tme 3 Tme 4 Tme 5 Tme.0000 0.8663 0.707 0.6048 0.4360 Tme 0.8663.0000 0.8408 0.7398 0.573 Tme 3 0.707 0.8408.0000 0.903 0.7376 Tme 4 0.6048 0.7398 0.903.0000 0.8577 Tme 5 0.4360 0.573 0.7376 0.8577.0000

9-0 / Chaper 9. Bar Depede Varables Furher readg More exesve roducos o homogeeous bar depede varable models are avalable Agres 00G ad Hosmer ad Lemshow 000G. For a ecoomercs perspecve see Camero ad Trved 998E. For dscussos of bar depede models wh edogeous explaaor varables see Wooldrdge 00E ad Arellao ad Hooré 00E. For models of bar depede varables wh radom erceps maxmum lkelhood esmaors ca be compued usg umercal egrao echques o approxmae he lkelhood. McCulloch ad Searle 00G dscuss umercal egrao for mxed effec models. Phero ad Baes 000S descrbe he adapve Gaussa quadraure mehod ha s used SAS PROC NLMIXED. Appedx 9. Lkelhood calculaos Appedx 9A.. Cossec of Lkelhood Esmaors Illusrao 9. - Icossec of maxmum lkelhood esmaes Coued Recall ha T K ad x 0 ad x. Thus from equao 9.3 we have α α + β L e e + 0 9A. α α + β α + e + e ad α + β L e 0. 9A. α + β β + e From equao 9A. s eas o see ha f + 0 he a mle -. Furher f + he a mle. For boh cases he corbuo o he sum equao 9A. s zero. Thus we cosder he case + ad le d be he dcaor varable ha +. I hs case we have ha a mle -b mle / from equao 9A.. Pug hs o equao 9A. elds exp a + b exp b / exp b / d mle mle d d + exp a + b mle mle mle + exp b mle / mle + exp b where Σ d s he umber of subjecs where +. Thus wh he oao + + d we have b mle l. + To esablsh he cossec sraghforward weak law of large umbers ca be used o β + e show ha he probabl lm of s. Thus he probabl lm of b β mle s β ad + e hece s a cosse esmaor of β. Illusrao 9.3 - Codoal maxmum lkelhood esmaor Coued Recall ha K x 0 ad x. The from equao 9.4 he codoal lkelhood s mle /.

Chaper 9. Bar Depede Varables / 9- β β β exp β d d + exp exp + + exp because + ad d s a varable o dcae +. Thus he codoal log-lkelhood s L β c d β { β l + e }. To fd he codoal maxmum lkelhood esmaor we have β L c β β e d l + e d 0. β β β + e + The roo of hs s b CMLE l. I Example 9. we used he fac ha he probabl lm + of + s e β + e β. Thus he probabl lm of b CMLE s β ad hece s cosse. Appedx 9A.. Compug Codoal Maxmum Lkelhood Esmaors Compug he Dsrbuo of Sums of Nodecall Idepedel Dsrbued Beroull Radom Varables We beg b preseg a algorhm for he compuao of he dsrbuo of sums of odecall depedel dsrbued Beroull radom varables. Thus we ake o be depede Beroull radom varables wh Prob πx β. For coveece we use he log form of π. Defe he sum radom varable S T + + + T. We wsh o evaluae ProbS T s for s 0 T. For oaoal coveece we om he subscrp o T. We frs oe ha s sraghforward o compue Prob S T 0 T usg a log form for π. Coug we have Prob S T T T π x β { π x rβ } r r T { π } { + β exp xβ } x T T π x β { π x rβ } π x β r Usg a log form for π we have πz/-πz e z. Thus wh hs oao we have T Prob S T Prob S T 0 exp x β. Le {j j j s } be a subse of { T} ad Σ st be he sum over all such subses. Thus for he ex sep he erao we have Prob S T T x β π x β π x β π j j { r } T Prob Coug hs geeral we have S T r r j r j π x j β π x j β 0. T π x β π x β j j.

9- / Chaper 9. Bar Depede Varables Prob S T s Prob S T π x j β π x j β s 0 L. s T π x β π x β j js Usg a log form for π we ma express he dsrbuo as Prob ST s Prob ST 0 exp x j +... + x j β. 9A.3 s s T Thus eve wh he log form for π we see he dffcul compug ProbS s s ha T volves he sum over quaes ΣsT. Ths expresses he dsrbuo erms of s ProbS 0. I s also possble o derve a smlar expresso erms of ProbS T; hs alerave expresso s more compuaoall useful ha equao 9A.3 for evaluao he dsrbuo a large values of s. Compug he Codoal Maxmum Lkelhood Esmaor From Seco 9.3 he logarhmc codoal lkelhood s T lcl + l π x β l π xβ l Prob ST sumt where we have ake α o be zero whou loss of geeral. As remarked Seco 9.3 whe T summg over all subjecs we eed o cosder hose subjecs where equal 0 or T because he codoal lkelhood s decall equall o oe. To fd hose values of β ha maxmze l CL oe could use he Newo-Raphso recursve algorhm see Appedx C.. To hs ed we requre he vecor of paral dervaves T lcl x π xβ l Prob ST sumt. β β The Newo-Raphso algorhm also requres he marx of secod dervaves bu compuaoal cosderaos of hs marx are smlar o he oes for he frs dervave ad are omed. From he form of he vecor of paral dervaves we see ha he ma ask s o compue he grade of l ProbS T s. Usg a log form for π ad equao 9A.3 droppg he subscrp o T we have β l Prob S T s x +... + x l Prob S T 0 + l exp s T j j β s β β x j +... + x j exp x j + + j s... x β s exp{ x j +... + x j β} s s T s T T x exp x + exp x β. β As wh he probabl equao 9A.3 hs s eas o compue for values of s ha are close o 0 or T. However geeral he calculao requres sum over quaes Σ s st boh he umeraor ad deomaor. Moreover hs s requred for each subjec a each sage of he T

Chaper 9. Bar Depede Varables / 9-3 lkelhood maxmzao process. Thus he calculao of codoal lkelhood esmaors becomes burdesome for large values of T. 9. Exercses ad Exesos Seco 9. 9.. Threshold erpreao of he prob regresso model Cosder a uderlg lear model * x β + ε * * where ε s ormall dsrbued * wh mea zero ad varace σ 0 0. Defe. Show ha p * > 0 x β Prob Φ where Φ s he sadard ormal dsrbuo fuco. σ 9.. Radom ul erpreao of he logsc regresso model Uder he radom ul erpreao a dvdual wh ul U j u V j + ε j where j ma be 0 or selecs caegor correspodg o j wh probabl p Prob Prob U 0 < U Prob u V0 + ε 0 < u V + ε Prob ε 0 ε < V V0. Suppose ha he errors are from a exreme value dsrbuo of he form a Prob ε < a exp e. j Show ha he choce probabl p has a log form. Tha s show p. + exp x β 9.3. Margal dsrbuo of he prob radom effecs model Cosder a ormal model for he codoal dsrbuo of a respose. Tha s Prob α Φ α + x β where Φ s he sadard ormal dsrbuo fuco. Assume furher ha α s ormall dsrbued wh mea zero ad varace τ. Show ha x β p Φ. + τ Emprcal Exercse 9.4. Choce of ogur brads These daa are kow as scaer daa because he are obaed from opcal scag of grocer purchases a check-ou. The subjecs coss of 00 households Sprgfeld Mssour. The respose of eres s he pe of ogur purchased. For hs exercse we cosder ol he brad Yopla or aoher choce. The households were moored over a wo-ear perod wh he umber of purchases ragg from 4 o 85; he oal umber of purchases s N 4. More exesve movao s provded Seco.. The wo markeg varables of eres are prce ad feaures. We use wo prce varables for hs sud PY he prce of Yopla ad PRICEOTHER he lowes prce of he oher brads. For feaures hese are bar varables defed o be oe f here was a ewspaper feaure adversg he brad a me of purchase ad zero oherwse. We use wo feaure varables FY

9-4 / Chaper 9. Bar Depede Varables he feaures assocaed wh Yopla ad FEATOTHER a varable o dcae f a oher brad has a feaured dsplaed. a. Basc summar sascs Creae he basc summar sascs for Yopla ad he four explaaor varables.. Wha are he odds of purchasg Yopla?. Deerme he odds rao whe assessg wheher or o Yopla s feaured. Ierpre our resul. b. Logsc regresso models Ru a logsc regresso model wh he wo explaaor varables PY ad FY. Furher ru a secod logsc regresso model wh four explaaor varables PY FY PRICEOTHER ad FEATOTHER. Compare hese wo models ad sa whch ou prefer. Jusf our choce b appealg o sadard sascal hpohess ess. c. Radom effecs model F a radom effecs model wh four explaaor varables. Ierpre he regresso coeffces of hs model. d. GEE model F a GEE model wh four explaaor varables.. Gve a bref descrpo of he heor behd hs model.. Compare he par d descrpo o he radom effecs model par c. I parcular how does each model address he heerogee?. For hs daa se descrbe wheher or o our model choce s robus o a mpora erpreaos abou he regresso coeffces.

Chaper 0. Geeralzed Lear Models / 0-003 b Edward W. Frees. All rghs reserved Chaper 0. Geeralzed Lear Models Absrac. Ths chaper exeds he lear model roduced Par I ad he bar depede varable model Chaper 9 o he geeralzed lear model formulao. Geeralzed lear models ofe kow b he acrom GLM represe a mpora class of olear regresso models ha have foud exesve use pracce. I addo o he ormal ad Beroull dsrbuos hese models clude he bomal Posso ad Gamma famles as dsrbuos for depede varables. Seco 0. begs hs chaper wh a revew of homogeeous GLM models ha s GLM models ha do o corporae heerogee. The Seco 0. example reforces hs revew. Seco 0.3 he descrbes margal models ad geeralzed esmag equaos a wdel appled framework for corporag heerogee. The Secos 0.4 ad 0.5 allow for heerogee b modelg subjec-specfc quaes as radom ad fxed effecs models respecvel. Seco 0.6 es ogeher fxed ad radom effecs uder he umbrella of Baesa ferece. 0. Homogeeous models Ths seco roduces he geeralzed lear model GLM; a more exesve reame ma be foud he classc work b McCullagh ad Nelder 989G. The GLM framework geeralzes lear models he followg sese. Lear model heor provdes a plaform for choosg approprae lear combaos of explaaor varables o predc a respose. I Chaper 9 we saw how o use olear fucos of hese lear combaos o provde beer predcors a leas for resposes wh Beroull bar oucomes. I GLM we wde he class of dsrbuos o allow us o hadle oher pes of o-ormal oucomes. Ths broad class cludes as specal cases he ormal Beroull ad Posso dsrbuos. As we wll see he ormal dsrbuo correspods o lear models he Beroull o he Chaper 9 bar respose models. To movae GLM we focus o he Posso dsrbuo; hs dsrbuo allows us o readl model cou daa. Our reame follows he srucure roduced Chaper 9; we frs roduce he homogeous verso of he GLM framework so ha hs seco does o use a subjec-specfc parameers or does roduce erms o accou for seral correlao. Subseque secos wll roduce echques for hadlg heerogee logudal ad pael daa. The seco begs wh a roduco of he respose dsrbuo Subseco 0.. ad he shows how o lk he dsrbuo s parameers o regresso varables Subseco 0... Subseco 0..3 he descrbes esmao prcples. 0.. Lear expoeal famles of dsrbuos Ths chaper cosders he lear expoeal faml of he form θ b θ p θ φ exp + S φ. 0. φ Here s a depede varable ad θ s he parameer of eres. The qua φ s a scale parameer ha we ofe wll assume s kow. The erm bθ depeds ol o he parameer

0- / Chaper 0. Geeralzed Lear Models θ o he depede varable. The sasc S φ s a fuco of he depede varable ad he scale parameer o he parameer θ. The depede varable ma be dscree couous or a mxure. Thus p. ma be erpreed o be a des or mass fuco depedg o he applcao. Table 0A. provdes several examples cludg he ormal bomal ad Posso dsrbuos. To llusrae cosder a ormal dsrbuo wh a probabl des fuco of he form µ µ µ / f µ σ exp exp l πσ. πσ σ σ σ Wh he choces θ µ φ σ bθ θ / ad S φ - / φ + l π φ/ we see ha he ormal probabl des fuco ca be expressed as equao 0.. For he fuco equao 0. some sraghforward calculaos show ha E bθ ad Var φ b θ. For referece hese calculaos appear Appedx 0A.. To llusrae he coex of he ormal dsrbuo example above s eas o check ha E bθ θ µ ad Var σ bµ σ as acpaed. 0.. Lk fucos I regresso modelg suaos he dsrbuo of vares b observao hrough he subscrps. Specfcall we allow he dsrbuo s parameers o var b observao hrough he oao θ ad φ. For our applcaos he varao of he scale parameer s due o kow wegh facors. Specfcall whe he scale parameer vares b observao s accordg o φ φ /w ha s a cosa dvded b a kow wegh w. Wh he relao Var φ b θ φ b θ/w we have ha a larger wegh mples a smaller varace oher hgs beg equal. I regresso suaos we wsh o udersad he mpac of a lear combao of explaaor varables o he dsrbuo. I he GLM coex s cusomar o call η x β he ssemac compoe of he model. Ths ssemac compoe s relaed o he mea hrough he expresso η gµ. 0. Here g. s kow as he lk fuco. As we saw he pror subseco we ca express he mea of as E µ bθ. Thus equao 0. serves o lk he ssemac compoe o he parameer θ. I s possble o use he de fuco for g. so ha µ x β. Ideed hs s he usual case lear regresso. However lear combaos of explaaor varables x β ma var bewee egave ad posve f whereas meas are ofe resrced o smaller rage. For example Posso meas var bewee zero ad f. The lk fuco serves o map he doma of he mea fuco oo he whole real le. Specal case: Lks for he Beroull dsrbuo Beroull meas are probables ad hus var bewee zero ad oe. For hs case s useful o choose a lk fuco ha maps he u erval 0 oo he whole real le. The followg are hree mpora examples of lk fucos for he Beroull dsrbuo: Log: gµ logµ l µ/ µ. Prob: gµ Φ - µ where Φ - s he verse of he sadard ormal dsrbuo fuco. Complemear log-log: gµ l -l µ.

Chaper 0. Geeralzed Lear Models / 0-3 Ths example demosraes ha here ma be several lk fucos ha are suable for a parcular dsrbuo. To help wh he seleco a uvel appealg case occurs whe he ssemac compoe equals he parameer of eres η θ. To see hs frs recall ha η gµ ad µ bθ droppg he subscrps for he mome. The s eas o see ha f g - b he η gbθ θ. The choce of g ha s he verse of b s called he caocal lk. Table 0. shows he mea fuco ad correspodg caocal lk for several mpora dsrbuos. Table 0. Mea fucos ad caocal lks for seleced dsrbuos Dsrbuo Mea fuco bθ Caocal lk gθ Normal θ θ Beroull e θ /+ e θ logθ Posso e θ l θ Gamma -/θ - -θ - 0..3 Esmao Ths seco preses maxmum lkelhood he cusomar form of esmao. To provde uo we beg wh he smpler case of caocal lks ad he exed he resuls o more geeral lks. Maxmum lkelhood esmao for caocal lks From equao 0. ad he depedece amog observaos he log-lkelhood s θ b θ l p + S φ. 0.3 φ Recall ha for caocal lks we have equal bewee he dsrbuo s parameer ad he ssemac compoe so ha θ η x β. Thus he log-lkelhood s xβ b xβ l p + S φ. 0.4 φ Takg he paral dervave wh respec o β elds he score fuco b x β l p x. β φ Because µ bθ b x β ad φ φ /w we ca solve for he maxmum lkelhood esmaors of β b MLE hrough he ormal equaos 0 w x µ. 0.5 Oe reaso for he wdespread use of GLM mehods s ha he maxmum lkelhood esmaors ca be compued quckl hrough a echque kow as eraed reweghed leas squares descrbed Appedx C.3. Noe ha lke ordar lear regresso ormal equaos we do o eed o cosder esmao of he varace scale parameer φ a hs sage. Tha s we ca frs compue b MLE ad he esmae φ.

0-4 / Chaper 0. Geeralzed Lear Models Maxmum lkelhood esmao for geeral lks For geeral lks we o loger assume he relaoshp θ x β bu assume ha β s relaed o θ hrough he relao µ bθ ad x β gµ. Usg equao 0.3 we have ha he jh eleme of he score fuco s θ µ l p β j β j φ because bθ µ. Now use he cha rule ad he relao Var φ bθ o ge µ b θ θ Var θ b θ. β j β j β j φ β j θ µ Thus we have. Ths elds β φ β Var j j µ l p Var µ β j β j whch s kow as he geeralzed esmag equaos form. Ths s he opc of Seco 0.3. Sadard errors for regresso coeffce esmaors As descrbed Appedx C. maxmum lkelhood esmaors are cosse ad asmpocall ormall dsrbued uder broad codos. To llusrae cosder he maxmum lkelhood esmaor deermed from equao 0.4 usg he caocal lk. The asmpoc varace-covarace marx of b MLE s he verse of he formao marx ha from equao C.4 s x x b x b MLE. 0.6 φ The square roo of he jh dagoal eleme of he verse of hs marx elds he sadard error for he jh row of b jmle whch we deoe as seb jmle. Exesos o geeral lks are smlar. Overdsperso A mpora feaure of several members of he lear expoeal faml of dsrbuos such as he Beroull ad he Posso dsrbuos s ha he varace s deermed b he mea. I coras he ormal dsrbuo has a separae parameer for he varace or dsperso. Whe fg models o daa wh bar or cou depede varables s commo o observe ha he varace exceeds ha acpaed b he f of he mea parameers. Ths pheomeo s kow as overdsperso. Several alerave probablsc models ma be avalable o expla hs pheomeo depedg o he applcao a had. See Seco 0.3 for a example ad McCullagh ad Nelder 989G for a more dealed veor. Alhough arrvg a a sasfacor probablsc model s he mos desrable roue ma suaos aalss are coe o posulae a approxmae model hrough he relao Var σ φ b x β / w. The scale parameer φ s specfed hrough he choce of he dsrbuo whereas he scale parameer σ allows for exra varabl. For example Table 0A. shows ha b specfg eher he Beroull or Posso dsrbuo we have φ. Alhough he scale parameer σ

Chaper 0. Geeralzed Lear Models / 0-5 allows for exra varabl ma also accommodae suaos whch he varabl s smaller ha specfed b he dsrbuoal form alhough hs suao s less commo. Fall oe ha for some dsrbuos such as he ormal dsrbuo he exra erm s alread corporaed he φ parameer ad hus serves o useful purpose. Whe he addoal scale parameer σ s cluded s cusomar o esmae b Pearso s ch-square sasc dvded b he error degrees of freedom. Tha s b x b MLE ˆ σ w. N K φ b x b MLE 0. Example: Tor flgs There s a wdespread belef ha he Ued Saes coeous pares have become creasgl wllg o go o he judcal ssem o sele dspues. Ths s parcularl rue whe oe par s from he surace dusr a dusr desged o spread rsk amog dvduals. Lgao he surace dusr arses from wo pes of dsagreeme amog pares breach of fah ad or. A breach of fah s a falure b a par o he corac o perform accordg o s erms. A or aco s a cvl wrog oher ha breach of corac for whch he cour wll provde a remed he form of aco for damages. A cvl wrog ma clude malce waoess oppresso or caprcous behavor b a par. Geerall large damages ca be colleced for or acos because he award ma be large eough o sg he gul par. Because large surace compaes are vewed as havg deep pockes hese awards ca be que large. The respose ha we cosder s he umber of flgs NUMFILE of or acos agas surace compaes. For each of sx ears 984-989 he daa were obaed from 9 saes wh wo observaos uavalable for a oal of N observaos. The ssue s o udersad was whch sae legal ecoomc ad demographc characerscs affec he umber of flgs. Table 0. descrbes hese characerscs. More exesve movao s provded Lee 994O ad Lee Browe ad Schm 994O. Table 0. Sae Characerscs Depede Varable NUMFILE Number of flgs of or acos agas surace compaes. Sae Legal Characerscs JSLIAB A dcaor of jo ad several labl reform. COLLRULE A dcaor of collaeral source reform. CAPS A dcaor of caps o o-ecoomc reform. PUNITIVE A dcaor of lms of puve damage Sae Ecoomc ad Demographc Characerscs POP The sae populao mllos. POPLAWYR The populao per lawer. VEHCMILE Number of auomobles mles per mle of road housads. POPDENSY Number of people per e square mles of lad. WCMPMAX Maxmum workers compesao weekl beef. URBAN Perceage of populao lvg urba areas. UNEMPLOY Sae uemplome rae perceages. Source: A Emprcal Sud of he Effecs of Tor Reforms o he Rae of Tor Flgs upublshed Ph.D. Dsserao Ha-Duck Lee Uvers of Wscos 994O. Tables 0.3 ad 0.4 summarze he sae legal ecoomc ad demographc characerscs. To llusrae Table 0.3 we see ha 3. perce of he sae-ear observaos were uder lms caps o o-ecoomc reform. Those observaos o uder

0-6 / Chaper 0. Geeralzed Lear Models lms o o-ecoomc reforms had a larger average umber of flgs. The correlaos Table 0.4 show ha several of he ecoomc ad demographc varables appear o be relaed o he umber of flgs. I parcular we oe ha he umber of flgs s hghl relaed o he sae populao. Table 0.3 Averages wh Explaaor Idcaor Varables Explaaor Varable JSLIAB COLLRULE CAPS PUNITIVE Average Explaaor Varable 0.49 0.304 0.3 0.3 Average NUMFILE Whe Explaaor Varable 0 5530 077 468 7693 Whe Explaaor Varable 5967 007 677 6469 Table 0.4. Summar Sascs for Oher Varables Varable Mea Meda Mmum Maxmum Sadard devao Correlao wh NUMFILE NUMFILE 054 9085 5 37455 9039.0 POP 6.7 3.4 0.5 9.0 7. 0.90 POPLAWYR 377.3 38.5.0 537.0 75.7-0.378 VEHCMILE 654.8 50.5 63.0 899.0 55.4 0.58 POPDENSY 68. 63.9 0.9 043.0 43.9 0.368 WCMPMAX 350.0 39.0 03.0 40.0 5.7-0.65 URBAN 69.4 78.9 8.9 00.0 4.8 0.550 UNEMPLOY 6. 6.0.6 0.8.6 0.008 NUMFILE 40000 0000 00000 80000 60000 40000 0000 0 984 985 986 987 988 989 YEAR Fgure 0. Mulple me seres plo of NUMFILE.

Chaper 0. Geeralzed Lear Models / 0-7 Fgure 0. s a mulple me seres plo of he umber of flgs. The sae wh he larges umber of flgs s Calfora. Ths plo shows he sae level heerogee of flgs. Whe daa represe cous such as he umber of or flgs s cusomar o cosder he Posso model o represe he dsrbuo of resposes. From mahemacal sascs heor s kow ha he sums of depede Posso radom varables have a Posso dsrbuo. Thus f λ λ represe depede Posso radom varables each wh parameer E λ µ he λ + + λ s Posso dsrbued wh parameer µ where µ µ + L + µ /. We assume ha s Posso dsrbuo wh parameer POP expx β where POP s he populao of he h sae a me. To accou for hs kow relaoshp wh populao we assume ha he aural logarhmc populao s oe of he explaaor varables e has a kow regresso coeffce equal o oe. I GLM ermolog such as varable s kow as a offse. Thus our Posso parameer for s exp l POP + x β + L + x K β K exp l POP + x β POP exp x β. A alerave approach s o use he average umber of or flgs as he respose ad assume approxmae ormal. Ths was he approach ake b Lee e al. 994O; he reader has a opporu o pracce hs approach Exercses.8 ad 3.. Noe ha he Posso model above he expecao of he average respose s E / POP exp x β whereas he varace s Var / POP exp x β / POP. Thus o make hese wo approaches compable oe mus use weghed regresso usg esmaed recprocal varaces as he weghs. Table 0.5 summarzes he f of hree Posso models. Wh he basc homogeeous Posso model all explaaor varables ur ou o be sascall sgfca as evdeced b he small p-values. However he Posso model assumes ha he varace equals he mea; hs s ofe a resrcve assumpo for emprcal work. Thus Table 0.5 also summarzes a homogeous Posso model wh a esmaed scale parameer o accou for poeal overdsperso. Table 0.5 emphaszes ha alhough he regresso coeffce esmaes do o chage wh he roduco of he scale parameer esmaed sadard errors ad hus p-values do chage. Ma varables such as CAPS ur ou o be sascall sgfca predcors of he umber of flgs whe a more flexble model for he varace s roduced. Subseque secos wll roduce models of he sae level heerogee. Alhough o as mpora for hs daa se s sll eas o exame emporal heerogee hs coex hrough ear bar dumm varables. The goodess of f sascs he devace ad Pearso ch-square favor cludg he me caegorcal varable see Appedx C.8 for defos of hese goodess of f sascs.

0-8 / Chaper 0. Geeralzed Lear Models Table 0.5 Tor Flgs Model Coeffce Esmaes Based o N observaos from 9 saes ad T 6 ears. Logarhmc populao s used as a offse. Homogeeous Posso model Model wh esmaed scale parameer Model wh scale parameer ad me caegorcal varable Varable Parameer p-values Parameer p-values Parameer p-values esmae esmae esmae Iercep -7.943 <.000-7.943 <.000 POPLAWYR/000.63 <.000.63 0.000.3 0.0004 VEHCMILE/000 0.86 <.000 0.86 <.000 0.856 <.000 POPDENSY/000 0.39 <.000 0.39 0.0038 0.384 0.0067 WCMPMAX/000-0.80 <.000-0.80 0.6-0.86 0.53 URBAN/000 0.89 <.000 0.89 0.887 0.977 0.8059 UNEMPLOY 0.087 <.000 0.087 0.0005 0.086 0.004 JSLIAB 0.77 <.000 0.77 0.09 0.30 0.705 COLLRULE -0.030 <.000-0.030 0.7444-0.03 0.8053 CAPS -0.03 <.000-0.03 0.7457-0.056 0.6008 PUNITIVE 0.030 <.000 0.030 0.663 0.053 0.4986 Scale.000 35.857 36.383 Devace 8309.0 8309.0 5496.4 Pearso Ch-Square 9855.7 9855.7 7073.9 0.3 Margal models ad GEE Margal models uderp a wdel appled framework for hadlg heerogee logudal daa. As roduced Seco 9.4 margal models are semparamerc. Specfcall we model he frs ad secod momes as fucos of ukow parameers whou specfg a full paramerc dsrbuo for he resposes. For hese models parameers ca be esmaed usg exesos of he mehod of mome esmao echque kow as geeralzed esmag equaos or GEE. Margal models To re-roduce margal models aga use β o be a K vecor of parameers assocaed wh he explaaor varables x. Furher le τ be a q vecor of varace compoe parameers. Assume ha he mea of each respose s E µ µ β τ where µ s a kow fuco. Furher deoe he varace of each respose as Var v v β τ where v s a kow fuco. To complee he secod mome specfcao we eed assumpos o he covaraces amog resposes. We assume ha observaos amog dffere subjecs are depede ad hus have zero covarace. For observaos wh a subjec we assume ha he correlao bewee wo observaos wh he same subjec s a kow fuco. Specfcall we wll deoe corr r s ρ rs µ r µ s τ

Chaper 0. Geeralzed Lear Models / 0-9 where ρ rs. s a kow fuco. As Chaper 3 margal models use correlaos amog observaos wh a subjec o represe heerogee ha s he edec for observaos from he same subjec o be relaed. Specal case Geeralzed lear model wh caocal lks As we saw Seco 0. he coex of he geeralzed lear model wh caocal lks we have µ E bx β ad v Var φ b x β / w. Here b. s a kow fuco ha depeds o he choce of he dsrbuo of resposes. There s ol oe varace compoe so q ad τ φ. For he cases of depede observaos amog subjecs dscussed Seco 0. he correlao fuco ρ rs s f rs ad 0 oherwse. For hs example oe ha he mea fuco depeds o β bu does o deped o he varace compoe τ. Specal case Error compoes model For he error compoes model roduced Seco 3. we ma use a geeralzed lear model wh a ormal dsrbuo for resposes so ha bθ θ /. Thus from he above example we have µ E x β ad v Var φ σ α + σ ε. Ulke he above example observaos wh subjecs are o depede bu have a exchageable correlao srucure. Specfcall Seco 3. we saw ha for r s we have σ α corr r s ρ. σ α + σ ε Wh hs specfcao he vecor of varace compoes s τ φ σ α where φ σ α + σ ε. I Seco 0. he paramerc form of he frs wo momes was a cosequece of he assumpo of expoeal faml dsrbuo. Wh margal models he paramerc form s ow a basc assumpo. We ow gve marx oao o express hese assumpos more compacl. To hs ed defe µ β τ µ µ µ L µ T o be he vecor of meas for he h subjec. Le V be he T T varace covarace marx Var where he rs h eleme of V s gve b Cov corr v v. r s As Seco 9.4 for esmao purposes we wll also requre he K T grade marx µ G β µ T µ τ L. 0.7 β β Geeralzed esmag equaos Margal models provde suffce srucure o allow for a mehod of momes esmao procedure called geeralzed esmag equaos GEE a pe of geeralzed mehod of momes GMM esmao. As wh geeralzed leas squares we frs descrbe hs esmao mehod assumg he varace compoe parameers are kow. We he descrbe exesos o corporae varace compoe esmao. The geeral GEE procedure s descrbed Appedx C.6; recall ha a roduco was provded Seco 9.4. Assumg ha he varace compoes τ are kow he GEE esmaor of β s he soluo of r s r s

0-0 / Chaper 0. Geeralzed Lear Models 0 G µ b V b µ b. 0.8 K These are he geeralzed esmag equaos. We wll deoe hs soluo as b EE. Uder mld regular codos hs esmaor s cosse ad asmpocall ormal wh varacecovarace marx Var b EE G b V b G µ µ b. 0.9 Specal case Geeralzed lear model wh caocal lks - Coued µ v Because µ bx β s eas o see ha ha x b x β w x. Thus β φ w x v L w x v G φ depedece amog observaos wh a subjec we have V dagv v T. Thus we ma express he geeralzed esmag equaos as µ β τ T T T s our K T marx of dervaves. Assumg 0 K G T µ b V b µ b w x µ b. φ Ths elds he same soluo as he maxmum lkelhood ormal equaos equao 0.5. Thus he GEE esmaors are equal o he maxmum lkelhood esmaors for hs specal case. Noe ha hs soluo does o deped o kowledge of he varace compoe φ. GEEs wh ukow varace compoes To deerme GEE esmaes he frs ask s o deerme a al esmaor of β sa b 0EE. To llusrae oe mgh use he GLM model wh depedece amog observaos as above o ge a al esmaor. Nex we use he al esmaor b 0EE o compue resduals ad deerme a al esmaor of he varace compoes sa τ 0EE. The a he +s sage recursvel:. Use τ EE ad he soluo of he equao 0 K G µ b τ EE V b τ EE µ b τ EE o deerme a updaed esmaor of β sa b +EE.. Use he resduals { - µ b +EE τ EE } o deerme a updaed esmaor of τ sa τ +EE. 3. Repea seps ad ul covergece. Aleravel for he frs sage we ma updae esmaors usg a oe-sep procedure such as a Newo-Raphso erao. As aoher example he sascal package SAS uses a Fsher scorg pe updae of he form: b + + EE b EE G µ b EE V b EE G µ b EE G µ b EE V b EE µ b EE

Chaper 0. Geeralzed Lear Models / 0- For GEEs more complex problems wh ukow varace compoes Prece 988B suggess usg a secod esmag equao of he form: * E * * W E 0. τ Here * s a vecor of squares ad cross-producs of observaos wh a subjec of he form * L L L. 3 T T For he varace of * Dggle e al. 00S sugges usg he de marx for W wh mos dscree daa. However for bar resposes he oe ha he las T observaos are reduda because. These should be gored; Dggle e al. recommed usg W dag Var L Var T T. Robus esmao of sadard errors As dscussed Secos.5.3 ad 3.4 for lear models oe mus be cocered wh ususpeced seral correlao ad heeroscedasc. Ths s parcularl rue wh margal models where he specfcao s based ol o he frs wo momes ad o a full probabl dsrbuo. Because he specfcao of he correlao srucure ma be suspec hese are ofe called workg correlaos. Sadard errors ha rel o he correlao srucure specfcao from he relao equao 0.9 are kow as model-based sadard errors. I coras emprcal sadard errors ma be calculaed usg he followg esmaor of he asmpoc varace of b EE µ V G µ G µ V µ µ V G µ G µ V G µ G. 0.0 Specfcall he sadard error of he jh compoe of b EE seb jee s defed o be he square roo of he jh dagoal eleme of he varace-covarace marx he above dspla. Example: Tor flgs coued To llusrae margal models ad GEE esmaors we reur o he Seco 0. Tor flg example. The fed models whch appear Table 0.6 were esmaed usg he SAS procedure GENMOD. Assumg a depede workg correlao we arrve a he same parameer esmaors as Table 0.5 uder he homogeous Posso model wh a esmaed scale parameer. Furher alhough o dsplaed he fs provde he same model-based sadard errors. Compared o he emprcal sadard errors we see ha almos all emprcal sadard errors are larger ha he correspodg model-based sadard errors he excepo s CAPS. To es he robusess of hs model f we f he same model wh a auoregressve model of order oe AR workg correlao. Ths model f also appears Table 0.6. Because we use he workg correlao as a wegh marx ma of he coeffces have dffere values ha he correspodg f wh he depede workg correlao marx. Noe ha he asersk dcaes whe a esmae exceeds wce he emprcal sadard error; hs gves a rough dea of he sascal sgfcace of he varable. Uder he AR workg correlao varables ha are sascall sgfca usg emprcal sadard errors are also sascall sgfca usg model-based sadard errors. Aga CAPS s a borderle excepo. Comparg he depede o he AR workg correlao resuls we see ha VEHCMILE UNEMPLOY ad JSLIAB are cossel sascall sgfca whereas POPLAWYR POPDENSY WCMPMAX URBAN COLLRULE ad PUNITIVE are cossel sascall sgfca. However he judgme of sascal sgfcace depeds o he model seleco for CAPS.

0- / Chaper 0. Geeralzed Lear Models Parameer Esmae Emprcal Sadard Error Table 0.6 Comparso of GEE Esmaors. All models use a esmaed scale parameer. Logarhmc populao s used as a offse. Idepede Workg Correlao AR Workg Correlao Model-Based Sadard Error Esmae Emprcal Sadard Error Model-Based Sadard Error Iercep -7.943* 0.6 0.435-7.840* 0.870 0.806 POPLAWYR/000.63.0 0.589.3.306 0.996 VEHCMILE/000 0.86* 0.65 0.0 0.748* 0.66 0.80 POPDENSY/000 0.39* 0.75 0.35 0.400* 0.8 0.3 WCMPMAX/000-0.80 0.895 0.59-0.764 0.664 0.506 URBAN/000 0.89 5.367 3.89 3.508 7.5 7.30 UNEMPLOY 0.087* 0.04 0.05 0.048* 0.08 0.0 JSLIAB 0.77* 0.089 0.08 0.39* 0.00 0.049 COLLRULE -0.030 0.0 0.09-0.04 0.079 0.065 CAPS -0.03 0.098 0.098 0.4* 0.068 0.066 PUNITIVE 0.030 0.5 0.068-0.043 0.054 0.049 Scale 35.857 35.857 AR Coeffce 0.854 The asersk * dcaes ha he esmae s more ha wce he emprcal sadard error absolue value. 0.4 Radom effecs models A mpora mehod for accoug for heerogee logudal ad pael daa s hrough a radom effecs formulao. As Secos 3. ad 3.3. for lear models hs model s eas o roduce ad erpre he followg herarchcal fasho: Sage. Subjec effecs {α } are a radom sample from a dsrbuo ha s kow up o a vecor of parameers. Sage. Codoal o {α } he resposes { L } a GLM wh ssemac compoe η z α + x β. T are a radom sample from Addoal movao ad samplg ssues regardg radom effecs were roduced Chaper 3 for lear models; he also pera o geeralzed lear models. Thus he radom effecs model roduced above s a geeralzao of: The Chaper 3 lear radom effecs model. Here we used a ormal dsrbuo for he GLM compoe. The Seco 9. bar depede varables model wh radom effecs. Here we used a Beroull dsrbuo for he GLM compoe. I Seco 9. we also focused o he case z. B assumg a dsrbuo for he heerogee compoes α here s pcall a much smaller umber of parameers o be esmaed whe compared o he fxed effecs pael daa models ha wll be descrbed Seco 0.5. The maxmum lkelhood mehod of esmao s avalable alhough s compuaoall esve. To use hs mehod he cusomar pracce s o assume ormall dsrbued radom effecs. Oher esmao mehods are also avalable he leraure. For example he EM for expecao-maxmzao algorhm for esmao s descrbed Dggle e al. 00S ad McCulloch ad Searle 00G.

Chaper 0. Geeralzed Lear Models / 0-3 Radom effecs lkelhood To develop he lkelhood we frs oe ha from he secod samplg sage codoal o α he lkelhood for he h subjec a he h observao s θ b θ p ; β α exp + S φ φ where bθ E α ad η z α + x β ge α. Because of he depedece amog resposes wh a subjec codoal o α he codoal lkelhood for he h subjec s θ b θ p ; β α exp + S φ. φ Takg expecaos over α elds he ucodoal lkelhood. To see hs explcl we use he caocal lk so ha θ η. Thus he ucodoal lkelhood for he h subjec s z a + x β b z a + x β p ; β exp S φ exp d Fα a. 0. φ Here F α. represes he dsrbuo of α whch we wll assume o be mulvarae ormal wh mea zero ad varace-covarace marx Σ α. Cosse wh he oao Chaper 3 le τ deoe he vecor of parameers assocaed wh scalg sage φ ad he sage parameers he marx D. Wh hs oao we ma wre he oal log-lkelhood as l p β τ l p β. From equao 0. we see ha evaluag he log-lkelhood requres umercal egrao ad hus s more dffcul o compue ha lkelhoods for homogeeous models. Specal case Posso dsrbuo To llusrae assume q ad z so ha ol erceps α var b subjec. Assumg a Posso dsrbuo for he codoal resposes we have φ ba e a ad S φ - l!. Thus from equao 0. he log-lkelhood for he h subjec s l p β l! + l exp a + x β exp a + x β fα a da l + +! x β l exp a exp a + x β fα a da where f α. s he probabl des fuco of α. As before evaluag ad maxmzg he log-lkelhood requres umercal egrao.

0-4 / Chaper 0. Geeralzed Lear Models Seral correlao ad overdsperso The radom effecs GLM model roduces erms ha accou for heerogee he daa. However he model also roduces cera forms of seral correlao ad overdsperso ha ma or ma o be evde he daa. Recall ha Chaper 3 we saw ha permg subjec-specfc effecs α o be radom duced seral correlao he resposes { L T }. Ths s also rue for he olear GLM models as show he followg example. Specal case Posso dsrbuo coued To llusrae we use he assumpos of Example 0. ad recall ha for a caocal lk we have E α bθ bη bα + x β. For he Posso dsrbuo we have ba e a so ha µ E E E α E bα + x β expx β E e α. Here we have dropped he subscrp o α because he dsrbuo s decal over. To see he seral correlao we exame he covarace bewee wo observaos for example ad. B he codoal depedece we have Cov ECov α + CovE α E α Covbα + x β bα + x β Cove α expx β e α expx β expx +x β Var e α. Ths covarace s alwas oegave dcag ha we ca acpae posve seral correlao usg hs model. Smlar calculaos show ha Var EVar α + VarE α E φ bα + x β + Var bα + x β E e α expx β + Var expα + x β µ + exp x β Var e α. Thus he varace alwas exceeds he mea. Compared o he usual Posso models ha requre equal bewee he mea ad he varace he radom effecs specfcao duces a larger varace. Ths s a specfc example of he pheomeo kow as overdsperso.. Example: Tor flgs coued To llusrae he radom effecs esmaors we reur o he Seco 0. Tor flg example. Table 0.7 summarzes a radom effecs model ha was f usg he SAS sascal procedure NLMIXED. For comparso he Table 0.5 fs from he homogeeous Posso model wh a esmaed scale parameer are cluded Table 0.7. The radom effecs model assumes a codoal Posso dsrbued respose wh a scalar homogee parameer ha has a ormal dsrbuo.

Chaper 0. Geeralzed Lear Models / 0-5 Table 0.7 Tor Flgs Model Coeffce Esmaes Radom Effecs Logarhmc populao s used as a offse. Homogeeous Model wh Radom Effecs Model esmaed scale parameer Varable Parameer p-values Parameer esmae p-values esmae Iercep -7.943 <.000 -.753 <.000 POPLAWYR/000.63 0.000 -.694 <.000 VEHCMILE/000 0.86 <.000-0.83 0.0004 POPDENSY/000 0.39 0.0038 9.547 <.000 WCMPMAX/000-0.80 0.6 -.900 <.000 URBAN/000 0.89 0.887-47.80 <.000 UNEMPLOY 0.087 0.0005 0.035 <.000 JSLIAB 0.77 0.09 0.546 0.3695 COLLRULE -0.030 0.7444 -.03 0.984 CAPS -0.03 0.7457 0.39 0.5598 PUNITIVE 0.030 0.663 0.0 0.89 Sae Varace.7 - Log Lkelhood 9576 563 Compuaoal cosderaos As s evde from equao 0. maxmum lkelhood esmao of regresso coeffces requres oe or more q-dmesoal umercal egraos for each subjec ad each erao of a opmzao roue. As we have see our Chaper 9 ad 0 examples hs compuaoal complex s maageable for radom ercep models where q. Accordg o McCulloch ad Searle 00G hs drec mehod s also avalable for applcaos wh q or 3; however for hgher-order models such as wh crossed-radom effecs alerave approaches are ecessar. We have alread meoed he EM expecao-maxmzao algorhm Chaper 9 as oe alerave see McCulloch ad Searle 00G or Dggle e al. 00S for more deals. Aoher alerave s o use smulao echques. McCulloch ad Searle 00G summarze a Moe Carlo Newo-Raphso approach for approxmag he score fuco a smulaed maxmum lkelhood approach for approxmag he egraed lkelhood ad a sochasc approxmao mehod for a more effce ad sequeal approach of smulao. The mos wdel used se of aleraves are based o Talor-seres expasos geerall abou he lk fuco or he egraed lkelhood. There are several jusfcaos for hs se of aleraves. Oe s ha a Talor-seres s used o produce adjused varables ha follow a approxmae lear mxed effecs model. Appedx C.3 descrbes hs adjusme he lear case. Aoher jusfcao s hese mehods are deermed hrough a pealzed quaslkelhood fuco where here s a so-called peal erm for he radom effecs. Ths se of aleraves s he bass for he SAS macro GLM800.sas ad for example he S-plus a sascal package procedure lme for olear mxed effecs. The dsadvaage of hs se of aleraves s ha he do o work well for dsrbuos ha are far from ormal such as Beroull dsrbuos L ad Breslow 996B. The advaage s ha he approxmao procedures work well eve for relavel large umber of radom effecs. We refer he reader o McCulloch ad Searle 00G for furher dscusso. We also oe ha geeralzed lear models ca be expressed as specal cases of olear regresso models. Here b olear we mea ha he regresso fuco eed o be a lear fuco of he predcors bu ca be expressed as a olear fuco of he form fx

0-6 / Chaper 0. Geeralzed Lear Models β. The expoeal faml of dsrbuos provdes a specal case of olear regresso fucos. Ths s releva o compug cosderaos because ma compuaoal roues have bee developed for olear regresso ad ca be used drecl he GLM coex. Ths s also rue of exesos o mxed effecs models as suggesed b he referece o mle procedure above. For dscussos of olear regresso models wh radom effecs we refer he reader o Davda ad Gla 995S Voesh ad Chchll 997S ad Phero ad Baes 000S. 0.5 Fxed effecs models As we have see aoher mehod for accoug for heerogee logudal ad pael daa s hrough a fxed effecs formulao. Specfcall o corporae heerogee we allow for a ssemac compoe of he form η z α + x β. Thus hs seco exeds he Seco 9.3 dscusso where we ol permed varg erceps α. However ma of he same compuaoal dffcules arse. Specfcall wll ur ou ha maxmum lkelhood esmaors are geerall cosse. Codoal maxmum lkelhood esmaors are cosse bu dffcul o compue for ma dsrbuoal forms. The wo excepos are he ormal ad Posso famles where we show ha s eas o compue codoal maxmum lkelhood esmaors. Because of he compuaoal dffcules we resrc cosderao hs seco o caocal lks. 0.5. Maxmum lkelhood esmao for caocal lks Assume a caocal lk so ha θ η z α + x β. Thus wh equao 0.3 we have he log-lkelhood z α + x β b z α + x β l p + S φ. 0. φ To deerme maxmum lkelhood esmaors of α ad β we ake dervaes of l p se he dervaves equal o zero ad solve for he roos of hese equaos. Takg he paral dervave wh respec o α elds 0 b z α + x β z µ z φ because µ bθ b z α + x β. Takg he paral dervave wh respec o β elds x b z α + x β x µ 0. φ φ Thus we ca solve for he maxmum lkelhood esmaors of α ad β hrough he ormal equaos 0 Σ z - µ ad 0 Σ x - µ. 0.3 Ths s a specal case of he mehod of momes. Uforuael as we have see Seco 9.3 hs procedure ma produce cosse esmaes of β. The dffcul s ha he umber of parameer esmaors q + K grows wh he umber of subjecs. Thus he usual asmpoc heorems ha esure our dsrbuoal approxmaos are o loger vald. φ

Chaper 0. Geeralzed Lear Models / 0-7 Example: Tor flgs coued To llusrae he fxed effecs esmaors we reur o he Seco 0. Tor flg example. Table 0.8 summarzes he f of he fxed effecs model. For comparso he Table 0.5 fs from he homogeeous Posso model wh a esmaed scale parameer are cluded Table 0.8. Table 0.8 Tor Flgs Model Coeffce Esmaes Fxed Effecs All models have a esmaed scale parameer. Logarhmc populao s used as a offse. Homogeeous Model Model wh sae caegorcal varable Model wh sae ad me caegorcal varables Varable Parameer p-values Parameer p-values Parameer p-values esmae esmae esmae Iercep -7.943 <.000 POPLAWYR/000.63 0.000 0.788 0.5893-0.48 0.7869 VEHCMILE/000 0.86 <.000 0.093 0.7465 0.303 0.340 POPDENSY/000 0.39 0.0038 4.35 0.565 3.3 0.4385 WCMPMAX/000-0.80 0.6 0.546 0.379.50 0.0805 URBAN/000 0.89 0.887-33.94 0.3567-3.90 0.4080 UNEMPLOY 0.087 0.0005 0.08 0.784 0.04 0.500 JSLIAB 0.77 0.09 0.3 0.0065 0.0 0.059 COLLRULE -0.030 0.7444-0.04 0.6853-0.06 0.7734 CAPS -0.03 0.7457 0.079 0.053 0.040 0.564 PUNITIVE 0.030 0.663-0.0 0.6377 0.039 0.479 Scale 35.857 6.779 6.35 Devace 8309.0 463.4 9834. Pearso Ch-Square 9855.7 3366. 0763.0 0.5. Codoal maxmum lkelhood esmao for caocal lks Usg equao 0. we see ha cera parameers deped o he resposes o hrough cera summar sascs. Specfcall usg he facorzao heorem descrbed Appedx 0A. we have ha he sascs Σ z are suffce for α ad ha he sascs Σ x are suffce for β. Ths covee proper of caocal lks s o avalable for oher choces of lks. Recall ha suffcec meas ha he dsrbuo of he resposes wll o deped o a parameer whe codoed b he correspodg suffce sasc. Specfcall we ow cosder he lkelhood of he daa codoal o { Σ z } so ha he codoal lkelhood wll o deped o {α }. B maxmzg hs codoal lkelhood we acheve cosse esmaors of β. To hs ed le S be he radom varable represeg Σ z ad le sum be he realzao of Σ z. The codoal lkelhood of he resposes s p α β p α β Lp T α β 0.4 p S sum where p S sum s he probabl des or mass fuco of S evaluao a sum. Ths lkelhood does o deped o {α } ol o β. Thus whe evaluag we ca ake α o be a zero vecor whou loss of geeral. Uder broad codos maxmzg equao 0.4 wh

0-8 / Chaper 0. Geeralzed Lear Models respec o β elds roo- cosse esmaors see for example McCullagh ad Nelder 989G. Sll as Seco 9.3 for mos paramerc famles s dffcul o compue he dsrbuo of S. Clearl he ormal dsrbuo s oe excepo because f he resposes are ormal he he dsrbuo of S s also ormal. The followg subseco descrbes aoher mpora applcao where he compuao s feasble uder codos lkel o be ecouered appled daa aalss. 0.5.3 Posso dsrbuo Ths subseco cosders Posso dsrbued daa a wdel used dsrbuo for cou resposes. To llusrae we cosder he caocal lk wh q ad z so ha θ α + x β. The Posso dsrbuo s gve b p α β µ exp - µ wh µ bθ exp α + x β. Coversel because l µ α + x β we have ha he logarhmc mea s a lear combao of explaaor varables. Ths s he bass of he socalled log-lear model. As oed Seco 0A. Σ s a suffce sasc for α. Furher s eas o check ha he dsrbuo of Σ urs ou o be Posso wh mea Σ expα + x β. I he subseque developme we wll use he rao of meas µ exp x β π β 0.5 µ exp x β ha does o deped o α. Now usg equao 0.4 he codoal lkelhood for he h subjec s p α β p α β Lp T α β Pr ob S T µ L µ exp - µ + L+ µ! µ µ T T! L T! L!! T π! β exp -! where π β s gve equao 0.5. Ths s a mulomal dsrbuo. Thus he jo dsrbuo of { L T } gve has a mulomal dsrbuo. Usg equao 0.4 he codoal lkelhood s

Chaper 0. Geeralzed Lear Models / 0-9! CL π β.! L T! Takg paral dervaves of he log codoal lkelhood elds l CL lπ β x xrπ r β. β β r Thus he codoal maxmum lkelhood esmae b CMLE s he soluo of T T x x b 0 rπ r CMLE. r Example Ar qual regulao Becker ad Hederso 000E vesgaed he effecs of ar qual regulaos o frm decsos cocerg pla locaos ad brhs four major pollug dusres: dusral orgac chemcals meal coaers mscellaeous plasc producs ad wood furure. The focused o ar qual regulao of groud-level ozoe a major compoe of smog. Wh he 977 amedmes o he Clea Ar Ac each cou he Ued Saes s classfed as beg eher or ou of aame of he aoal sadards for each of several polluas cludg ozoe. Ths bar varable s he prmar varable of eres. Becker ad Hederso examed he brh of plas a cou usg aame saus ad several corol varables o evaluae hpoheses cocerg he locao shf from o-aame o aame areas. The used pla ad dusr daa from he Logudal Research Daabase developed b he US Cesus Bureau o derve he umber of ew phscal plas brhs b cou for T 6 fve-ear me perods from 963 o 99. The corol varables coss of me dummes cou maufacurg wages ad cou scale as measured b all oher maufacurg emplome he cou. Becker ad Hederso emploed a codoal lkelhood Posso model; hs wa he could allow for fxed effecs ha accou for me-cosa umeasured cou effecs. The foud ha o-aame saus reduces he expeced umber of brhs a cou b 5-45 perce depedg o he dusr. Ths was a sascall sgfca fdg ha corroboraed a ma research hpohess ha here has bee a shf brhs from o-aame o aame coues. Seco 0.6 Baesa ferece I Seco 4.4 Baesa ferece was roduced he coex of he ormal lear model. Ths seco provdes a roduco a more geeral coex ha s suable for hadlg geeralzed lear models. Ol he hghlghs wll be preseed; we refer o Gelma e al. 004G for a more dealed reame.

0-0 / Chaper 0. Geeralzed Lear Models Beg b recallg Baes rule p daa parameers p parameers p parameers daa p daa where pparameers s he dsrbuo of he parameers kow as he pror dsrbuo. pdaa parameers s he samplg dsrbuo. I a freques coex s used for makg fereces abou he parameers ad s kow as he lkelhood. pparameers daa s he dsrbuo of he parameers havg observed he daa kow as he poseror dsrbuo. pdaa s he margal dsrbuo of he daa. I s geerall obaed b egrag or summg he jo dsrbuo of daa ad parameers over parameer values. Ths s ofe he dffcul sep Baesa ferece. I a regresso coex we have wo pes of daa he respose varable ad he se of explaaor varables X. Le θ ad ψ deoe he ses of parameers ha descrbe he samplg dsrbuos of ad X respecvel. Moreover assume ha θ ad ψ are depede. The usg Baes rule he poseror dsrbuo s p X θψ p θψ p X θψ p θ p X θψ p ψ p θψ X p X p X { p X θ p θ } { p X ψ p ψ } θ X p ψ X p. Here he smbol meas s proporoal o. Thus he jo poseror dsrbuo of he parameers ca be facored o wo peces oe for he resposes ad oe for he explaaor varables. Assumg o depedeces bewee θ ad ψ here s o loss of formao he radoal regresso seg b gorg he dsrbuos assocaed wh he explaaor varables. B he radoal regresso seg we mea ha oe esseall reas he explaaor varables as o-sochasc. Mos sascal ferece ca be accomplshed readl havg compued he poseror. Wh hs ere dsrbuo summarzg lkel values of he parameers hrough cofdece ervals or ulkel values hrough hpohess ess s sraghforward. Baesa mehods are also especall suable for forecasg. I he regresso coex suppose we wsh o summarze he dsrbuo of a se of ew resposes ew gve ew explaaor varables X ew ad prevousl observed daa ad X. Ths dsrbuo p ew X ew X s a pe of predcve dsrbuo. We have ha X X p θ X X dθ p θ X X p θ X X dθ. p ew ew ew ew ew ew Ths assumes ha he parameers θ are couous. Here ew θ X X ew dsrbuo of ew ad p θ X X p θ X ew p s he samplg ew s he poseror dsrbuo assumg ha values of he ew explaaor varables are depede of θ. Thus he predcve dsrbuo ca be compued as a weghed average of he samplg dsrbuo where he weghs are gve b he poseror. A dffcul aspec of Baesa ferece ca be he assgme of prors. Classcal assgmes of prors are geerall eher o-formave or cojugae. No-formave prors are dsrbuos ha are desged o erjec he leas amou of formao possble. Two

Chaper 0. Geeralzed Lear Models / 0- mpora pes of o-formave prors are uform also kow as fla prors ad Jeffre s pror. A uform pror s smpl a cosa value; hus o value s more lkel ha a oher. A drawback of hs pe of pror s ha o vara uder rasformao of he parameers. To llusrae cosder he ormal lear regresso model so ha X β σ ~ N X β σ I. A wdel used o-formave pror s a fla pror o β log σ so ha he jo dsrbuo of β σ urs ou o be proporoal o σ -. Thus alhough uform log σ hs pror gves heaver wegh o small values of σ. A Jeffre s pror s oe ha s vara uder rasformao. Jeffre s prors are complex he case of muldmesoal parameers ad hus we wll o cosder hem furher here. Cojugac of a pror s acuall a proper ha depeds o boh he pror as well as samplg dsrbuo. Whe he pror ad poseror dsrbuos come from he same faml of dsrbuos he he pror s kow as a cojugae pror. Appedx 0A.3 gves several examples of he more commol used cojugae prors. For logudal ad pael daa s covee o formulae Baesa models hree sages: oe for he parameers oe for he daa ad oe sage for he lae varables used o represe he heerogee. Thus exedg Seco 7.3. we have: Sage 0. Pror dsrbuo Draw a realzao of a se of parameers from a populao. The parameers coss of regresso coeffces β ad varace compoes τ. Sage. Heerogee effecs dsrbuo Codoal o he parameers from sage 0 draw a radom sample of subjecs from a populao. The vecor of subjec-specfc effecs α s assocaed wh he h subjec. Sage. Codoal samplg dsrbuo Codoal o α β ad τ draw realzaos of { z x } for T for he h subjec. Summarze hese draws as { Z X }. A commo mehod of aalss s o combe sages 0 ad ad o rea β* β α α as he regresso parameers of eres. Also commo s o use a ormal pror dsrbuo for hs se of regresso parameers. Ths s he cojugae pror whe he samplg dsrbuo s ormal. Whe he samplg dsrbuo s from he GLM faml bu o ormal here s o geeral recpe for cojugae prors. A ormal pror s useful because s flexble ad compuaoall covee. To be specfc we cosder. Sage *. Pror dsrbuo Assume ha β α Eβ Σβ 0 β* ~ N. M 0 0 D I α Thus boh β ad α α are ormall dsrbued. Sage. Codoal samplg dsrbuo Codoal o β* ad {Z X} { } are depede ad he dsrbuo of s from a geeralzed lear model GLM faml wh parameer η z α + x β. For some pes of GLM famles such as he ormal a addoal scale parameer s used ha s pcall cluded he pror dsrbuo specfcao. Wh hs specfcao prcple oe smpl apples Baes rule o deerme he poseror dsrbuo of β* gve he daa { X Z}. However as a praccal maer hs s dffcul o do whou cojugae prors. Specfcall o compue he margal dsrbuo of he daa oe mus use umercal egrao o remove parameer dsrbuos; hs s compuaoall esve for ma problems of eres. To crcumve hs dffcul moder

0- / Chaper 0. Geeralzed Lear Models Baesa aalss regular emplos smulao echques kow as Markov Cha Moe Carlo MCMC mehods ad a especall mpora specal case he Gbbs sampler. MCMC mehods produce smulaed values of he poseror dsrbuo ad are avalable ma sascal packages cludg he shareware ha s favored b ma Baesa aalss BUGS/WINBUGS avalable a hp://www.mrc-bsu.cam.ac.uk/bugs/. There are ma specalzed reames ha dscuss he heor ad applcaos of hs approach; we refer he reader o Gelma e al. 004G Gll 00G ad Cogdo 003G. For some applcaos such as predco he eres s he full jo poseror dsrbuo of he global regresso parameers ad he heerogee effecs β α α. For oher applcaos he eres s he poseror dsrbuo of he global regresso coeffces β. I hs case oe egraes ou he heerogee effecs from he jo poseror dsrbuo of β*. Example Respraor fecos Zeger ad Karm 99S roduced Baesa ferece for GLM logudal daa. Specfcall Zeger ad Karm cosdered he sage GLM codoal samplg dsrbuo ad for he heerogee dsrbuo assumed ha α are codoall..d. N0 D. The allowed for a geeral form for he pror dsrbuo. To llusrae hs se-up Zeger ad Karm examed fecous dsease daa o 50 Idoesa chldre who were observed up o T 6 cosecuve quarers for a oal of N 00 observaos. The goal was o assess deermas of a bar varable ha dcaes he presece of a respraor dsease. The used ormall dsrbued radom erceps q ad z wh a logsc codoal samplg dsrbuo. Example Paes Chb Greeberg ad Wkelma 998E dscussed parameerzao of Baesa Posso models. The cosdered a codoal Posso model wh a caocal lk so ha α β ~ Possoθ where θ E α β z α + x β. Ulke pror work he dd o assume ha he heerogee effecs have zero mea bu sead used α ~ N η α D. Wh hs specfcao he dd o use he usual coveo of assumg ha z s a subse of x. Ths he argued leads o more sable covergece algorhms whe compug poseror dsrbuos. To complee he specfcao Chb e al. used pror dsrbuos: β ~ N µ β Σ β η α ~ Nη 0 Σ η ad D - ~ Wshar. The Wshar dsrbuo s a mulvarae exeso of he ch-square dsrbuo see for example Aderso 958G. To llusrae Chb e al. used pae daa frs cosdered b Hausma Hall ad Grlches 984E. These daa clude he umber of paes receved b 64 frms over T 5 ears 975-979. The explaaor varables cluded he logarhm of research ad developme R&D expedures as well as her ad 3 ear lags ad me dumm varables. Chb e al. use a varable ercep ad a varable slope for he logarhmc R&D expedures.

Chaper 0. Geeralzed Lear Models / 0-3 Furher readg McCullagh ad Nelder 989S provde a more exesve roduco o homogeeous geeralzed lear models. Codoal lkelhood esg coeco wh expoeal famles was suggesed b Rasch 96EP ad asmpoc properes were developed b Aderse 970S movaed par b he presece of fel ma usace parameers of Nema ad Sco 948E. Pael daa codoal lkelhood esmao was roduced b Chamberla 980E for bar log models ad b Hausma Hall ad Grlches 984E for Posso as well as egave bomal cou models. Three excelle sources for furher dscussos of olear mxed effecs models are Davda ad Gla 995S Voesh ad Chchll 997S ad Phero ad Baes 000S. For addoal dscussos o compug aspecs we refer o McCulloch ad Searle 00G ad Phero ad Baes 000S.

0-4 / Chaper 0. Geeralzed Lear Models

Chaper 0. Geeralzed Lear Models / 0-5 Appedx 0A. Expoeal famles of dsrbuos The dsrbuo of he radom varable ma be dscree couous or a mxure. Thus p. equao 0. ma be erpreed o be a des or mass fuco depedg o he applcao. Table 0A. provdes several examples cludg he ormal bomal ad Posso dsrbuos. Table 0A. Seleced Dsrbuos of he Oe-Parameer Expoeal Faml Dsr- Parameers Des Compoes E Var buo or Mass Fuco Geeral θ φ θ b θ θ φ bθ Sφ bθ b θ φ exp + S φ φ Normal µ σ µ µ σ θ lπφ θ µ φ σ exp σ π σ + φ Bomal π π π π l π Posso λ exp λ Gamma α β θ l + e θ e l θ + e π e θ θ + e π π λ l λ e θ -l! e θ λ e θ λ! α β β -l -θ α φ lφ exp β Γ α α α α l Γ φ θ φ α β θ β + φ l 0A. Mome Geerag Fuco To assess he momes of expoeal famles s covee o work wh he mome geerag fuco. For smplc we assume ha he radom varable s couous. Defe he mome geerag fuco s θ b θ s s M E e exp + + S φ d φ b θ + sφ b θ θ + sφ b θ + sφ exp exp + S φ d φ φ * * b θ + sφ b θ θ b θ b θ + sφ b θ exp exp + S φ d exp φ φ. φ Wh hs expresso we ca geerae he momes. Thus for he mea we have b θ + sφ b θ b θ + sφ b θ E M 0 exp b θ + φ exp φ s φ s s 0 s 0 b θ. Smlarl for he secod mome we have

0-6 / Chaper 0. Geeralzed Lear Models b θ + sφ b θ M s b θ + sφ exp s φ b θ + sφ b θ b θ + sφ b θ φ b θ + s φ exp + b θ + sφ exp. φ φ Ths elds E M 0 φ b θ + b θ ad Var φ b θ. 0A. Suffcec I complex suaos s covee o be able o decompose he lkelhood o several peces ha ca be aalzed separael. To accomplsh hs he cocep of suffcec s useful. A sasc T T s suffce for a parameer θ f he dsrbuo of codoal o T does o deped o θ. Whe checkg wheher or o a sasc s suffce a mpora resul s he facorzao heorem. Ths resul dcaes uder cera regular codos ha a sasc T s suffce for θ f ad ol f he des mass fuco of ca be decomposed o he produc of wo compoes p θ p T θ p. 0A. Here he frs poro p ma deped o θ bu depeds o he daa ol hrough he suffce sasc T. The secod poro p ma deped o he daa bu does o deped o he parameer θ. See for example Bckel ad Doksum 977G. To llusrae f { } are depede ad follow he dsrbuo equao 0. he he jo dsrbuo s θ b θ p θ φ exp + S φ. φ Thus wh p exp S φ ad θ T b θ p T θ exp he sasc φ T s suffce for θ. 0A.3 Cojugae Dsrbuos Assume ha he parameer θ s radom wh dsrbuo πθ τ where τ s a vecor of parameers ha descrbe he dsrbuo of θ. I Baesa models he dsrbuo π s kow as he pror ad reflecs our belef or formao abou θ. The lkelhood p θ p θ s a probabl codoal o θ. The dsrbuo of θ wh kowledge of he radom varables πθ τ s called he poseror dsrbuo. For a gve lkelhood dsrbuo prors ad poserors ha come from he same paramerc faml are kow as cojugae famles of dsrbuos. For a lear expoeal lkelhood here exss a aural cojugae faml. For he lkelhood equao 0. defe he pror dsrbuo πθ τ C exp θ a τ - bθ a τ 0A. where C s a ormalzg cosa. Here a τ ad a τ are fucos of he parameers τ. The jo dsrbuo of ad θ s gve b pθ p θ πθ τ. Usg Baes Theorem he poseror dsrbuo s

Chaper 0. Geeralzed Lear Models / 0-7 + + φ θ θ φ θ a b a exp π τ τ τ C where C s a ormalzg cosa. Thus we see ha πθ τ has he same form as πθ τ. Specal case 0A. Normal-Normal Model Cosder a ormal lkelhood equao 0. so ha bθ θ /. Thus wh equao 0A. we have a a exp π τ τ τ θ θ θ C a a a exp τ τ τ τ θ C. Thus he pror dsrbuo of θ s ormal wh mea a τ/a τ ad varace a τ -. The poseror dsrbuo of θ gve s ormal wh mea a τ+/φ/a τ+φ - ad varace a τ+φ - -. Specal case 0A. Bomal-Bea Model Cosder a bomal lkelhood equao 0. so ha bθ l+e θ. Thus wh equao 0A. we have θ θ θ e l a a exp π + τ τ τ C a a a e e e τ τ τ + + + C θ θ θ Thus we have ha pror of logθ s a bea dsrbuo wh parameers a τ ad a τ-a τ. The poseror of logθ s a bea dsrbuo wh parameers a τ+/φ ad a τ+φ - - a τ+/φ. Specal case 0A.3 Posso-Gamma Model Cosder a Posso lkelhood equao 0. so ha bθ e θ. Thus wh equao 0A. we have θ θ θ e a a exp π τ τ τ C a e exp e a τ τ θ θ C. Thus we have ha he pror of e θ s a Gamma dsrbuo wh parameers a τ ad a τ.the poseror of e θ s a Gamma dsrbuo wh parameers a τ+/φ ad a τ+φ -. 0A. 4 Margal Dsrbuos I ma cojugae famles he margal dsrbuo of he radom varable urs ou o be from a well-kow paramerc faml. To see hs cosder he pror dsrbuo equao 0A.. Because deses egrae sum he dscree case o we ma express he cosa C as θ θ θ d a a a a C C b exp a fuco of he parameers a a τ ad a a τ. Wh hs ad equao 0A. he margal dsrbuo of s θ φ θ φ θ φ θ θ π θ d a a S a a C d + + b exp exp p g

0-8 / Chaper 0. Geeralzed Lear Models exp φ φ φ + + a a C S a a C. 0A.3 I s of eres o cosder several specal cases. Specal case 0A. Normal-Normal Model - Coued Usg Table 0A. we have φ πφ φ exp exp S. Sraghforward calculaos show ha exp exp a a a a a C a a C π. Thus from equao 0A.3 he margal dsrbuo s + + + / exp exp exp g φ φ π φ φ πφ π a a a a a a + + + + / exp φ φ φ φ π a a a a a + + / exp φ φ π a a a a. Thus s ormall dsrbued wh mea a /a ad varace φ+a -. Specal case 0A. Bomal-Bea Model - Coued Usg Table 0A. we have S exp φ ad φ. Furher sraghforward calculaos show ha B a a a a a a a a a C Γ Γ Γ where B.. s called he Bea fuco. Thus from equao 0A.3 we have B B g a a a a a a + + +. Ths s called he bea-bomal dsrbuo.

Chaper 0. Geeralzed Lear Models / 0-9 Specal case 0A.3 Posso-Gamma Model - Coued exp S φ /! / Γ + ad φ. Furher Usg Table 0A. we have a a C a a Γ a a a a Γ a Γ + Γ a + a Γ a + a + a + a + Γ a Γ + a + a Γ Γ + a Γ a + sraghforward calculaos show ha g Ths s a egave bomal dsrbuo.. Thus from equao 0A.4 we have a + a a +. 0. Exercses ad Exesos Posso ad Bomals 0. Cosder ha has a Posso dsrbuo wh mea parameer µ ad suppose ha s depede of. a. Show ha + has a Posso dsrbuo wh mea parameer µ + µ. b. Show ha gve + has a Bomal dsrbuo. Deerme he parameers of he Bomal dsrbuo. c. Suppose ha ou ell he compuer o evaluae he lkelhood of assumg ha s depede of each has a Posso dsrbuo wh mea parameers µ ad µ - µ.. Evaluae he lkelhood of.. However he daa are such ha s bar ad. Evaluae he lkelhood ha he compuer s evaluag.. Show ha hs equals he lkelhood assumg has a Beroull dsrbuo wh mea µ up o a proporoal cosa. Deerme he cosa. d. Cosder a logudal daa se { x } where are bar wh mea p πx β where π. s kow. You would lke o evaluae he lkelhood bu ca ol do so erms of a package usg Posso dsrbuos. Usg he par c resul expla how o do so.

Chaper. Caegorcal Depede Varables ad Survval Models / - 003 b Edward W. Frees. All rghs reserved Chaper. Caegorcal Depede Varables ad Survval Models Absrac. Exedg Chaper 9 hs chaper cosders depede varables havg more ha wo possble caegorcal aleraves. As Chaper 9 we ofe erpre a caegorcal varable o be a arbue possessed or choce made b a dvdual household or frm. B allowg more ha wo aleraves we subsaall broade he scope of applcaos o complex socal scece problems of eres. The pedagogc approach of hs chaper follows he paer esablshed earler chapers; we beg wh homogeeous models Seco. followed b he Seco. model ha corporaes radom effecs hus provdg a heerogee compoe. Seco.3 roduces a alerave mehod for accommodag me paers hrough raso or Markov models. Alhough raso models are applcable he Chaper 0 geeralzed lear models he are parcularl useful he coex of caegorcal depede varables. Ma repeaed applcaos of he dea of rasog gves rse o survval models aoher mpora class of logudal models. Seco.4 develops hs lk.. Homogeeous models We ow cosder a respose ha s a uordered caegorcal varable. We assume ha he depede varable ma ake o values... c correspodg o c caegores. We frs roduce he homogeous verso so ha hs seco does o use a subjec-specfc parameers or does roduce erms o accou for seral correlao. I ma socal scece applcaos he respose caegores correspod o a arbue possessed or choces made b dvduals households or frms. Some applcaos of caegorcal depede varable models clude: emplome choce such as Vallea 999E mode of rasporao choce such as he classc work b McFadde 978E choce of polcal par afflao such as Brader ad Tucker 00O ad markeg brad choce such as Ja e al. 994O. Example. Polcal par afflao Brader ad Tucker 00O suded he choce made b Russa voers of polcal pares 995-996 elecos. Ther eres was assessg effecs of meagful polcal par aachmes durg Russa s raso o democrac. The examed a T 3 wave surve of 84 respodes ake: hree o four weeks before he 995 parlamear elecos mmedael afer he parlamear elecos ad 3 afer he 996 presdeal elecos. Ths surve desg was modeled o he Amerca Naoal Eleco Sudes see Appedx F. The depede varable was he polcal par voed for cossg of c 0 pares cludg he Lberal Democrac Par of Russa Commus Par of he Russa Federao Our Home s Russa ad ohers. The depede varables cluded socal characerscs such as educao geder relgo aoal age ad locao urba versus rural ecoomc

- / Chaper. Caegorcal Depede Varables ad Survval Models characerscs such as come ad emplome saus ad polcal audal characerscs such audes oward marke rasos ad prvazao... Sascal ferece For a observao from subjec a me deoe he probabl of choosg he jh caegor as π j Prob j so ha π + + π c. I he homogeeous framework we assume ha observaos are depede from oe aoher. I geeral we wll model hese probables as a kow fuco of parameers ad use maxmum lkelhood esmao for sascal ferece. We use he oao j o be a dcaor of he eve j. The lkelhood for he h subjec a he h me po s: π f c π f j π j. j M M π c f c Thus assumg depedece amog observaos he oal log-lkelhood s L j l π j. c j Wh hs framework sadard maxmum lkelhood esmao s avalable. Thus our ma ask s o specf a approprae form for π... Geeralzed log Lke sadard lear regresso geeralzed log models emplo lear combaos of explaaor varables of he form: V j β j.. x Because he depede varables are o umercal we cao model he respose as a lear combao of explaaor varables plus a error. Isead we use he probables exp V j Prob j π j j c. c exp V k k Noe here ha β j s he correspodg vecor of parameers ha ma deped o he alerave or choce whereas he explaaor varables x do o. So ha probables sum o oe a covee ormalzao for hs model s β c 0. Wh hs ormalzao ad he specal case of c he geeralzed log reduces o he log model roduced Seco 9.. Parameer erpreaos We ow descrbe a erpreao of coeffces geeralzed log models smlar o Seco 9.. for he logsc model. From equaos. ad. we have Prob j l V j V c x β j c. Prob The lef-had sde of hs equao s erpreed o be he logarhmc odds of choosg choce j compared o choce c. Thus as Seco 9.. we ma erpre β j as he proporoal chage he odds rao.

Chaper. Caegorcal Depede Varables ad Survval Models / -3 Geeralzed logs have a eresg esed srucure ha we wll explore brefl Seco 9..5. Tha s s eas o check ha codoal o o choosg he frs caegor he form of Prob j has a geeralzed log form equao.. Furher f j ad h are dffere aleraves we oe ha Prob j Prob j { j or h} Prob j + Prob h exp V j. exp V + exp V + exp x β β j h Ths has a log form ha was roduced Seco 9... h j Specal case Iercep ol model To develop uo we ow cosder he model wh ol erceps. Thus le x ad β j β 0 j α j. Wh he coveo α c 0 we have ad Prob j π j Prob e α + e α j e α j +... + e l α j Prob c. From he secod relao we ma erpre he jh ercep α j o be he logarhmc odds of choosg alerave j compared o alerave c. α c + Example. Job secur Ths s a couao of he Example 9. o he deermas of job urover based o he work of Valea 999E. The Chaper 9 aalss of hs daa cosdered ol he bar depede varable dsmssal he movao beg ha hs s he ma source of job secur. Valea 999E also preseed resuls from a geeralzed log model hs prmar movao beg ha he ecoomc heor descrbg urover mples ha oher reasos for leavg a job ma affec dsmssal probables. For he geeralzed log model he respose varable has c 5 caegores: dsmssal lef job because of pla closures qu chaged jobs for oher reasos ad o chage emplome. The laer caegor s he omed oe Table.. The explaaor varables of he geeralzed log are same as he prob regresso descrbed Example 9.; he esmaes summarzed Example 9. are reproduced here for coveece. Table. shows ha urover decles as eure creases. To llusrae cosder a pcal ma he 99 sample where we have me 6 ad focus o dsmssal probables. For hs value of me he coeffce assocaed wh eure for dsmssal s -0. + 6 0.008-0.093 due o he eraco erm. From hs we erpre a addoal ear of eure o mpl ha he dsmssal probabl s exp-0.093 9% of wha would be oherwse represeg a decle of 9%. Table. also shows ha he geeralzed coeffces assocaed wh dsmssal are smlar o he prob fs. The sadard errors are also qualavel smlar alhough hgher for he geeralzed logs whe compared o he prob model. I parcular we aga see ha he

-4 / Chaper. Caegorcal Depede Varables ad Survval Models coeffce assocaed wh he eraco bewee eure ad me red reveals a creasg dsmssal rae for expereced workers. The same s rue for he rae of qug. Table. Turover Geeralzed Log ad Prob Regresso Esmaes Prob Geeralzed Log Model Varable Regresso Dsmssed Pla Oher Qu Model Closed Reaso Teure -0.084-0. -0.086-0.068-0.7 0.00 0.05 0.09 0.00 0.0 Tme Tred -0.00-0.008-0.04 0.0-0.0 0.005 0.0 0.06 0.03 0.007 Teure Tme Tred 0.003 0.008 0.004-0.005 0.006 0.00 0.00 0.00 0.00 0.00 Chage Logarhmc Secor 0.094 0.86 0.459-0.0 0.333 Emplome 0.057 0.3 0.89 0.58 0.08 Teure Chage Logarhmc -0.00-0.06-0.053-0.005-0.07 Secor Emplome 0.009 0.03 0.05 0.05 0.0 Noes: Sadard errors pareheses. Omed caegor s o chage emplome. Oher varables corolled for cossed of educao maral saus umber of chldre race ears of full-me work experece ad s square uo membershp goverme emplome logarhmc wage he U.S. emplome rae ad locao...3 Mulomal codoal log Smlar o equao. a alerave lear combao of explaaor varables s x β..3 V j j Here x j s a vecor of explaaor varables ha depeds o he jh alerave whereas he parameers β do o. Usg he expresso equao. s he bass for he mulomal log model also kow as he codoal log model. Wh hs specfcao he oal log-lkelhood s c c c L β j l π j jx jβ l exp x kβ..4 j j k Ths sraghforward expresso for he lkelhood eables maxmum lkelhood ferece o be easl performed. The geeralzed log model s a specal case of he mulomal log model. To see hs cosder explaaor varables x ad parameers β j each of dmeso K. Defe 0 M β 0 β x j x ad β. M 0 M β c 0

Chaper. Caegorcal Depede Varables ad Survval Models / -5 Specfcall x j s defed as j- zero vecors each of dmeso K followed b x ad he followed b c-j zero vecors. Wh hs specfcao we have x j β x β j. Thus a sascal package ha performs mulomal log esmao ca also perform geeralzed log esmao hrough he approprae codg of explaaor varables ad parameers. Aoher cosequece of hs coeco s ha some auhors use he descrpor mulomal log whe referrg o he Seco.. geeralzed log model. Moreover hrough smlar codg schemes mulomal log models ca also hadle lear combaos of he form: Here x j V j x j β + x β..5 are explaaor varables ha deped o he alerave whereas x do o. Smlarl β j are parameers ha deped o he alerave whereas β do o. Ths pe of lear combao s he bass of a mxed log model. As wh codoal logs s cusomar o choose oe se of parameers as he basele ad specf β c 0 o avod redudaces. To erpre parameers for he mulomal log model we ma compare aleraves h ad k usg equaos. ad.3 o ge Prob h l x h x k β..6 Prob k Thus we ma erpre β j as he proporoal chage he odds rao where he chage s he value of he jh explaaor varable movg from he kh o he hh alerave. Wh equao. oe ha π / π expv / expv. Ths rao does o deped o he uderlg values of he oher aleraves V j for j3... c. Ths feaure called he depedece of rreleva aleraves ca be a drawback of he mulomal log model for some applcaos. Example.3 Choce of ogur brads We ow cosder a markeg daa se roduced b Ja e al. 994O ha was furher aalzed b Che ad Kuo 00S. These daa obaed from A. C. Nelse are kow as scaer daa because he are obaed from opcal scag of grocer purchases a check-ou. The subjecs coss of 00 households Sprgfeld Mssour. The respose of eres s he pe of ogur purchased cossg of four brads: Yopla Dao Wegh Wachers ad Hlad. The households were moored over a wo-ear perod wh he umber of purchases ragg from 4 o 85; he oal umber of purchases s N4. The wo markeg varables of eres are PRICE ad FEATURES. For he brad purchased PRICE s recorded as prce pad ha s he shelf prce e of he value of coupos redeemed. For oher brads PRICE s he shelf prce. FEATURES s a bar varable defed o be oe f here was a ewspaper feaure adversg he brad a me of purchase ad zero oherwse. Noe ha he explaaor varables var b alerave suggesg he use of a mulomal codoal log model. Tables. ad.3 summarze some mpora aspecs of he daa. Table. shows ha Yopla was he mos frequel 33.9% seleced pe of ogur our sample whereas Hlad was he leas frequel seleced.9%. Yopla was also he mos heavl adversed appearg ewspaper adversemes 5.6% of he me ha he brad was chose. Table.3 shows ha Yopla was also he mos expesve cosg 0.6 ces per ouce o average. Table.3 also shows ha here are several prces ha were far below he average suggesg some poeal flueal observaos. j

-6 / Chaper. Caegorcal Depede Varables ad Survval Models TABLE. Summar Sascs b Choce of Yogur Summar Sascs Yopla Dao Wegh Hlad Toals Wachers Number of Choces 88 970 553 7 4 Number of Choces Perce 33.9 40..9.9 00.0 Feaure Averages Perce 5.6 3.8 3.8 3.7 4.4 Table.3 Summar Sascs for Prces Varable Mea Meda Mmum Maxmum Sadard devao Yopla 0.07 0.08 0.003 0.93 0.09 Dao 0.08 0.086 0.09 0. 0.0 Wegh Wachers 0.079 0.079 0.004 0.04 0.008 Hlad 0.054 0.054 0.05 0.086 0.008 A mulomal log model was f o he daa usg he followg specfcao for he ssemac compoe α + β PRICE + FEATURE V j j j β j usg Hlad as he omed alerave. The resuls are summarzed Table.4. Here we see ha each parameer s sascall sgfcal dffere from zero. Thus he parameer esmaes ma be useful whe predcg he probabl of choosg a brad of ogur. Moreover a markeg coex he coeffces have mpora subsave erpreaos. Specfcall we erpre he coeffce assocaed wh FEATURES o sugges ha a cosumer s exp0.494.634 mes more lkel o purchase a produc ha s feaured a ewspaper ad compared o oe ha s o. For he PRICE coeffce a oe ce decrease prce suggess ha a cosumer s exp0.3666.443 mes more lkel o purchase a brad of ogur. Table.4 Yogur Mulomal Log Model Esmaes Varable Parameer -sasc esmae Yopla 4.450 3.78 Dao 3.76 5.55 Wegh Wachers 3.074.5 FEATURES 0.49 4.09 PRICE -36.658-5.04 - Log Lkelhood 048 AIC 038..4 Radom ul erpreao I ecoomc applcaos we hk of a dvdual choosg amog c caegores where prefereces amog caegores are deermed b a uobserved ul fuco. For he h dvdual a he h me perod use U j for he ul of he jh choce. To llusrae assume ha he dvdual chooses he frs caegor f U > U j for j... c ad deoe hs choce as. We model ul as a uderlg value plus radom ose ha s U j V j + ε j where V j s specfed equao.4. The ose varable s assumed o have a exreme-value dsrbuo of he form Fa Probε j a exp e -a.

Chaper. Caegorcal Depede Varables ad Survval Models / -7 Ths form s compuaoall covee. Omg he observao-level subscrps {} for he mome we have Prob Prob U > U j for j... c Prob ε j < ε + V V jfor j... c { Prob ε < ε + V V for j... c ε } E{ F ε + V V L F + V V } ε E j j c c where kv exp V j V j [ ε + V V + L + exp + V Vc ] Eexp[ k exp ε ] E exp exp ε V. Now s a pleasa exercse calculus o show wh he dsrbuo fuco gve above ha exp[ exp ] kv ε. Thus we have k + E j V j V exp Prob. c c + exp V j V exp V j Because hs argume s vald for all aleraves j c he radom ul represeao elds he mulomal log model...5 Nesed log To mgae he problem of depedece of rreleva aleraves we ow roduce a pe of herarchcal model kow as a esed log model. To erpre he esed log model he frs sage oe chooses a alerave sa he frs pe wh probabl exp V π Prob..7 c exp V + exp V / ρ ρ k k The codoal o o choosg he frs alerave he probabl of choosg he a oe of he oher aleraves follows a mulomal log model wh probables π j exp V j / ρ Prob j j c..8 c π exp V / ρ k k I equaos.7 ad.8 he parameer ρ measures he assocao amog he choces j c. The value of ρ reduces o he mulomal log model ha we erpre o mea depedece of rreleva aleraves. We also erpre Prob o be a weghed average of values from he frs choce ad he ohers. Codoal o o choosg he frs caegor he form of Prob j has he same form as he mulomal log. The advaage of he esed log s ha geeralzes he mulomal log model a wa such ha we o loger have he problem of depedece of rreleva aleraves. A dsadvaage poed ou b McFadde 98E s ha ol oe choce s observed; hus we do o kow whch caegor belogs he frs sage of he esg whou addoal heor regardg choce behavor. Noeheless he esed log geeralzes he mulomal log b allowg alerave depedece srucures. Tha s oe ma vew he esed log as a robus alerave o he mulomal log ad exame each oe of he caegores he frs sage of he esg.

-8 / Chaper. Caegorcal Depede Varables ad Survval Models..6 Geeralzed exreme value dsrbuo The esed log model ca also be gve a radom ul erpreao. To hs ed reur o he radom ul model bu assume ha he choces are relaed hrough a depedece wh he error srucure. McFadde 978E roduced he geeralzed exreme-value dsrbuo of he form: a a F a... a exp G e... e c { } c. Uder regular codos o G McFadde 978E showed ha hs elds U > U for k... c k j Prob j Prob j where G j x... xc G x... xc s he jh paral dervave of G. x Specal cases j. Le G x... xc x +... + xc. I hs case G j ad k V V V j c e G j e... e V Vc G e... e Prob j s he mulomal log case. ρ c / ρ. Le G x +... xc x x k. I hs case G ad k exp V Prob. c exp V + ρ ρ k exp V k / Addoal calculaos show ha Ths s he esed log case. Prob j exp c k V / ρ j exp V / ρ k. exp c k V j exp V k. Ths Thus he geeralzed exreme-value dsrbuo provdes a framework ha ecompasses he mulomal ad codoal log models. Amema 985E provdes backgroud o more complex esed models ha ulze he geeralzed exreme-value dsrbuo.. Mulomal log models wh radom effecs Repeaed observaos from a dvdual ed o be smlar; he case of caegorcal choces hs meas ha dvduals ed o make he same choces from oe observao o he ex. Ths seco models ha smlar hrough a commo heerogee erm. To hs ed we augme our ssemac compoe wh a heerogee erm ad smlar o equao.4 cosder lear combaos of explaaor varables of he form z α + x β..9 V j j j

Chaper. Caegorcal Depede Varables ad Survval Models / -9 As before α represes he heerogee erm ha s subjec-specfc. The form of equao.9 s que geeral ad cludes ma applcaos of eres. However o develop uo we focus o he specal case α + x β..0 V j j j Here erceps var b dvdual ad alerave bu are commo over me. Wh hs specfcao for he ssemac compoe he codoal o he heerogee probabl ha he h subjec a me chooses he jh alerave s exp V j exp αj + x jβ πj α j c. c c exp V exp α + x β k k k k k where we ow deoe he se of heerogee erms as α α α c. From he form of hs equao we see ha a heerogee erm ha s cosa over aleraves j does o affec he codoal probabl. To avod parameer redudaces a covee ormalzao s o ake α c 0. For sascal ferece we beg wh lkelhood equaos. Smlar o he developme Seco.. he codoal lkelhood for he h subjec s c T c T exp + j j j α j x jβ L α π j α c.. j exp α + k k x kβ We assume ha {α } s..d. wh dsrbuo fuco G α ha s pcall ake o be mulvarae ormal. Wh hs coveo he ucodoal lkelhood for he h subjec s L L a dg a α. Assumg depedece amog subjecs he oal log-lkelhood s L L. Relao wh olear radom effecs Posso model Wh hs framework sadard maxmum lkelhood esmao s avalable. However for appled work here are relavel few sascal packages avalable for esmag mulomal log models wh radom effecs. As a alerave oe ca look o properes of he mulomal dsrbuo ad lk o oher dsrbuos. Che ad Kuo 00S recel surveed hese lkages he coex of radom effecs models ad we ow prese oe lk o a olear Posso model. Sascal packages for olear Posso models are readl avalable; wh hs lk he ca be used o esmae parameers of he mulomal log model wh radom effecs. To hs ed a aals would sruc a sascal package o assume ha he bar radom varables j are Posso dsrbued wh codoal meas π j ad codoal o he heerogee erms are depede. Ths s a olear Posso model because from Seco 0.5.3 a lear Posso model akes he logarhmc codoal mea o be a lear fuco of explaaor varables. I coras from equao. log π j s a olear fuco. Of course hs assumpo s o vald. Bar radom varables have ol wo oucomes ad hus cao have a Posso dsrbuo. Moreover he bar varables mus sum o oe ha s j j ad hus are o eve codoall depede. Noeheless wh hs assumpo ad he Posso dsrbuo revewed Seco 0.5.3 he codoal lkelhood erpreed b he sascal package s:

-0 / Chaper. Caegorcal Depede Varables ad Survval Models L T c j T π j α exp - π j α c α π j j! j j α e. j Up o he cosa hs s he same codoal lkelhood as equao. see Exercse 0.. Thus a sascal package ha performs olear Posso models wh radom effecs ca be used o ge maxmum lkelhood esmaes for he mulomal log model wh radom effecs. See Che ad Kuo 00S for a relaed algorhm based o a lear Posso model wh radom effecs. Example.3 Choce of ogur brads - Coued To llusrae we used a mulomal log model wh radom effecs o he ogur daa roduced Example.. Followg Che ad Kuo 00S radom erceps for Yopla Dao ad Wegh Wachers were assumed o follow a mulvarae ormal dsrbuo wh a usrucured covarace marx. Table.5 shows resuls from fg hs model based o he olear Posso model lk ad usg SAS PROC NLMIXED. Here we see ha he coeffces for FEATURES ad PRICE are qualavel smlar o he model whou radom effecs reproduced for coveece from Table.4. The are qualavel smlar he sese ha he have he same sg ad same degree of sascal sgfcace. Overall he AIC sasc suggess ha he model wh radom effecs s he preferred model. Table.5 Yogur Mulomal Log Model Esmaes Whou Radom Wh Radom Effecs Effecs Varable Parameer -sasc Parameer -sasc esmae esmae Yopla 4.450 3.78 5.6 7.9 Dao 3.76 5.55 4.77 6.55 Wegh Wachers 3.074.5.896.09 FEATURES 0.49 4.09 0.849 4.53 PRICE -36.658-5.04-44.406 -.08 - Log Lkelhood 048 730.4 AIC 038 733.4 As wh bar depede varables codoal maxmum lkelhood esmaors have bee proposed; see for example Coawa 989S. Appedx A provdes a bref roduco o hese alerave esmaors..3 Traso Markov Models Aoher wa of accoug for heerogee s o race he developme of a depede varable over me ad represe he dsrbuo of s curre value as a fuco of s hsor. To hs ed defe H o be he hsor of he h subjec up o me. For example f he explaaor varables are assumed o be o-sochasc he we mgh use H { - }. Wh hs formao se we ma paro he lkelhood for he h subjec s f T f H L.3

Chaper. Caegorcal Depede Varables ad Survval Models / - where f H s he codoal dsrbuo of gve s hsor ad f s he margal dsrbuo of. To llusrae oe pe of applcao s hrough a codoal geeralzed lear model GLM of he form f H θ b θ exp + S φ φ where E H b θ ad Var H φ b θ. Assumg a caocal lk for he ssemac compoe oe could use θ g E H x β + ϕ. j See Dggle Heager Lag ad Zeger 00S Chaper 0 for furher applcaos of ad refereces o geeral raso GLMs. We focus o caegorcal resposes. Uordered caegorcal respose To smplf our dscusso of uordered caegorcal resposes we also assume dscree u me ervals. To beg we cosder Markov models of order. Thus he hsor H eed ol coa -. More formall we assume ha π Prob k j Prob k { j... }. jk Tha s gve he formao - here s o addoal formao coe { - } abou he dsrbuo of. Whou covarae formao s cusomar o orgaze he se of raso probables π jk as a marx of he form π π L π c π π L π c Π. M M O M π c π c L π cc Here each row sums o oe. Wh covarae formao ad a al sae dsrbuo Prob oe ca race he hsor of he process kowg ol he raso marx Π. We call he row defer j he sae of org ad he colum defer k he desao sae. For complex raso models ca be useful o graphcall summarze of he se of feasble rasos uder cosderao. To llusrae Fgure. summarzes a emploee rereme ssem wh c 4 caegores. Here deoes acve couao he peso pla deoes rereme from he peso pla 3 deoes whdrawal from he peso pla ad 4 deoes deah. For hs ssem he crcles represe he odes of he graph ad correspod o he respose caegores. The arrows or arcs dcae he modes of possble rasos. Ths graph dcaes ha moveme from sae o saes 3 or 4 s possble so ha we would assume π j 0 for j 3 4. However oce a dvdual s saes 3 or 4 s o possble o move from hose saes kow as absorbg saes. Thus we use π jj for j 3 4 ad π jk 0 for j 3 4 ad k j. Noe ha alhough deah s ceral possble ad eve eveuall cera for hose rereme we assume π 4 0 wh he udersadg ha he pla has pad peso beefs a reremes ad eed o loger be cocered wh addoal rasos afer exg j j

- / Chaper. Caegorcal Depede Varables ad Survval Models he pla regardless of he reaso. Ths s assumpo s ofe covee because s dffcul o rack dvduals havg lef acve membershp a beef pla. acve membershp rereme 3 whdrawal 3 4 4 deah Fgure. Graphcal Summar of a Traso Model for a Hpohecal Emplome Rereme Ssem. For aoher example cosder he modfcao summarzed Fgure.. Here we see ha rerees are ow permed o re-eer he workforce so ha π ma be posve. Moreover ow he raso from rereme o deah s also explcl accoued for so ha π 4 0. Ths ma be of eres a ssem ha pas rereme beefs as log as a reree lves. We refer o Haberma ad Pacco 999O for ma addoal examples of Markov raso models ha are of eres emploee beef ad oher pes of acuaral ssems. acve membershp rereme 3 whdrawal 3 4 4 deah Fgure. A Modfed Traso Model for a Emplome Rereme Ssem. We ca parameerze he problem b choosg a mulomal log oe for each sae of org. Thus we use exp V jk π j k c.4 jk c h exp V where he ssemac compoe V jk s gve b V β..5 jh jk x jk j

Chaper. Caegorcal Depede Varables ad Survval Models / -3 As dscussed he coex of emplome rereme ssems a gve problem oe assumes ha a cera subse of raso probables are zero hus cosrag he esmao of β j. For esmao we ma proceed as Seco.. Defe f k ad j jk. 0 oherwse Wh hs oao he codoal lkelhood s c c π jk j k jk f..6 Here he case ha π jk 0 b assumpo we have ha jk 0 ad use he coveo ha 0 0. To smplf maers we assume ha he al sae dsrbuo Prob s descrbed b a dffere se of parameers ha he raso dsrbuo f -. Thus o esmae hs laer se of parameers oe ol eeds o maxmze he paral log-lkelhood L l f.7 P T where f - s specfed equao.6. I some cases he eresg aspec of he problem s he raso. I hs case oe loses lle b focusg o he paral lkelhood. I oher cases he eresg aspec s he sae such as he proporo of reremes a a cera age. Here a represeao for he al sae dsrbuo akes o greaer mporace. I equao.5 we specfed separae compoes for each alerave. Assumg o mplc relaoshp amog he compoes hs specfcao elds a parcularl smple aalss. Tha s we ma wre he paral log-lkelhood as c L L β where from equaos.4-.6 we have L P j P j j T T c c P j β j lf j jkx jkβ j l exp x jkβ j k k as equao.4. Thus we ca spl up he daa accordg o each lagged choce ad deerme maxmum lkelhood esmaors for each alerave solao of he ohers. Example.3 Choce of ogur brads - Coued To llusrae we reur o he Yogur daa se. We ow explcl model he rasos bewee brad choces as deoed Fgure.3. Here purchases of ogur occur ermel over a wo-ear perod; he daa are o observed a dscree me ervals. B gorg he legh of me bewee purchases we are usg wha s somemes referred o as operaoal me. I effec we are assumg ha oe s mos rece choce of a brad of ogur has he same effec o oda s choce regardless as o wheher he pror choce was oe da or oe moh ago. Ths assumpo suggess fuure refemes o he raso approach modelg ogur choce.

-4 / Chaper. Caegorcal Depede Varables ad Survval Models Yopla Dao 3 Wegh Wachers 3 4 4 Hlad Fgure.3 A Traso Model for Yogur Brad Choce. Tables.6a ad.6b show he relao bewee he curre ad mos rece choce of ogur brads. Here we call he mos rece choce he org sae ad he mos curre choce he desao sae. Table.6a shows ha here are ol 3 observaos uder cosderao; hs s because al values from each of 00 subjecs are o avalable for he raso aalss. For mos observao pars he curre choce of he brad of ogur s he same as chose mos recel exhbg brad loal. Oher observao pars ca be descrbed as swchers. Brad loal ad swchg behavor s more appare Table.6b where we rescale cous b row oals o gve rough emprcal raso probables. Here we see ha cusomers of Yopla Dao ad Wegh Wachers exhb more brad loal compared o hose of Hlad who are more proe o swchg. Table.6a Yogur Traso Cous Desao Sae Org Sae Yopla Dao Wegh Hlad Toal Wachers Yopla 654 65 4 7 777 Dao 7 8 9 6 98 Wegh Wachers 44 8 473 5 540 Hlad 4 7 6 30 67 Toal 783 9 539 68 3 Table.6b Yogur Traso Emprcal Probables Perce Desao Sae Org Sae Yopla Dao Wegh Hlad Toal Wachers Yopla 84. 8.4 5.3. 00.0 Dao 7.7 88.6.0.7 00.0 Wegh Wachers 8. 3.3 87.6 0.9 00.0 Hlad 0.9 5.4 9.0 44.8 00.0 Of course Tables.6a ad.6b do o accou for chagg aspecs of prce ad feaures. I coras hese explaaor varables are capured he mulomal log f dsplaed Table.7. Table.7 shows ha purchase probables for cusomers of Dao Wegh Wachers ad Hlad are more resposve o a ewspaper ad ha Yopla cusomers.

Chaper. Caegorcal Depede Varables ad Survval Models / -5 Moreover compared o he oher hree brads Hlad cusomers are o prce sesve ha chages PRICE have relavel mpac o he purchase probabl s o eve sascall sgfca. Table.6b suggess ha pror purchase formao s mpora whe esmag purchase probables. To es hs s sraghforward use a lkelhood rao es of he ull hpohess H 0 : β j β ha s he compoes do o var b org sae. Table.7 shows ha he oal mus wo mes he paral log-lkelhood s 379.8 + + 8.5 6850.3. Esmao of hs model uder he ull hpohess elds a correspodg value of 974.. Thus he lkelhood rao es sasc s LRT 890.9. There are 5 degrees of freedom for hs es sasc. Thus hs provdes srog evdece for rejecg he ull hpohess corroborag he uo ha he mos rece pe of purchase has a srog fluece he curre brad choce. Table.7 Yogur Traso Model Esmaes Sae of Org Yopla Dao Wegh Wachers Hlad Varable Esmae -sa Esmae -sa Esmae -sa Esmae -sa Yopla 5.95.75 4.5 9.43 4.66 6.83 0.5 0.3 Dao.59 7.56 5.458 6.45.40 4.35 0.0 0.4 Wegh Wachers.986 5.8.5 3.9 5.699.9 -.05 -.93 FEATURES 0.593.07 0.907.89 0.93.39.80 3.7 PRICE -4.57-6.8-48.989-8.0-37.4-5.09-3.840 -. - Log Lkelhood 397.8 608.8 56. 8.5 Example.3 o he choce of ogur brads llusraed a applcao of a codoal mulomal log model where he explaaor varables FEATURES ad PRICE deped o he alerave. To provde a example where hs s o so we reur o Example 9..3 o he choce of a professoal ax preparer. Here facal ad demographc characerscs do o deped o he alerave ad so we appl a sraghforward log model. Example.4 Icome ax pames ad ax preparers Ths s a couao of he Chaper 9 Example 9..3 o he choce of wheher or o a ax fler wll use a professoal ax preparer. Our Chaper 9 aalss of hs daa dcaed srog smlares resposes wh a subjec. I fac our Seco 9.3 fxed effecs aalss dcaed ha 97 ax flers ever used a preparer he fve ears uder cosderao ad 89 alwas dd ou of 58 subjecs. We ow model hese me rasos explcl leu of usg a me cosa lae varable α o accou for hs smlar. Table.8 shows he relaoshp bewee he curre ad mos rece choce of preparer. Alhough we bega wh 90 observaos 58 al observaos are o avalable for he raso aalss reducg our sample sze o 90 58 03. Table.8 srogl suggess ha he mos rece choce s a mpora predcor of he curre choce. Table.8 Tax Preparers Traso Emprcal Probables Perce Org Sae Desao Sae Cou PREP 0 PREP PREP 0 546 89.4 0.6 PREP 486 8.4 9.6

-6 / Chaper. Caegorcal Depede Varables ad Survval Models Table.9 provdes a more formal assessme wh a f of a log raso model. To assess wheher or o he raso aspec s a mpora pece of he model we ca use a lkelhood rao es of he ull hpohess H 0 : β j β ha s he coeffces do o var b org sae. Table.9 shows ha he oal mus wo mes he paral log-lkelhood s 36.5 + 64.6 66.. Esmao of hs model uder he ull hpohess elds a correspodg value of 380.3. Thus he lkelhood rao es sasc s LRT 754.. There are 4 degrees of freedom for hs es sasc. Thus hs provdes srog evdece for rejecg he ull hpohess corroborag he uo ha mos rece choce s a mpora predcor of he curre choce. To erpre he regresso coeffces Table.9 we use he summar sascs Seco 9..3 o descrbe a pcal ax fler ad assume ha LNTPI 0 MR 3 ad EMP 0. If hs ax fler had o prevousl chose o use a preparer he esmaed ssemac compoe s V -0.704 +.040 0.073 + 0.350 -.. Thus he esmaed probabl of choosg o use a preparer s exp-./+exp-. 0.07. Smlar calculaos show ha f hs ax fler had chose o use a preparer he he esmaed probabl s 0.9. These calculaos are accord wh he esmaes Table.8 ha do o accou for he explaaor varables. Ths llusrao pos ou he mporace of he ercep deermg hese esmaed probables. Table.9 Tax Preparers Traso Model Esmaes Sae of Org PREP 0 PREP Varable Esmae -sa Esmae -sa Iercep -0.704-3.06 0.08 0.8 LNTPI.04.50 0.04 0.73 MR -0.07 -.37 0.047.5 EMP 0.35 0.85 0.750.56 - Log Lkelhood 36.5 64.6 Hgher order Markov models There are srog seral relaoshps he Taxpaer daa ad hese ma o be compleel capured b smpl lookg a he mos rece choce. For example ma be ha a ax fler who uses a preparer for wo cosecuve perods has a subsaall dffere choce probabl ha a comparable fler who does o use a preparer oe perod bu elecs o use a preparer he subseque perod. I s cusomar Markov modelg o smpl expad he sae space o hadle hgher order me relaoshps. To hs ed we ma defe a ew caegorcal respose * { - }. Wh hs ew respose he order raso probabl f * - * s equvale o a order raso probabl of he orgal respose f - -. Ths s because he codog eves are he same - * { - - } ad because - s compleel deermed b he codog eve - *. Expasos o hgher orders ca be readl accomplshed a smlar fasho. To smplf he exposo we cosder ol bar oucomes so ha c. Examg he raso probabl we are ow codog o four saes - * { - - } {00 0 0 }. As above oe ca spl up he paral lkelhood o four compoes oe for each sae. Aleravel oe ca wre he log model as Prob - - log V wh

Chaper. Caegorcal Depede Varables ad Survval Models / -7 V 0 0 + x β I 0 β 0 + x β I x β I + 3 3 I 4 4 x where I. s he dcaor fuco of a se. The advaage of rug he model hs fasho as compared o splg up o four dsc compoes s ha oe ca es drecl he equal of parameers ad cosder a reduced parameer se b combg hem. The advaage of he alerave approach s compuaoal coveece; oe performs a maxmzao procedure over a smaller daa se ad a smaller se of parameers albe several mes. Example.4 Icome ax pames ad ax preparers - Coued To vesgae he usefuless of a secod order compoe - he raso model we beg Table.0 wh emprcal raso probables. Table.0 suggess ha here are mpora dffereces he raso probables for each lag org sae - Lag PREP bewee levels of he lag wo org sae - Lag PREP. Table. provdes a more formal aalss b corporag poeal explaaor varables. The oal mus wo mes he paral log-lkelhood s 469.4. Esmao of hs model uder he ull hpohess elds a correspodg value of 067.7. Thus he lkelhood rao es sasc s LRT 567.3. There are degrees of freedom for hs es sasc. Thus hs provdes srog evdece for rejecg he ull hpohess. Wh hs daa se esmao of he model corporag lag oe dffereces elds oal paral mus wo mes log-lkelhood of 490.. Thus he lkelhood rao es sasc s LRT 0.8. Wh 8 degrees of freedom comparg hs es sasc o a ch-square dsrbuo elds a p-value of 0.0077. Thus he lag wo compoe s sascall sgfca corbuo o he model. Table.0 Tax Preparers Order Traso Emprcal Probables Perce Org Sae Desao Sae Lag Lag Cou PREP 0 PREP PREP PREP 0 0 376 89. 0.9 0 8 67.9 3. 0 43 5.6 74.4 37 6. 93.9 Table. Tax Preparers Order Traso Model Esmaes Sae of Org Lag PREP 0 Lag PREP 0 Lag PREP 0 Lag PREP Lag PREP Lag PREP 0 Lag PREP Lag PREP Varable Esmae -sa Esmae -sa Esmae -sa Esmae -sa Iercep -9.866 -.30-7.33-0.8.69 0.5-0.5-0.9 LNTPI 0.93.84 0.675 0.63-0.0-0.7 0.97. MR -0.066 -.79-0.00-0.0 0.065 0.89 0.040.4 EMP 0.406 0.84 0.050 0.04 NA NA.406.69 - Log Lkelhood 54. 33.4 4.7 39.

-8 / Chaper. Caegorcal Depede Varables ad Survval Models Jus as oe ca corporae hgher order lags o a Markov srucure s also possble o brg he me spe a sae. Ths ma be of eres a model of healh saes where we mgh wsh o accommodae he me spe a healh sae or a a-rsk sae. Ths pheomeo s kow as lagged durao depedece. Smlarl he raso probables ma deped o he umber of pror occurreces of a eve kow as occurrece depedece. For example whe modelg emplome we ma wsh o allow raso probables o deped o he umber of prevous emplome spells. For furher cosderaos of hese ad oher specalzed raso models see Lacaser 990E ad Haberma ad Pacco 999O..4 Survval Models Caegorcal daa raso models where oe models he probabl of moveme from oe sae o aoher are closel relaed o survval models. I survval models he depede varable s he me ul a eve of eres. The classc example s me ul deah he compleme of deah beg survval. Survval models are ow wdel appled ma scefc dscples; oher examples of eves of eres clude he ose of Alzhemer s dsease bomedcal me ul bakrupc ecoomcs ad me ul dvorce socolog. Lke he daa suded elsewhere hs ex survval daa are logudal. The crosssecoal aspec pcall cosss of mulple subjecs such as persos or frms uder sud. There ma be ol oe measureme o each subjec bu he measureme s ake wh respec o me. Ths combao of cross-secoal ad emporal aspecs gves survval daa her logudal flavor. Because of he mporace of survval models s o ucommo for researchers o equae he phrase logudal daa wh survval daa. Some eves of eres such as bakrupc or dvorce ma o happe for a specfc subjec. I s commo ha a eve of eres ma o have e occurred wh he sud perod so ha he daa are rgh cesored. Thus he complee observao mes ma o be avalable due o he desg of he sud. Moreover frms ma merge or be acqured b oher frms ad subjecs ma move from a geographcal area leavg he sud. Thus he daa ma be complee due o eves ha are exraeous o he research queso uder cosderao kow as radom cesorg. Cesorg s a regular feaure of survval daa; large values of a depede varable are more dffcul o observe ha small values oher hgs beg equal. I Seco 7.4 we roduced mechasms ad models for hadlg complee daa. For he repeaed cross-secoal daa descrbed hs ex models for compleeess have become avalable ol relavel recel alhough researchers have log bee aware of hese ssues focusg o aro. I coras models of compleeess have bee hsorcall mpora ad oe of he dsgushg feaures of survval daa. Some survval models ca be wre erms of he Seco.3 raso models. To llusrae suppose ha Y s he me ul a eve of eres ad for smplc assume ha s dscree posve eger. From kowledge of Y we ma defe o be oe f Y ad zero oherwse. Wh hs oao we ma wre he lkelhood Prob Y Prob 0... 0 Prob 0 Prob 0 0 Prob 0 erms of raso probables Prob - ad he al sae dsrbuo Prob. Noe ha Seco.3 we cosdered o be he o-radom umber of me us uder cosderao whereas here s a realzed value of a radom varable.

Chaper. Caegorcal Depede Varables ad Survval Models / -9 Example.5 Tme ul bakrupc Shumwa 00O examed he me o bakrupc for 38 frms lsed o Compusa Idusral Fle ad he CRSP Dal Sock Reur Fle for he New York Sock Exchage over he perod 96-99. Several explaaor facal varables were examed cludg workg capal o oal asses reaed o oal asses eargs before eres ad axes o oal asses marke equ o oal lables sales o oal asses e come o oal asses oal lables o oal asses ad curre asses o curre lables. The daa se cluded 300 bakrupces from 39745 frm ears. See also Km e al. 995O for a smlar sud o surace solveces. Survval models are frequel expressed erms of couous depede radom varables. To summarze he dsrbuo of Y defe he hazard fuco Prob Y > probabl des fuco h l Prob Y > survval fuco Prob Y > he saaeous probabl of falure codoal o survvorshp up o me. Ths s also kow as he force of moral acuaral scece as well as he falure rae egeerg. A relaed qua of eres s he cumulave hazard fuco H h s ds. 0 Ths qua ca also be expressed as he mus he log survval fuco ad coversel ProbY > -exph. Survval models regularl allow for o-formave cesorg. Thus defe δ o be a dcaor fuco for rgh-cesorg ha s f Y s cesored δ. 0 oherwse The he lkelhood of a realzao of Y δ sa d ca be expressed erms of he hazard fuco ad cumulave hazard as Prob Y > f Y s cesored Prob Y > oherwse d Prob Y > h Prob Y > d h d exp H There are wo commo mehods for roducg regresso explaaor varables oe s he acceleraed falure me model ad he oher s he Cox proporoal hazard model. Uder he former oe esseall assumes a lear model he logarhmc me o falure. We refer o a sadard reame of survval models for more dscusso of hs mechasm. Uder he laer oe assumes ha he hazard fuco ca be wre as he produc of some basele hazard ad a fuco of a lear combao of explaaor varables. To llusrae we use h h 0 exp x β.8.

-0 / Chaper. Caegorcal Depede Varables ad Survval Models where h 0 s he basele hazard. Ths s kow as a proporoal hazards model because f oe akes he rao of hazard fucos for a wo ses of covaraes sa x ad x oe ges h x h 0 exp x β exp x x β h x h exp x β 0 ha he rao s depede of me. To express he lkelhood fuco for he Cox model le H 0 be he cumulave hazard fuco assocaed wh he basele hazard fuco h 0. Le Y δ Y δ be depede ad assume ha Y follows a Cox proporoal hazard model wh regressors x. The he lkelhood s L 0 Y Y δ δ β h h exp H h Y exp x β exp H Y exp x β Maxmzg hs erms of h 0 elds he paral lkelhood 0 0 δ exp x β LP β.9 exp x jβ j R Y where R s he se of all {Y Y } such ha Y ha s he se of all subjecs sll uder sud a me. From equao.9 we see ha ferece for he regresso coeffces depeds ol o he raks of he depede varables {Y Y } o her acual values. Moreover equao.9 suggess ad s rue ha large sample dsrbuo heor has properes smlar o he usual desrable full paramerc heor. Ths s mldl surprsg because he proporoal hazards model s sem-paramerc; equao.8 he hazard fuco has a full paramerc compoe exp x β bu also coas a oparamerc basele hazard h 0. I geeral oparamerc models are more flexble ha paramerc couerpars for model fg bu resul less desrable large sample properes specfcall slower raes of covergece o a asmpoc dsrbuo. A mpora feaure of he proporoal hazards model s ha ca readl hadle me depede covaraes of he form x. I hs case oe ca wre he paral lkelhood as exp x Y β LP β. exp x j Y β j R Y Maxmzao of hs lkelhood s somewha complex bu ca be readl accomplshed wh moder sascal sofware. To summarze here s a large overlap bewee survval models ad he logudal ad pael daa models cosdered hs ex. Survval models are cocered wh depede varables ha are me ul a eve of eres whereas he focus of logudal ad pael daa models s broader. Because he cocer me survval models usg codog argumes exesvel model specfcao ad esmao. Also because of he me eleme survval models heavl volve cesorg ad rucao of varables s ofe more dffcul o observe large values of a me varable oher hgs beg equal. Lke logudal/pael daa models survval models address repeaed observaos o a subjec. Ulke logudal/pael daa models survval models also address repeaed occurreces of a eve such as marrage. To rack eves over me survval models ma be expressed erms of sochasc processes. Ths δ.

Chaper. Caegorcal Depede Varables ad Survval Models / - formulao allows oe o model ma complex daa paers of eres. There are ma excelle appled roducos o survval modelg see for example Kle ad Moeschberger 997B ad Sger ad Wlle 003EP. For a more echcal reame see Hougaard 000B. Appedx A. Codoal lkelhood esmao for mulomal log models wh heerogee erms To esmae he parameers β he presece of he heerogee erms α j we ma aga look o codoal lkelhood esmao. The dea s o codo he lkelhood usg suffce sascs roduced Appedx 0A.. From equaos.4 ad. he log-lkelhood for he h subjec s c c π j l L α j lπj j l + lπc j j πc c j j α j + x j x c β + l πc because l π j /π c V j - V c α j + x j - x c β. Thus usg he facorzao heorem Appedx 0A. Σ j s suffce for α j. We erpre Σ j o be he umber of choces of alerave j T me perods. To calculae he codoal lkelhood we le S j be he radom varable represeg Σ j ad le sum j be he realzao of Σ j. Wh hs he dsrbuo of he suffce sasc s c πj Prob S sum j j where B s he sum over all ses of he form { j : Σ j sum j. B suffcec we ma ake o α j 0 whou loss of geeral. Thus he codoal lkelhood of he h subjec s c L + j j j c c α exp x x β lπ c Prob S j j sum j π exp B j B j j c j x j x c β + lπ c j c. exp + B j x j x c β lπ c j. As Appedx 9A. hs ca be maxmzed β. However s compuaoall esve. j

Appedces / A- 003 b Edward W. Frees. All rghs reserved Appedces Appedx A. Elemes of Marx Algebra A. Basc Defos marx - a recagular arra of umbers arraged rows ad colums he plural of marx s marces. dmeso of he marx - he umber of rows ad colums of he marx.. Cosder a marx A ha has dmeso m k. Le a j be he smbol for he umber he h row ad jh colum of A. I geeral we work wh marces of he form a a L ak a a L ak A. M M O M am am L amk vecor - a colum vecor s a marx coag ol oe row m. row vecor - a marx coag ol oe colum k. raspose - raspose of a marx A s defed b erchagg he rows ad colums ad s deoed b A or A T. Thus f A has dmeso m k he A has dmeso k m. square marx - a marx where he umber of rows equals he umber of colums ha s mk. dagoal eleme he umber he rh row ad colum of a square marx r dagoal marx - a square marx where all o-dagoal umbers are equal o zero. de marx - a dagoal marx where all he dagoal elemes are equal o oe ad s deoed b I. smmerc marx - a square marx A such ha he marx remas uchaged f we erchage he roles of he rows ad colums ha s f A A. Noe ha a dagoal marx s a smmerc marx. grade vecor a vecor of paral dervaves. If f. s a fuco of he vecor x x x m he he grade vecor s fx/ x. The h row of he grade vecor s s fx/ x. Hessa marx a marx of secod dervaves. If f. s a fuco of he vecor x x x m he he Hessa marx s fx/ x x. The eleme he h row ad jh colum of he Hessa marx s fx/ x x j. A. Basc Operaos scalar mulplcao. Le c be a real umber called a scalar. Mulplg a scalar c b a marx A s deoed b c A ad defed b ca ca L cak ca ca L cak ca. M M O M cam cam L camk marx addo ad subraco. Le A ad B be marces each wh dmeso m k. Use a j ad b j o deoe he umbers he h row ad jh colum of A ad B respecvel. The he marx C A + B s defed o be he marx wh he umber a j +b j o deoe he umber he h row ad jh colum. Smlarl he marx C A - B s defed o be he marx wh he umber a j -b j o deoe he umbers he h row ad jh colum.

A- / Appedces marx mulplcao. If A s a marx of dmeso m c ad B s a marx of dmeso c k he C A B s a marx of dmeso m k. The umber he h row ad jh colum of C s c s a s b sj. deerma - a fuco of a square marx deoed b dea or A. For a marx he deerma s dea a. To defe deermas for larger marces we eed wo addoal coceps. Le A rs be he m- m- submarx of A defed be removg he rh row ad sh colum. The deerma of A rs dea rs s called he mor of he eleme a rs. Furher - r+s s called he mor of he eleme a rs. Recursvel defe m de A a s rs de A rs for a r. For example for we have he deerma s dea a a a a. marx verse. I marx algebra here s o cocep of dvso. Isead we exed he cocep of recprocals of real umbers. To beg suppose ha A s a square marx of dmeso k k such ha dea 0. Furher le I be he k k de marx. If here exss a k k marx B such ha A B I B A he B s called he verse of A ad s wre B A -. A.3 Furher Defos learl depede vecors a se of vecors c c k s sad o be learl depede f oe of he vecors he se ca be wre as a lear combao of he ohers. learl depede vecors a se of vecors c c k s sad o be learl depede f he are o learl depede. Specfcall a se of vecors c c k s sad o be learl depede f ad ol f he ol soluo of he equao x c + + x k c k 0 s x x k 0. rak of a marx he larges umber of learl depede colums or rows of a marx. sgular marx a square marx A such ha dea 0. o-sgular marx a square marx A such ha dea 0. posve defe marx a smmerc square marx A such ha x A x > 0 for x 0. o-egave defe marx a smmerc square marx A such ha x A x 0 for x 0. orhogoal wo marces A ad B are orhogoal f A B 0 a zero marx. orhogoal marx - a marx A such ha A A I. dempoe a square marx such ha A A A. race he sum of all dagoal elemes of a square marx. egevalues he soluos of he h degree polomal dea λ I 0. Also kow as characersc roos ad lae roos. egevecor a vecor x such ha A x λ x where λ s a egevalue of A. Also kow as a characersc vecor ad lae vecor. geeralzed verse - of a marx A s a marx B such ha A B A A. We use he oao A o deoe he geeralzed verse of A. I he case ha A s verble he A s uque ad equals A -. Alhough here are several defos of geeralzed verses he above defo suffces for our purposes. See Searle 987G for furher dscusso of alerave defos of geeralzed verses. A.4 Marx Decomposos Le A be a m m smmerc marx. The A has m pars of egevalues ad egevecors λ e λ m e m. The egevecors ca be chose o have u legh e j e j ad o be orhogoal o oe aoher e e j 0 for j. The egevecors are uque uless wo or more egevalues are equal.

Appedces / A-3 Specral decomposo - Le A be a m m smmerc marx. The we ca wre A λ e e + + λ m e m e m where he egevecors are have u legh ad are muuall orhogoal. Suppose ha A s posve defe. The each egevalue s posve. Smlarl f A s oegave defe he each egevalue s o-egave. Usg he specral decomposo we ma wre a m m smmerc marx A as A P Λ P where P [e : : e m ] ad Λ dagλ λ m. Square roo marx for a o-egave defe marx A we ma defe he square roo marx as A / P Λ / P where Λ / dagλ / λ / m. The marx A / s smmerc ad s such ha A / A / A. Marx power for a posve defe marx A we ma defe A c P Λ c P where Λ c dagλ c λ c m for a scalar c. Cholesk facorzao - Suppose ha A s posve defe. The we ma wre A L L where L s a lower ragular marx l j 0 for < j. Furher L s verble so ha A - L - L -. L s kow as he Cholesk square-roo marx. Le A be a m k marx. The here exss a m m orhogoal marx U ad a k k orhogoal marx V such ha A U Λ V. Here Λ s a m k marx such ha he er of Λ s λ 0 for mm k ad he oher eres are zero. The elemes λ are called he sgular values of A. Sgular value decomposo For λ > 0 le u ad v be he correspodg colums of U r ad V respecvel. The sgular value decomposo of A s A λ u v where r s he rak of A. QR decomposo Le A be a m k marx wh m k of rak k. The here exss a m R m orhogoal marx Q ad a k k upper ragular marx R such ha A Q. We ca 0 also wre A Q k R where Q k cosss of he frs k colums of Q. A.5 Paroed Marces A sadard resul o verses of paroed marces s B B B B C CB B B B C C A. where C B BB B ad C B B B B. A relaed resul o deermas of paroed marces s B B de de B de C de B de B B B B Aoher sadard resul o verses of paroed marces s B B B B B B + B B B B B B B B. A. B A.3 To llusrae wh R B Z B -Z B ad D - B we have The relaed deerma resul s R + Z D Z R R Z D + Z R Z Z V R. A.4 D + Z R l de V l de R l de D + l de Z. A.5

A-4 / Appedces Suppose ha A s a verble p p marx ad c d are a p vecors. The from for example Grabll 983G Theorem 8.9.3 we have A cd A A + cd A. A.6 + d A c To check hs resul smpl mulpl A+cd b he rgh had sde o ge I he de marx. Le P Q be dempoe ad orhogoal marces. Le a b be posve coas. The a P + b Q c a c P + b c Q for scalar c. A.7 Balag 00E. A.6 Kroecker Drec Produc Le A be a m marx ad B be a m marx. The drec produc s defed o be ab ab L ab ab ab L ab A B a m m marx. M M O M amb amb L amb Some properes of Drec Producs. A B F G A F B G A B A B If P Q are orhogoal marces he P Q s a orhogoal marx. See Grabll 969G Chaper 8.

Appedces / A-5 Appedx B. Normal Dsrbuo Uvarae ormal dsrbuo Recall ha he probabl des fuco of Nµ σ s gve b µ f exp. σ π σ If µ 0 ad σ he N0 s sad o be sadard ormal. The sadard ormal probabl des fuco s φ exp. π The correspodg sadard ormal dsrbuo fuco s deoed b Φ φ z dz. Mulvarae ormal dsrbuo A vecor of radom varables... s sad o be mulvarae ormal f all lear combaos of he form a Σ a are ormall dsrbued where a s are cosas. I hs case we wre Nµ V where µ E s he expeced value of ad V Var s he varace-covarace marx of. From he defo we have ha Nµ V mples ha a Na µ a V a. The mulvarae probabl des fuco of Nµ V s gve b / / f f... π de V exp µ V µ. For mxed lear models he mea s a fuco of lear combaos of parameers such ha µ X β. Thus he probabl des fuco of NX β V s gve b / / f f... π de V exp Xβ V Xβ. Normal lkelhood A logarhmc probabl des fuco evaluaed usg he observaos s kow as a log-lkelhood. Suppose ha hs des depeds o he mea parameers β ad varace compoes τ. The he log-lkelhood for he mulvarae ormal ca be expressed as L β τ l π + l de V + Xβ V Xβ. B. Codoal dsrbuos Suppose ha s a mulvarae ormall dsrbued vecor such ha µ Σ Σ N. µ Σ Σ The he codoal dsrbuo of s also ormal. Specfcall we have µ + Σ Σ µ Σ Σ Σ Σ N. B. Thus E µ + Σ Σ µ ad Var Σ Σ Σ Σ.

A-6 / Appedces Appedx C. Lkelhood-Based Iferece Beg wh a radom vecor whose jo dsrbuo s kow up o a vecor of parameers θ. Ths jo probabl des mass fuco s deoed as p; θ. The loglkelhood fuco s l p; θ L; θ Lθ whe evaluaed a a realzao of. Tha s he log-lkelhood s a fuco of he parameers wh he daa fxed raher ha a fuco of he daa wh he parameers fxed. C. Characerscs of Lkelhood Fucos Two basc characerscs of lkelhood fucos are: E L θ 0 C. θ ad L θ L θ E θ L + 0 θ θ E C. θ θ The dervave of he log-lkelhood fuco L θ / θ s called he score fuco. From equao C. we see ha has mea zero. To see equao C. uder suable regular codos we have p ; θ E L E θ θ 0 θ p ; θ p ; θ d θ θ θ p ; d. θ For coveece hs demosrao assumes a des for ; exesos o mass ad mxures are sraghforward. The proof of equao C. s smlar ad s omed. Some suable regular codos are requred o allow he erchage of he dervave ad egral sg. Usg equao C. we ma defe L θ L θ I θ E E L θ θ θ θ θ he formao marx. Ths qua s used he scorg algorhm for parameer esmao. Uder broad codos we have ha L θ / θ s asmpocall ormal wh mea 0 ad varace Iθ. C. Maxmum Lkelhood Esmaors Maxmum lkelhood esmaors are values of he parameers θ ha are mos lkel o have bee produced b he daa. Cosder radom varables wh probabl fuco p; θ. The value of θ sa θ MLE ha maxmzes p; θ s called he maxmum lkelhood esmaor. We ma also deerme θ MLE b maxmzg L; θ Lθ he loglkelhood fuco. I ma applcaos hs ca be doe b fdg roos of he score fuco L θ / θ. Uder broad codos we have ha θ MLE s asmpocall ormal wh mea θ ad varace Iθ -. Moreover maxmum lkelhood esmaors are he mos effce he followg sese. Suppose ha θˆ s a alerave ubased esmaor. The Cramer-Rao heorem saes uder mld regular codos for all vecors c ha Var c θ MLE Var c θˆ for suffcel large.

Appedces / A-7 We also oe ha Lθ MLE -Lθ has a ch square dsrbuo wh degrees of freedom equal o he dmeso of θ. Example - Oe parameer expoeal faml Le be depede draws from a oe parameer expoeal faml dsrbuo as equao 9. θ b θ p θ φ exp + S φ. C.3 φ The score fuco s L θ l p θ φ θ θ θ θ b θ + S φ φ b θ b θ φ. φ Thus seg hs equal zero elds b θ or θ b. The formao marx s b θ I θ E L θ θ. φ MLE I geeral maxmum lkelhood esmaors are deermed eravel. For geeral lkelhoods wo basc procedures are used: Newo-Raphso uses he erave algorhm θ NEW MLE θ OLD L θ θ Fsher scorg uses he erave algorhm θ θ + I θ s he formao marx. NEW OLD OLD L θ L θ θ θold θ θold. where Iθ Example - Geeralzed lear model Le be depede draws from a oe parameer expoeal faml wh dsrbuo equao C.3. Suppose ha he radom varable has mea µ wh ssemac compoe η gµ x β ad caocal lk so ha η θ. Assume ha he scale parameer vares b observao so ha φ φ / w where w s a kow wegh fuco. The he score fuco s L β xβ b xβ + b x β S φ w x. β β φ φ The marx of secod dervaves s L β xx w x β β β b. C.4 φ Thus he Newo-Raphso algorhm s β NEW β OLD w b xβold xx w b xβold x.

/ Appedces A-8 Because he marx of secod dervaves s o-sochasc we have ha I L β β β β ad hus he Newo-Raphso s equvale o he Fsher scorg algorhm. C.3 Ieraed Reweghed Leas Squares Coue wh he pror example cocerg he geeralzed lear model ad defe a adjused depede varable β x β x β x β + b b *. Ths has varace [ ] [ ] b / b b b Var Var * x β x β x β β x β w φ φ. Use he ew wegh as he recprocal of he varace w β w b x β/φ. The wh he expresso x β β β x β β x β β x w w w b b * * φ from he Newo-Raphso erao we have OLD OLD OLD NEW w w x x β x x x β β β b b OLD OLD OLD OLD OLD w w x β x β β x x β β * φ φ OLD OLD OLD OLD OLD OLD w w w β x x β β x β x x β β * OLD OLD OLD w w * β x β x x β. Thus hs provdes a mehod for erao usg weghed leas squares. C.4 Profle Lkelhood Spl he vecor of parameers o wo compoes sa θ θ θ. Here erpre θ o be he parameers of eres whereas θ are auxlar or usace parameers of secodar eres. Le θ MLE θ be he maxmum lkelhood esmaor for a fxed value of θ. The he profle lkelhood for θ s θ θ θ θ θ ; p sup ; p MLE. Ths s o a lkelhood he usual sese. To llusrae s sraghforward o check ha equao C. does o hold for θ θ ; p MLE. C.5 Quas-Lkelhood Suppose ha s dsrbued accordg o he oe parameer expoeal faml equao C.3. The E µ bθ ad Var φ b θ. Thus φ θ θ θ θ µ / Var b b. Usg he cha rule we have V b Var l p l p µ µ φ θ φ φ θ θ µ θ φ θ µ.

Appedces / A-9 Here we have explcl deoed he varace of as a fuco of he mea µ b usg he oao Var Vµ. We wre µ - µ / Vµ a fuco of ad µ. Smlar o he score fuco hs fuco has he followg properes: E µ 0 ad E µ µ Var µ. V µ Sce hese properes are he oes ha make he asmpocs of lkelhood aalss go we ma hk of µ Q µ s ds as a quas log-lkelhood. The parameer µ s kow up o a fe dmeso vecor of parameers β ad hus we wre µβ for µ. Thus he quas-score fuco s Q µ µ β µ. β β Esmao proceeds as he lkelhood case Tha s ma applcaos we assume ha { } are depede observaos wh mea µ ad varace Vµ. The he quasscore fuco s Q µ µ µ β. β β C.6 Esmag Equaos A alerave mehod for parameer esmao s usg he oo of a esmag equao ha exeds he dea of mome esmao. I he followg we summarze reames due o McCullagh ad Nelder 989G Chaper 9 ad Dggle e al. 00S. From a ecoomercs perspecve where hs procedure s kow as geeralzed mehod of momes see Haash 000E. A esmag fuco s a fuco g. of a vecor of radom varables ad θ a p vecor of parameers such ha E g; θ 0 for all θ a parameer space of eres C.5 where 0 s a vecor of zeroes. For example ma applcaos we ake g; θ - µθ where E µθ. The choce of he g fuco s crcal applcaos. I ecoomercs equao C.5 s kow as he mome codo. Le H be a p marx ad defe he esmaor as he soluo of he equao H g; θ 0 p deoed as θ EE. Wha s he bes choce of H? Usg a Talor-seres expaso we see ha g ; θ 0 p H g; θ EE H { g; θ + θ EE - θ } θ so ha g ; θ θ EE θ H H g ; θ. θ Thus he asmpoc varace s ; g θ ; Var Var ; g θ θ EE H H g θ H H. θ θ

/ Appedces A-0 The choce of H ha elds he mos effce esmaor s θ θ g θ g ; ; Var. Ths elds ; ; Var ; Var θ θ g θ g θ θ g θ EE. For he case g; θ - µθ we have θ θ µ V H / where V Var ad Var θ θ µ V θ θ µ θ EE. I hs case he esmag equaos esmaor θ EE s he soluo of he equao θ µ V θ θ µ 0 p. For depede observaos hs represeao s decal o he quas-lkelhood esmaors preseed Appedx C.5. As aoher example suppose ha w x are..d. ad le g; θ g ; θ g ; θ where g ; θ w - x θ ad E w x x θ. Assume ha Var g ; θ Var w - x θ σ E w w σ Σ w. Thus x w w x Σ I θ θ g θ g H w M ; ; Var σ w x Σ w x Σ w w M σ. Thus he esmaor s a soluo of x θ w w Σ x θ x w θ x w Σ w x w Σ x 0 w w w σ σ M L. Ths elds EE w w Σ x w x w Σ x θ w w. Usg w w place of Σ w elds he srumeal varable esmaor. For he case of logudal daa mxed models we wll assume ha he daa vecor ca be decomposed as where E µ θ ad Var V V θ τ. Here he r vecor τ s our vecor of varace compoes. Assumg depedece amog subjecs we cosder G θ µ V θ θ µ τ θ θ. C.6 The esmag equaos esmaor of θ deoed b θ EE s he soluo of he equao 0 p G θ θ τ where 0 p s a p vecor of zeroes.

Appedces / A- To compue θ EE we requre esmaors of he varace compoes τ. For secod momes we wll use as our prmar daa source. For oao le vechm deoe he colum vecor creaed b sackg he colums of he marx M. Thus for example vech T + T + T +. Thus we use h vech as our daa vecor ad le η E h. Thus he esmag equao for τ s η G τ θ τ h η. C.7 τ The esmag equaos esmaor of τ deoed b τ EE s he soluo of he equao 0 r G τ θ τ. To summarze we frs compue al esmaors of θ τ sa θ 0EE τ 0EE pcall usg basc mome codos. The a he h sage recursvel:. Use τ EE ad equao C.6 o updae he esmaor of θ; ha s θ +EE s he soluo of he equao G θ θ τ EE 0 p.. Use θ +EE ad equao C.7 o updae he esmaor of τ; ha s τ +EE s he soluo of he equao G τ θ +EE τ 0 r. 3. Repea seps ad ul covergece. Uder mld regular codos θ EE τ EE s cosse ad asmpocall ormal; see for example Dggle e al. 00S. Uder mld regular codos Goureroux Mofor ad Trogo 984E show ha he esmaor θ EE calculaed usg he esmaed τ s jus as effce asmpocall as f τ were kow. Lag ad Zeger 986B also provde he followg esmaor of he asmpoc varace-covarace marx of θ EE τ EE µ µ G G G G G G C.8 h η h η where µ V 0 G θ ad η 0 τ G µ θ η θ µ τ η τ. C.7 Hpohess Tess We cosder esg he ull hpohess H 0 : hθ d where d s a kow vecor of dmeso r ad h. s kow ad dffereable. There are hree wdel used approaches for esg he ull hpohess called he lkelhood rao Wald ad Rao ess. The Wald approach evaluaes a fuco of he lkelhood a θ MLE. The lkelhood rao approach uses θ MLE ad θ Reduced. Here θ Reduced s he value of θ ha maxmzes Lθ uder he resrco ha hθ d. The Rao approach also uses θ Reduced bu deermes b maxmzg Lθ - λhθ - d where λ s a vecor of Lagrage mulplers. Hece Rao's es s also called he Lagrage mulpler es. The hree sascs are: LRT Lθ MLE - Lθ Reduced Wald TS W θ MLE where TS W θ hθ - d{ hθ -Iθ - hθ} - hθ - d. Rao TS R θ Reduced where TS R θ Lθ -Iθ - Lθ. Here hθ hθ/ θ s he grade of hθ ad Lθ Lθ/ θ s he grade of Lθ he score fuco.

A- / Appedces The ma advaage of he Wald sasc s ha ol requres compuao of θ MLE ad o θ Reduced. Smlarl he ma advaage of he Rao sasc s ha ol requres compuao of θ Reduced ad o θ MLE. I ma applcaos compuao of θ MLE s oerous. Uder broad codos all hree es sascs are asmpocall ch-square wh r degrees of freedom uder H 0. All asmpoc mehods work well whe he umber of parameers s fe dmesoal ad he ull hpohess specfes ha θ s o he eror of he parameer space. I he usual fxed effecs model he umber of dvdual-specfc parameers s he same order as he umber of subjecs. Here he umber of parameers eds o f as he umber of subjecs eds o f ad he usual asmpoc approxmaos are o vald. Isead specal codoal maxmum lkelhood esmaors ejo he asmpoc properes smlar o maxmum lkelhood esmaors. Whe a hpohess specfes ha θ s o he boudar he he asmpoc dsrbuo s o loger vald whou correcos. A example s H 0 : θ σ α 0. Here he parameer space s [0. B specfg he ull hpohess a 0 we are o he boudar. Self ad Lag 987S provde some correcos ha mprove he asmpoc approxmao. C.8 Goodess of F Sascs I lear regresso models he mos wdel ced goodess of f sasc s he R measure ha s based o he decomposo ˆ + ˆ + ˆ ˆ. I he laguage of Seco.3 hs decomposo s: Toal SS Error SS + Regresso SS + Sum of Cross-Producs. The dffcul wh olear models s ha he Sum of Cross-Producs erm rarel equals zero. Thus oe ges dffere sascs whe defg R as Regresso SS/Toal SS as compared o -Error SS/Toal SS. A alerave wdel ced goodess of f measure s he Pearso ch-square sasc. To defe hs sasc suppose ha E µ Var Vµ for some fuco V. ad ha µˆ s a esmaor of µ. The he Pearso ch-square sasc s defed as ˆ µ. For Posso V ˆ µ models of cou daa hs formulao reduces o he form I he coex of geeralzed lear models a goodess of f measure s he devace sasc. To defe hs sasc suppose E µ µθ ad wre L µˆ for he log-lkelhood evaluaed a µˆ µ θˆ. The scaled devace sasc s defed as D* µˆ L L µˆ. I lear expoeal famles we mulpl b he scalg facor φ o defe he devace sasc D µˆ φ D* µˆ. Ths mulplcao acuall removes he varace scalg facor from he defo of he sasc. Usg Appedx 9A s sraghforward o check ha he devace sasc reduces o he followg forms for hree mpora dsrbuos: Normal: D µ ˆ ˆ µ ˆ µ ˆ µ. Beroull: D πˆ l πˆ + l πˆ

Appedces / A-3 Posso: D µ ˆ l + ˆ µ. ˆ µ Here we use he coveo ha l 0 whe 0. C.9 Iformao Crera Lkelhood rao ess are useful for choosg bewee wo models ha are esed ha s where oe model s a subse of he oher. How do we compare models whe he are o esed? Oe wa s o use he followg formao crera. The dsace bewee wo probabl dsrbuos gve b probabl des fuco g ad f θ ca be summarzed b g KLg fθ Eg l. fθ Ths s he Kullback-Lebler dsace ha urs ou o be oegave. Here we have dexed f b a vecor of parameers θ. Pckg he fuco g o be f θ 0 he mmzg KLfθ 0 fθ s equvale o he maxmum lkelhood prcple. I geeral we have o esmae g.. Akake showed ha a reasoable alerave s o mmze AIC - Lθ MLE + umber of parameers kow as Akake s Iformao Crero. Ths sasc s used b whe comparg several alerave oesed models. Oe pcks he model ha mmzes AIC. If he models uder cosderao have he same umber of parameers hs s equvale o choosg he model ha maxmzes he log-lkelhood. We remark ha me seres aalss he AIC s rescaled b he umber of parameers. The sasc AIC s also useful ha reduces o he C p a sasc ha s wdel used regresso aalss. Ths sasc mmzes a bas ad varace rade-off whe selecg amog lear regresso models. Schwarz derved a alerave crero usg Baesa mehods. Hs measure s kow as he Baesa Iformao Crero defed as BIC - Lθ MLE + l umber of parameers. Ths measure gves greaer wegh o he umber of parameers. Tha s oher hgs beg equal BIC wll sugges a more parsmoous model ha AIC.

A-4 / Appedces Appedx D. Sae Space Model ad he Kalma Fler D. Basc Sae Space Model Cosder he observao equao W δ + ε T D. where s a vecor ad δ s a m vecor. The raso equao s δ T δ - + η. D. Togeher equaos D. ad D. defe he sae space model. To complee he specfcao defe Var - ε H ad Var - η Q where Var s a varace codoal o formao up o ad cludg me ha s { }. Smlarl le E deoe he codoal expecao ad assume ha E - ε 0 ad E - η 0. Furher defe d 0 E δ 0 P 0 Var δ 0 ad P Var δ. Assume ha {ε } ad {η } are muuall depede. I subseque secos wll be useful o summarze equao D.. Thus we defe W δ ε W 0 L 0 δ ε Wδ M M T WT δ Wh he oao N T T + ε M εt 0 M 0 W M 0 L O L 0 δ + ε W δ + ε. D.3 M M M WT δt εt we have ha s a N vecor of radom varables δ s a Tm vecor of sae varables W s a N Tm marx of kow varables ad ε s a N vecor of dsurbace erms. D. Kalma Fler Algorhm Takg a codoal expecao ad varace of equao D. elds he predco equaos d /- E - δ T d - D.4a ad P /- Var - δ T P - T + Q. D.4b Takg a codoal expecao ad varace of equao D. elds E - W d /- ad F Var - W P /- W + H. The updag equaos are d d /- + P /- W F - - W d /- ad P P /- - P /- W F - W P /-. D.5a D.5b D.6a D.6b The updag equaos ca be movaed b assumg ha δ ad are jol ormall dsrbued. Wh hs assumpo ad equao B. of Appedx B we have E δ E - δ E - δ + Cov - δ Var - - - E - ad Var δ Var - δ - Cov - δ Var - - Cov - δ. These expressos eld he updag equaos mmedael.

Appedces / A-5 For compuaoal coveece he Kalma fler algorhm equaos D.4-D.6 ca be expressed more compacl as d +/ T + d /- + K - W d /- D.7 ad P +/ T + P /- - P /- W F - W P /- T + + Q + D.8 where K T + P /- W F - s kow as he ga marx. To sar hese recursos from D.4 we have d /0 T d 0 ad P /0 T P 0 T + Q. D.3 Lkelhood Equaos Assumg he codoal varaces Q ad H ad al codos d 0 ad P 0 are kow equaos D.7 ad D.8 allow oe o recursvel compue E - ad Var - F. These quaes are mpora because he allow us o drecl evaluae he lkelhood of { T }. Tha s assume ha { T } are jol mulvarae ormal ad le f be used for he jo ad codoal des. The wh equaos B. ad B. of Appedx B he logarhmc lkelhood s L l f T l f + f... T T T N l π + l de F + E F E. D.9 From he Kalma fler algorhm equaos D.4-D.6 we see ha E - s a lear combao of { - }. Thus we ma wre E 0 L L E D.0 M M T T ET T where L s a N N lower ragular marx wh oe s o he dagoal. Elemes of he marx L do o deped o he radom varables. The advaages of hs rasformao are ha he compoes of he rgh had sde of equao D.0 are mea zero ad are muuall ucorrelaed. D.4 Exeded Sae Space Model ad Mxed Lear Models To hadle he logudal daa model descrbed Seco 6.5 we ow exed he sae space model o mxed lear models so ha hadles fxed ad radom effecs. B corporag hese effecs we wll also be able o roduce al codos ha are esmable. Specfcall cosder he observao equao X β + Z α + W δ + ε. D. Here β s a K vecor of ukow parameers called fxed effecs. Furher α s a q* radom vecor kow as radom effecs. We assume ha α has mea 0 varace-covarace marx σ B Var α ad s depede of {δ } ad {ε }. The raso equao s as equao D. ha s we assume δ T δ - + η. D. Smlar o equao D.3 we summarze equao D. as

A-6 / Appedces X β + Z α + W δ + ε. D.3 Here W δ ad ε are defed equao D.3 ad X X X X T ad Z Z Z Z T. D.5 Lkelhood Equaos for Mxed Lear Models To hadle fxed ad radom effecs beg wh he rasformao marx L defed equao D.0. I developg he lkelhood he mpora po o observe s ha he rasformed sequece of radom varables - X β + Z α has he same properes as he basc se of radom varables Appedx D.. Specfcall wh he rasformao marx L he T compoes of he vecor v L X β + Z α v v are mea zero ad muuall ucorrelaed. Furher codoal o α ad { - } he h compoe of hs marx v has varace F. Wh he Kalma fler algorhm D.7 ad D.8 we defe he rasformed varables * L X* L X ad Z* L Z. To llusrae for he jh colum of X sa X j oe would recursvel calculae d +/ X j T + d /- X j + K X j - W d /- X j D.7* place of equao D.7. We beg he recurso equao D.7* wh d /0 X j 0. Equao D.8 remas uchaged. The aalogous o expresso D.5a he h compoe of X* s X * j X j - W d /- X j. Wh hese rasformed varables we ma express he rasformed radom varables as * L v + X*β + Z* α. Recall σ B Var α ad oe ha v F 0 L 0 Var v Var v 0 F L 0 σ Λ. D.4 M M M O M vt 0 0 L F T Ths elds E * X*β ad Var * σ Λ + Z* B Z* σ V. We use τ o deoe he vecor of ukow quaes ha parameerze V. From equao B. of Appedx B he logarhmc lkelhood s Lβσ τ - {N l π + N l σ + σ - * - X*β V - * - X*β + l de V }. D.5 M v T The correspodg resrced log-lkelhood s L R βσ τ - {l de X* V - X* K l σ } + Lβσ τ + cosa. D.6 Eher D.5 or D.6 ca be maxmzed o deerme a esmaor of β. The resul s equvale o he geeralzed leas squares esmaor b GLS X* V - X* - X* V - *. D.7

Appedces / A-7 Usg b GLS for β equaos D.6 ad D.7 elds coceraed lkelhoods. To deerme he REML esmaor of σ we maxmze L R b GLS σ τ holdg τ fxed o ge s REML N-K * - X* b GLS V - * - X* b GLS. D.8 Thus he logarhmc lkelhood evaluaed a hese parameers s Lb GLS REML s τ - {N l π + N l s + N-K + l de V }. D.9 The correspodg resrced logarhmc lkelhood s L REML - {l de X* V - X* K l s REML REML } + Lb GLS s REML τ + cosa. D.0 The lkelhood expressos equaos D.9 ad D.0 are uvel sraghforward. However because of he umber of dmesos he ca be dffcul o compue. We ow provde alerave expressos ha alhough more complex are smpler o compue wh he Kalma fler algorhm. From equaos A.3 ad A.5 of Appedx A we have ad V - Λ - - Λ - Z* B - + Z* Λ - Z* - Z* Λ - l de V l de Λ - l de B - + l deb - + Z* Λ - Z*. D. D. Wh equao D. we mmedael have he expresso for b GLS equao 6.3. From equao D.8 he resrced maxmum lkelhood esmaor of σ ca be expressed as s REML N-K {* V - * - * V - X* b GLS } whch s suffce for equao 6.34. Ths equao D.0 ad D. are suffce for he equao 6.35.

A-8 / Appedces Appedx E. Smbols ad Noao Smbol Descrpo Chaper defed dces for he h subjec h me perod T umber of observaos for he h subjec umber of subjecs N oal umber of observaos N T + T + + T respose for he h subjec h me perod T vecor of resposes for he h subjec.... T x j jh explaaor varable assocaed wh global parameers for he h subjec h me perod K umber of explaaor varables assocaed wh global parameers x K vecor of explaaor varables assocaed wh global parameers for he h subjec h me perod x x x x K X T K marx of explaaor varables assocaed wh global parameers for he h subjec X x x... x z j q z Z T jh explaaor varable assocaed wh subjec-specfc parameers for he h subjec h me perod umber of explaaor varables assocaed wh subjec-specfc parameers q vecor of explaaor varables assocaed wh subjec-specfc parameers for he h subjec h me perod z z z z q T q marx of explaaor varables assocaed wh subjec-specfc parameers for he h subjec Z z z... z T ε error erm for he h subjec h me perod ε T vecor of error erms for he h subjec ε ε ε... ε. T β j jh global parameer assocaed wh x j β K vecor of global parameers β β β L β K α subjec-specfc ercep parameer for he h subjec α j jh subjec-specfc parameer for he h subjec assocaed wh z j α q vecor of subjec-specfc parameer for he h subjec α α... α q λ me-specfc parameer σ varace of he error erm uder he homoscedasc model σ Var ε ρ correlao bewee wo error erms ρ corrε r ε s b j b esmaors of β j β a a j a esmaors of α α j α s ubased esmaor of σ e resdual for he h subjec h me perod e T vecor of resduals for he h subjec e e e... e. T

Appedces / A-9 Smbol Descrpo Chaper defed average over me of he resposes from he h subjec T T K vecor of averages over me of he explaaor varables assocaed wh he global parameers from he h subjec x x T T x W a K K wegh marx for he h subjec W x x x x T r he rak of he h resdual e from he vecor of resduals {e... e T }. Spearma's rak correlao coeffce bewee he h ad jh subjecs T r + + T r j T / / srj T r T + / sr j R he average of Spearma's rak correlaos R AVE AVE / { < sr j} j R AVE he average of squared Spearma's rak correlaos RAVE < sr / { j} j R T T varace-covarace marx R Var ε. R rs he eleme he rh row ad sh colum of R R rs Cov ε r ε s R T T marx varace-covarace for he h subjec R Var ε T vecor of oes I Ide marx geerall of dmeso T T I T T de marx J marx of oes geerall of dmeso T T J T T marx of oes τ vecor of varace compoes Q marx ha projecs a vecor of resposes o OLS resduals Q I Z Z Z Z marx ha projecs a vecor of resposes o GLS resduals Q Z Q / Z R Z Z R / Z I R Z 3 σ α varace of he subjec-specfc ercep he oe-wa error compoes model Var α s α ubased esmaor of σ α 3 b EC geeralzed leas squares GLS esmaor of β he error compoes 3 model TS a es sasc for assessg homogee he error compoes model 3 δ g a group effec varable 3

A-0 / Appedces Smbol Descrpo Chaper defed D varace-covarace marx of subjec-specfc effecs he logudal 3 daa mxed model Var α V varace-covarace marx of he h subjec he logudal daa mxed 3 model V Z D Z + R. X N K marx of explaaor varables assocaed wh fxed effecs he 3 mxed lear model β K vecor of fxed effecs he mxed lear model 3 Z N q marx of explaaor varables assocaed wh radom effecs he 3 mxed lear model α q vecor of radom effecs he mxed lear model 3 N vecor of resposes he mxed lear model 3 ε N vecor of dsurbaces he mxed lear model 3 b GLS GLS esmaor of β he mxed lear model 3 b MLE maxmum lkelhood esmaor MLE of β he mxed lear model 3 l. logarhmc lkelhood of he h subjec 3 L. logarhmc lkelhood of he ere daa se 3 b W weghed leas square esmaor of β he mxed lear model 3 seb W robus sadard error of b W 3 L R logarhmc resrced maxmum lkelhood of he ere daa se 3 b OLS ordar leas squares esmaor of α + β he radom coeffces model 3 D SWAMY Swam s esmaor of D 3 s shrkage esmaor of µ +α he oe-wa radom effecs ANOVA 4 model ζ weghg credbl facor used o compue he shrkage esmaor 4 BLUP bes lear ubased predcor BLUP of µ +α he oe-wa radom 4 effecs ANOVA model m α GLS GLS esmaor of µ he oe-wa radom effecs ANOVA model 4 e BLUP BLUP resdual predcor of ε 4 w geerc radom varable o be predced 4 w BLUP BLUP predcor of w 4 a BLUP BLUP predcor of α he oe-wa error compoes model 4 β j level depede varable a hree-level model 5 γ level 3 depede varable a hree-level model 5 χ q ch square radom varable wh q degrees of freedom 5 A pcal depede varable a k-level model 5... k k Idex se. The se of all dces... k such ha... k s observed. 5 k A pcal eleme of k of he form... k. 5 k-s Idex se. The se of all dces... k s such ha... k s jk s... j s 5 + k observed for some { jk-s +... jk }. k-s A pcal eleme of k-s. 5

Appedces / A- Smbol Descrpo Chaper defed g Z Level g covaraes marces he hgh order mullevel model aalogous 5 k+ g o Chaper ad 3 Z ad X marces. g X k+ g β g kg β g g β k+ g ε g k+ g Level g parameer marces he hgh order mullevel model aalogous o Chaper ad 3 α ad β. Level g respose ad dsurbace erms he hgh order mullevel model aalogous o Chaper ad 3 ad ε. Lε x A lear projeco of ε o x. 6 f. T A geerc jo probabl des or mass fuco { T x x T }. 6 x.. x T. θ ψ Vecors of parameers for f.. 6 w a se of predeermed varables. 6 W a marx of srumeal varables. 6 P W a projeco marx P W W W W - W. 6 X* a colleco of all observed explaaor varables X* {X Z X 6 Z }. o o o s a q+k vecor of observed effecs o z x 6 o o o T. frs dfferece operaor for example - - 6 K a T T upper ragular marx such ha K 0. 6 b IV srumeal varable esmaor of β 6 Σ IV varace-covarace marx used o compue he varace of b IV 6 Σ E W K ε ε K W. IV Y X Γ resposes explaaor varables ad parameers he mulvarae 6 regresso model Σ varace of he respose he mulvarae regresso model. 6 G OLS ordar leas squares esmaor of Γ he mulvarae regresso model 6 B Γ regresso coeffces for edogeous ad exogeous regressors 6 respecvel he smulaeous equaos model. Π reduced form regresso coeffces he smulaeous equaos model 6 Π I-B - Γ. η reduced form dsurbace erm he smulaeous equaos model η 6 I-B - ε Ω varace of he reduced form dsurbace erm he smulaeous equaos model. 6 5 5

A- / Appedces Smbol Descrpo Chaper defed τ x Λ x regresso coeffces he x-measureme equao of smulaeous 6 equaos wh lae varables. ξ δ lae explaaor varable ad dsurbace erm respecvel he x- 6 measureme equao of smulaeous equaos wh lae varables. µ ξ expeced value of ξ E ξ µ ξ. 6 Φ Θ δ varaces of ξ ad δ respecvel. Var ξ Φ ad Var δ Θ δ. 6 τ Λ regresso coeffces he -measureme equao of smulaeous 6 equaos wh lae varables. η ε lae explaaor varable ad dsurbace erm respecvel he - 6 measureme equao of smulaeous equaos wh lae varables. Θ ε varaces ε Var ε Θ ε. 6 τ η B Γ regresso coeffces he srucural equao of smulaeous equaos 6 wh lae varables. φ. Sadard ormal probabl des fuco 7 Φ. Sadard ormal probabl dsrbuo fuco 7 o o o s a q+k vecor of observed effecs o z x 7 o o o T. U a T g marx of uobserved depede varables 7 γ a g vecor of parameers correspodg o U 7 η a T..d. ose vecor 7 G a T g marx of augmeed depede varables 7 M a T T marx used o specf he avalabl of observaos 7 r j r r j s a dcaor of he jh observao a dcaes he observao s 7 avalable r r r T r r T Y Y s he vecor of all poeall observed resposes Y T T 7

Appedces / A-3 Smbol Descrpo Chaper defed R AR ρ Correlao marx correspodg o a AR process 8 R RW ρ Correlao marx correspodg o a geeralzed radom walk process 8 M T T desg marx ha descrbes mssg observaos 8 ρu correlao fuco for spaal daa 8 H spaal varace marx Var ε 8 V H σ λ J + H 8 r umber of me-varg coeffces per me perod 8 λ λ vecors of me-varg coeffces λ λ λ r λ λ λ T 8 z αj z α Explaaor varables assocaed wh α ; he are he same as Chaper 8 Z α Z α z j z Z ad Z. z λj z λ Explaaor varables assocaed wh λ ; smlar o z αj z α Z α ad Z α. 8 Z λ Z λ γ parameer assocaed wh lagged depede varable model 8 Φ marx of me-varg parameers for a raso equao 8 T marx of me-varg parameers for a geeralzed raso equao 8 φ... φ p parameers of he ARp process auoregressve of order p 8 Φ marx represeao of φ... φ p 8 ξ p vecor of uobservables Kalma fler algorhm 8 δ vecor of uobservables Kalma fler algorhm δ λ ξ 8 p probabl of equalg oe 9 πz dsrbuo fuco for bar depede varables 9 u ul fuco for he h subjec a me 9 U j V j uobserved ul ad value of he jh choce for he h subjec a me 9 logp log fuco defed as logp l p/-p 9 LRT lkelhood rao es sasc 9 R ms max-scaled coeffce of deermao R 9 p α codoal dsrbuo gve he radom effec α 9 a MLE maxmum lkelhood esmaor of α 9 µ mea of he he h subjec a me µ E 9 µ T vecor of meas µ µ µ T 9 G τ marx of dervaves 9 µ β b EE τ EE esmag equaos esmaors of β ad τ 9

A-4 / Appedces APPENDIX F. Seleced Logudal ad Pael Daa Ses I ma dscples logudal ad pael daa ses ca be readl cosruced usg a radoal pre- ad pos-eve sud; ha s subjecs are observed pror o some codo of eres as well as afer hs codo or eve. I s also commo o use logudal ad pael daa mehods o exame daa ses ha follow subjecs observed a a aggregae level such as a goverme e sae provce or ao for example or frm. Logudal ad pael daa ses are also avalable ha follow dvduals over me. However hese daa ses are geerall expesve o cosruc ad are coduced hrough he sposorshp of a goverme agec. Thus alhough avalable daa provders are geerall boud b aoal laws requrg some form of user agreeme o proec cofdeal formao regardg he subjecs. Because of he wde eres hese daa ses mos daa provders make formao abou he daa ses avalable o he Iere. Despe he expese ad cofdeal requremes ma coures have coduced or are he process of coducg household pael sudes. Soco-demographc ad ecoomc formao s colleced abou a household as well as dvduals wh he household. Iformao ma relae o come wealh educao healh geographc mobl axes ad so forh. To llusrae oe of he oldes ogog aoal paels he US Pael Sud of Icome Damcs PSID collecs 5000 varables. Table F. ces some major eraoal household pael daa ses. Educao s aoher dscple ha has a log hsor of eres logudal mehods. Table F. ces some major educaoal logudal daa ses. Smlarl because of he damc aure of agg ad rereme aalss rel o logudal daa for aswers o mpora socal scece quesos cocered wh hese ssues. Table F.3 ces some major agg ad rereme logudal daa ses. Logudal ad pael daa mehods are used ma scefc dscples. Table F.4 ces some oher wdel used daa ses. The focus of Table F.4 s o polcal scece hrough eleco surves ad logudal surves of he frm. Table F. Ieraoal Household Pael Sudes Pael Sud ears avalable Sample Descrpo plus web se Ausrala Household Icome ad Labor Frs wave s scheduled o be colleced lae 00. Damcs 00- www.melbouresue.com/hlda Belga Socoecoomc Pael 985- A represeave sample of 647 Belga households www.ufsa.ac.be/~csb/eg/sepab.hm 985 3800 988 ad 3800 99 ha cludes 900 ew households. Brsh Household Pael Surve 99- Aual surve of prvae households Bra. A www.rc.essex.ac.uk/bhps Caada Surve of Labor Icome Damcs 993- www.saca.ca Duch Soco-Ecoomc Pael 984-997 www.osa.kub.l Frech Household Pael 985-900 www.ceps.lu/paco/pacofrpa.hm Germa Socal Ecoomc Pael 984- www.dw.de/soep Hugara Household Pael 99-996 www.ark.hu aoal represeave sample of 5000 households. Approxmael 5000 households or 3000 dvduals. A aoal represeave sample of 5000 households. There were 75 households a he basele creased o 09 he secod wave. Frs wave colleced 984 cluded 59 Wes Germa households cossg of 45 dvduals. Sample coas 059 households.

Appedces / A-5 Table F. Ieraoal Household Pael Sudes - Coued Sud ears avalable Sample Descrpo plus web se Idoesa Faml Lfe Surve 993- I 993 74 households were ervewed. www.rad.org/fls/ifls/ Japaese Pael Surve o Cosumers 994- Naoal represeave sample of 500 wome age 4- www.kakeke.or.jp 34 993; 997 500 wome were added. Korea Labor ad Icome Pael Sud Sample coas 5000 households. 998- www.kl.re.kr/klps Luxembourg Pael Soco-Ecoomque Represeave sample of 0 households ad 60 985- dvduals 985-994. I 994 expaded o 978 www.ceps.lu/psell/pselpres.hm households ad 83 dvduals. Mexca Faml Lfe Surve Wll coa abou 8000 households. Plas are o collec 00- Polsh Household Pael 987-990 www.ceps.lu/paco/pacopopa.hm Russa Logudal Moorg Surve 99- www.cpc.uc.edu/projecs/rlms/home.hml Souh Afrca KwaZulu-Naal Icome Damcs Sud 993-998 www.fpr.cgar.org/hemes/mp7/safkz.hm Swedsh Pael Sud Marke ad Nomarke Acves 984- www.ek.uu.se/facul/klevmark/hus.hm Swss Household Pael 999- www.ue.ch/psm Tawaese Pael Sud of Faml Damcs 999- U.S. Pael Sud of Icome Damcs 968- www.sr.umch.edu/src/psd/dex.hml Sources: Isue for Socal Research Uvers of Mchga www.sr.umch.edu/src/psd/paelsudes.hml Haske-DeNew J. P. 00E. daa a wo pos me 00 ad 004. Four waves avalable of a sample of persos lvg prvae households excludg polce offcers mlar persoel ad members of he omeklaura. Sample coas 700 households. 400 households from 70 rural ad urba commues were surveed 993 ad re-ervewed 998. Sample coas 000 dvduals surveed 984. Frs wave 999 cosss of 5074 households comprsg 7799 dvduals. Frs wave 999 cosss of dvduals aged 36-45. Secod wave 000 cludes dvduals aged 46-65. Bega wh 480 famles wh a oversamplg of poor famles. Aual ervews were coduced over 5000 varables were colleced o roughl 3000 dvduals. Sud ears avalable plus web se Earl Chldhood Logudal Sud 998- www.ces.ed.gov/ecls/ Table F. Youh ad Educao Sample Descrpo Icludes a Kdergare cohor ad a Brh Cohor. The Kdergare cohor cosss of a aoall represeave sample of approxmael 3000 kdergarers from abou 000 kdergare programs. The Brh Cohor cludes a aoall represeave sample of approxmael 5000 chldre bor he caledar ear 000.

A-6 / Appedces Sud ears avalable plus web se Hgh School ad Beod 980-99 www.ces.ed.gov/surves/hsb/ Naoal Educaoal Logudal Surve: 988 988- www.ces.ed.gov/surves/els88/ Naoal Logudal Sud of he Hgh School Class of 97 97-86 www.ces.ed.gov/surves/ls7/ The Naoal Logudal Surve of Youh 997 NLS www.bls.gov/ls/ls97.hm Table F. Youh ad Educao - Coued Sample Descrpo The Hgh School ad Beod surve cluded wo cohors: he 980 seor class ad he 980 sophomore class. Boh cohors were surveed ever wo ears hrough 986 ad he 980 sophomore class was also surveed aga 99. Surve of 4599 8h graders 988. I he frs followup 990 9363 were subsampled due o budgear reasos. Subseque follow-ups were coduced 99 994 ad 000. Ths surve followed he 97 cohor of hgh school seors hrough 986. The orgal sample was draw 97; follow-up surves were coduced 973 974 976 979 ad 986. A aoall represeave sample of approxmael 9000 ouhs who were o 6 ears old 996. Sud ears avalable plus web se Framgham Hear Sud 948- www.hlb.h.gov/abou/framgham/dex.hml Healh ad Rereme Sud HRS hrsole.sr.umch.edu/ Table F.3 Elderl ad Rereme Sample Descrpo I 948 509 me ad wome bewee he ages of 30 ad 6 were recrued o parcpaed hs hear sud. The are moored ever oher ear. I 97 54 of he orgal parcpas' adul chldre ad her spouses were recrued o parcpae smlar examaos. The orgal HRS cohor bor 93-94 ad frs ervewed 99 ages 5-6. The AHEAD cohor bor before 93 ad frs ervewed 993 ages 70 ad above. Spouses were cluded regardless of age. Logudal Rereme Hsor Sud 969-979 Socal Secur Admsrao www.cpsr.umch.edu/dex.hml These cohors were merged he 998 wave ad clude over 000 parcpas. Varables colleced clude come emplome wealh healh codos healh saus healh surace coverage ad so forh. Surve of 53 me ad omarred wome age 58-63 969. Follow-up surves were admsered a wo-ear ervals 97 973 975 977 ad 979.

Appedces / A-7 Sud ears avalable plus web se The Naoal Logudal Surves of Labor Marke Experece NLS www.bls.gov/ls/ Table F.3 Elderl ad Rereme - Coued Sample Descrpo The NLS follows fve dsc labor markes: 500 older me bewee 45 ad 49 966 55 oug me bewee 4 ad 4 966 5083 maure wome bewee 30 ad 44 967 559 oug wome bewee 4 ad 968 ad 686 ouhs bewee 4 ad 4 979 addoal cohors were added 986 ad 997. The ls of varables s he housads wh a emphass o he suppl sde of he labor marke. Table F.4 Oher Logudal ad Pael Sudes Sud ears avalable Sample Descrpo plus web se Naoal Eleco Sudes 956 958 960 A sample of 54 voers who were ervewed Amerca Pael Sud a mos fve mes. www.umch.edu/~es/sudres/es56_60/es56_60.hm Naoal Eleco Sudes 97 974 976 Seres Fle A sample of 4455 voers who were ervewed www.umch.edu/~es/sudres/es7_76/es7_76.hm a mos fve mes. Naoal Eleco Sudes 980 Pael Sud Over 000 voers were ervewed four mes www.umch.edu/~es/sudres/es80pa/es80pa.hm over he course of he 980 presdeal eleco. Naoal Eleco Sudes 990-99 Full Pael Fle Voer opos are raced o follow he forues www.umch.edu/~es/sudres/es90_9/es90_9.hm Cesus Bureau Logudal Research Daabase 980- www.cesus.gov/pub/eco/www/ma0800.hml Medcal Expedure Pael Surve 996- www.meps.ahrq.gov Naoal Assocao of Isurace Commssoers NAIC www.ac.org/dbproducs/ of he Bush presdec. Lks esablshme level daa from several cesuses ad surves of maufacurers ad ca respod o dverse ecoomc research prores. Surves of households medcal care provders as well as busess esablshmes ad govermes o healh care use ad coss. Maas aual ad quarerl daa for more ha 6000 Lfe/Healh Proper/Casual Fraeral Healh ad Tle compaes.

A-8 / Appedces Appedx G. Refereces Bologcal Sceces Logudal Daa Refereces Gbbos R. D. ad D. Hedeker 997. Radom effecs prob ad logsc regresso models for hree-level daa. Bomercs 53 57-537. Grzzle J. E. ad Alle M. D. 969. Aalss of growh ad dose respose curves. Bomercs 5 357-8. Hederso C. R. 953. Esmao of varace compoes. Bomercs 9 6-5. Hederso C. R. 973 Sre evaluao ad geec reds Proceedgs of he Amal Breedg ad Geecs Smposum Hoor of Dr. Ja L. Lush 0-4. Amer. Soc. Amal Sc.-Amer. Dar Sc. Ass. Poulr Sc. Ass. Champag Illos. Hougaard P. 000. Aalss of Mulvarae Survval Daa. Sprger-Verlag New York. Jerch R.I. ad Schlucer M.D. 986. Ubalaced repeaed-measures models wh srucured covarace marces. Bomercs 4 805-80. Joes R. H. ad Boad-Boeg F. 99. Uequall spaced logudal daa wh seral correlao. Bomercs 47 6-75. Keward M. G. ad J. H. Roger 997. Small sample ferece for fxed effecs from resrced maxmum lkelhood. Bomercs 53 983-997. Kle J. P. ad M. L. Moeschberger 997. Survval Aalss: Techques for Cesored ad Trucaed Daa. Sprger-Verlag New York. Lard N. M. 988. Mssg daa logudal sudes. Sascs Medce 7 305-5. Lard N. M. ad Ware J. H. 98. Radom-effecs models for logudal daa. Bomercs 38 963-74. Lag K. -Y. ad McCullagh P. 993. Logudal daa aalss usg geeralzed lear models. Bomerka 73 3-. Lag K.-Y. ad S.L. Zeger 986. Logudal daa aalss usg geeralzed lear models. Bomerka 73 -. Ldsrom M. J. ad Baes D. M. 990. Nolear mxed effecs models for repeaed measures daa. Bomercs 46 673-87. Pala M. ad Yao T.-J. 99. Aalss of logudal daa wh umeasured cofouders. Bomercs 47 355-369. Pala M. Yao T.-J. ad Velu R. 994. Tesg for omed varables ad o-lear regresso models for logudal daa. Sascs Medce 3 9-3. Paerso H. D. ad Thompso R. 97. Recover of er-block formao whe block szes are uequal. Bomerka 58 545-54. Pohoff R.F. ad S.N. Ro 964. A geeralzed mulvarae aalss of varace moel useful especall for growh curve problems. Bomerka 5 33-36. Prece R. L. 988. Correlaed bar regresso wh covaraes specfc o each bar observao. Bomercs 44 033-48. Prece R. L. ad Zhao L. P. 99. Esmag equaos for parameers meas ad covaraces of mulvarae dscree ad couous resposes. Bomercs 47 85-39. Rao C. R. 965. The heor of leas squares whe he parameers are sochasc ad s applcao o he aalss of growh curves. Bomerka 5 447-58. Rao C. R. 987. Predco of fuure observaos growh curve models. Sascal Scece 434-47. Sraell R. Lard N. ad Ware J. H. 984. Radom effecs models for seral observaos wh bar resposes. Bomercs 40 96-7. Sullva L.M. Dukes K.A. ad Losa E. 999. A roduco o herarchcal lear modellg Sascs Medce 8 855-888.

Appedces / A-9 Wshar J. 938. Growh-rae deermaos uro sudes wh he baco pg ad her aalss. Bomerka 30 6-8. Wrgh S. 98. O he aure of sze facors. Geecs 3 367-374. Zeger S. L. Lag K.-Y. ad Alber P. S. 988. Models for logudal daa: a geeralzed esmag equao approach. Bomercs 44 049-60. Zeger. S. L. ad Lag K.-Y. 986. Logudal daa aalss for dscree ad couous oucomes. Bomercs 4-30. Ecoomercs Pael Daa Refereces Amema T. 985. Advaced Ecoomercs. Harvard Uvers Press Cambrdge MA. Aderso T. W. ad C. Hsao 98. Formulao ad esmao of damc models usg pael daa. Joural of Ecoomercs 8 47-8. Adrews D.W. K. 00. Tesg whe a parameer s o he boudar of he maaed hpohess. Ecoomerca 69 683-734. Arellao M. 993. O he esg of correlaed effecs wh pael daa. Joural of Ecoomercs 59 87-97. Arellao M. 003. Pael Daa Ecoomercs. Oxford Uvers Press Oxford. Arellao M. ad O. Bover 995. Aoher look a he srumeal-varable esmao of error compoes models. Joural of Ecoomercs 68 9-5. Arellao M. ad B. Hooré 00. Pael Daa models: Some rece developmes. I Hadbook of Ecoomerc volume 5 ed. J. J. Heckma ad E. Leamer pp. 33-396. Ashefeler O. 978. Esmag he effec of rag programs o eargs wh logudal daa. Revew of Ecoomcs ad Sascs 60 47-57. Aver R. B. 977. Error compoes ad seemgl urelaed regressos. Ecoomerca 45 99-09. Balesra P. ad Nerlove M. 966. Poolg cross-seco ad me-seres daa he esmao of a damc model: he demad for aural gas. Ecoomerca 34 585-6. Balag B.H. 980. O seemgl urelaed regressos ad error compoes. Ecoomerca 48 547-55. Balag B.H. 00. Ecoomerc Aalss of Pael Daa Secod Edo. Wle New York. Balag B.H. ad Y.J. Chag 994. Icomplee paels: A comparave sud of alerave esmaors for he ubalaced oe-wa error compoe regresso model Joural of Ecoomercs 6 No. pp. 67. Balag B.H. ad Q. L 990. A Lagrage mulpler es for he error compoes model wh complee paels. Ecoomerc Revews 9 03-07. Balag B.H. ad Q. L 99. Predco he oe-wa error compoe model wh seral correlao. Joural of Forecasg 56-567. Becker R. ad Hederso V. 000. Effecs of ar qual regulaos o pollug dusres. Joural of Polcal Ecoom 08 379-4. Bludell R. ad S. Bod 998. Ial codos ad mome resrcos damc pael daa models. Joural of Ecoomercs 87 5-43. Bhargava A. Fraz L. ad W. Naredraaha 98. Seral correlao ad he fxed effecs model. Revew of Ecoomc Sudes 49 533-549. Boud J. D. A. Jaeger ad R. M. Baker 995. Problems wh srumeal varables esmao whe he correlao bewee he srumes ad edogeous explaaor varables s weak. Joural of he Amerca Sascal Assocao 90 443-450. Breusch T. S. ad Paga A. R. 980. The Lagrage mulpler es ad s applcaos o model specfcao ecoomercs. Revew of Ecoomc Sudes 47 39-53. Breusch T. S. Mzo G. E. ad Schmd P. 989. Effce esmao usg pael daa. Ecoomerca 57 695-700.

A-30 / Appedces Camero A.C. ad P.K. Trved 998. Regresso Aalss of Cou Daa. Cambrdge Uvers Press Cambrdge U.K. Card D. 995. Usg geographc varao college proxm o esmae he reur o schoolg. I Aspecs of Labour Marke Behavor: Essas Hoour of Joh Vaderkamp ed. L. N. Chrsophdes E. K. Gra ad R. Swdsk. Uvers of Toroo Press 0-. Chamberla G. 980. Aalss of covarace wh qualave daa. Revew of Ecoomc Sudes 47 5-38. Chamberla G. 98. Mulvarae regresso models for pael daa. Joural of Ecoomercs 8 5-46. Chamberla G. 984. Pael daa. I Hadbook of Ecoomercs Eds. Z. Grlches ad M. Irllgaor 47-38. Norh-Hollad Amserdam. Chamberla G. 99. Comme: Sequeal mome resrcos pael daa. Joural of Busess ad Ecoomc Sascs 0 0-6. Chb S. E. Greeberg ad R. Wkelma 998. Poseror smulao ad Baes facor pael cou daa models. Joural of Ecoomercs 86 33-54. Davdso R. ad MacKo J.G. 990. Specfcao ess based o arfcal regressos. Joural of he Amerca Sascal Assocao 85 0-7. Egle R. F. D. F. Hedr ad J. F. Rchard 983. Exogee. Ecoomerca 5 77-304. Feberg S. E. M. P. Keae ad M. F. Bogao 998. Trade lberalzao ad delocalzao : ew evdece from frm-level pael daa. Caada Joural of Ecoomcs 3 749-777. Frees E. W. 995. Assessg cross-secoal correlaos logudal daa. Joural of Ecoomercs 69 393-44. Glaeser E. L. ad D. C. Maré 00. Ces ad sklls. Joural of Labor Ecoomcs 9 36-34. Goldberger A. S. 96. Bes lear ubased predco he geeralzed lear regresso model. Joural of he Amerca Sascal Assocao 57 369-75. Goldberger A. S. 97. Srucural equao mehods he socal sceces. Ecoomerca 40 979-00. Goldberger A. S. 99. A Course Ecoomercs. Harvard Uvers Press Cambrdge MA. Goureroux C. Mofor A. ad Trogo A. 984. Pseudo-maxmum lkelhood mehods: heor. Ecoomerca 5 68-700. Greee W. H. 00. Ecoomerc Aalss Ffh Edo. Prece-Hall NJ. Haavelmo T. 944. The probabl approach o ecoomercs. Suppleme o Ecoomerca. Haske-DeNew J. P. 00. A hchhker s gude o he world s household pael daa ses. The Ausrala Ecoomc Revew 343 356-366. Haash F. 000. Ecoomercs. Prceo Uvers Press Prceo New Jerse. Hausma J. A. 978. Specfcao ess ecoomercs. Ecoomerca 46 5-7. Hausma J. A. B. H. Hall ad Z. Grlches 984. Ecoomerc models for cou daa wh a applcao o he paes-r&d relaoshp. Ecoomerca 5 909-938. Hausma J. A. ad Talor W. E. 98. Pael daa ad uobservable dvdual effecs. Ecoomerca 49 377-98. Hausma J. A. ad Wse D. 979. Aro bas expermeal ad pael daa: he Gar come maeace experme. Ecoomerca 47 455-73. Heckma J. J. 976. The commo srucure of sascal models of rucao sample seleco ad lmed depede varables ad a smple esmaor for such models. A. Eco. Soc. Meas. 5 475-49. Heckma J. J. 98a. Sascal models for dscree pael daa. I Srucural Aalss Of Dscree Daa Wh Ecoomerc Applcaos Eds. C. F. Mask ad D. McFadde 4-78. MIT Press Cambrdge.

Appedces / A-3 Heckma J. J. ad Sger B. 985. Logudal Aalss of Labor Marke Daa. Cambrdge Uvers Press Cambrdge. Hoch I. 96. Esmao of produco fuco parameers combg me-seres ad crossseco daa. Ecoomerca 30 34-53. Holl A. 98. A remark o Hausma s specfcao es. Ecoomerca 50 749-59. Holz-Eak D. Newe W. ad Rose H. S. 988. Esmag vecor auoregressos wh pael daa. Ecoomerca 56 37-95. Hsao C. 986. Aalss of Pael Daa. Cambrdge Uvers Press Cambrdge. Johso P.R. 960. Lad subsues ad chages cor elds. Joural of Farm Ecoom. 4 94-306. Judge G. G. Grffhs W. E. Hll R. C. Lukepohl H. ad Lee T. C. 985. The Theor ad Pracce Of Ecoomercs. Wle New York. Keae M. P. ad D. E. Rukle 99. O he esmao of pael daa models wh seral correlao whe srumes are o srcl exogeous. Joural of Busess ad Ecoomc Sascs 0-9. Kefer N. M. 980. Esmao of fxed effecs models for me seres of cross secos wh arbrar eremporal covarace. Joural of Ecoomercs 4 95-0. Kuh E. 959. The vald of cross-secoall esmaed behavor equao me seres applcao. Ecoomerca 7 97-4. Lacaser T. 990. The Ecoomerc Aalss of Traso Daa. Cambrdge Uvers Press New York. Maddala G. S. 97. The use of varace compoes models poolg cross seco ad me seres daa. Ecoomerca 39 34-58. Maddala G. S. ed. 993. The Ecoomercs of Pael Daa. Volumes I ad II Edward Elgar Publshg Cheleham. Mask C. F. 987. Semparamerc aalss of radom effecs lear models from bar pael daa. Ecoomerca 55 357-6. Mask C. 99. Comme: The mpac of socologcal mehodolog o sascal mehodolog b C. C. Clogg. Sascal Scece 7 0-03. Máás L. ad Sevesre P. eds. 996. The Ecoomercs of Pael Daa: Hadbook of Theor ad Applcaos. Kluwer Academc Publshers Dordrech. McFadde D. 974. Codoal log aalss of qualave choce behavor. I Froers of Ecoomercs 05-4 ed. P. Zarembka. Academc Press New York. McFadde D. 978. Modelg he choce of resdeal locao. I Spaal Ieraco Theor ad Plag Models 75-96 ed. A Karlqvs e al. Norh-Hollad Publshg Amserdam. McFadde D. 98. Ecoomerc models of probablsc choce. I Srucural Aalss of Dscree Daa wh Ecoomerc Applcaos 98-7 ed. C. Mask ad D. McFadde MIT Press. Mudlak Y. 96. Emprcal produco fuco free of maageme bas. Joural of Farm Ecoomcs 43 44-56. Mudlak Y. 978a. O he poolg of me seres ad cross-seco daa. Ecoomerca 46 69-85. Mudlak Y. 978b. Models wh varable coeffces: egrao ad exesos. Aales de L Isee 30-3 483-509. Nerlove M. 967. Expermeal evdece o he esmao of damc ecoomc relaos from a me-seres of cross-secos. Ecoomc Sudes Quarerl 8 4-74. Nerlove M. 97a. Furher evdece o he esmao of damc ecoomc relaos from a me-seres of cross-secos. Ecoomerca 39 359-8. Nerlove M. 97b. A oe o error compoes models. Ecoomerca 39 383-96. Nckell S. 98. Bases damc models wh fxed effecs. Ecoomerca 49 399-46.

A-3 / Appedces Newe W. 985. Geeralzed mehod of momes specfcao esg. Joural of Ecoomercs 9 9-56. Nema J. ad Sco E. L. 948. Cosse esmaes based o parall cosse observaos. Ecoomerca 6-3. Polachek S.W. ad M. Km 994. Pael esmaes of he geder eargs gap. Joural of Ecoomercs 6 3-4. Schmd P. 983. A oe o a fxed effec model wh arbrar erpersoal covarace. Joural of Ecoomercs 39-93. Solo G. S. 989. The value of pael daa ecoomc research. I Pael surves Eds. D. Kasprzk G. J. Duca G. Kalo ad M. P. Sgh 486-96. Joh Wle New York. Swam P. A. V. B. 970. Effce ferece a radom coeffce regresso model Ecoomerca 38: 3-33. Swam P. A. V. B. 97. Sascal Iferece Radom Coeffce Regresso Models. Sprger-Verlag New York. Thel H. ad A. Goldberger 96. O pure ad mxed esmao ecoomcs. Ieraoal Ecoomc Revew 65-78. Vallea R. G. 999. Declg job secur. Joural of Labor Ecoomcs 7 S70-S97. Wallace T. ad A. Hussa 969. The use of error compoes combg cross seco wh me seres daa. Ecoomerca 37 55-7. Whe H. 980. A heeroskedasc-cosse covarace marx esmaor ad a drec es for heeroskedasc. Ecoomerca 48 87-38. Whe H. 98. Maxmum lkelhood esmao of msspecfed models. Ecoomerca 50-5. Whe H. 984. Asmpoc Theor for Ecoomercas. Academc Press Orlado Florda. Wooldrdge J. M. 00. Ecoomerc Aalss of Cross Seco ad Pael Daa. The MIT Press Cambrdge Massachuses. Zeller A. 96. A effce mehod of esmag seemgl urelaed regresso ad ess for aggregao bas. Joural of he Amerca Sascal Assocao 57 348-68. Educaoal Scece ad Pscholog Refereces Bales P. B. ad J. R. Nesselroade 979. Hsor ad Raoal of Logudal Research. Chaper of Logudal Research he Sud of Behavor ad Developme eded b Bales ad Nesselroade Academc Press New York. Bolle K.A. 989. Srucural Equaos wh Lae Varables. New York: Wle. de Leeuw J. ad I. G. G. Kref 00. Sofware for mullevel aalss. I A. H. Lelad & H. Goldse edors Mullevel Modelg of Healh Sascs pp. 87-04. Wle. Duca O. D. 969. Some lear models for wo wave wo varable pael aalss. Pschologcal Bulle 7 77-8. Duca T. E. S. C. Duca L. A. Srakar F. L ad A. Alper 999. A Iroduco o Lae Varable Growh Curve Modelg. Lawrece Erlbaum Mahwah NJ. Goldse H. 00. Mullevel Sascal Models Thrd Edo. Edward Arold Lodo. Guo S. ad D. Husse 999. Aalzg logudal rag daa: A hree-level herarchcal lear model. Socal Work Research 34 58-68. Kref I. ad J. de Leeuw 998. Iroducg Mullevel Modelg. Sage New York. Lee V. 000. Usg herarchcal lear modelg o sud socal coexs: The case of school effecs. Educaoal Pschologs 35 5-4. Lee V ad J. B. Smh 997. Hgh school sze: Whch works bes ad for whom? Educaoal Evaluao ad Polc Aalss 93 05-7. Logford N.T. 993. Radom Coeffce Models. Oxford: Claredo Press. Rasch G. 96. O geeral laws ad he meag of measureme pscholog. Proc. Fourh Berkele Smposum 4 434-458.

Appedces / A-33 Raudebush S. W. ad A. S. Brk 00. Herarchcal lear models: Applcaos ad daa aalss mehods Secod Edo. Sage Publcaos Lodo. Rub D. R. 974. Esmag causal effecs of reames radomzed ad oradomzed sudes. Joural of Educaoal Pscholog 66 688-70. Sger J. D. 998. Usg SAS PROC MIXED o f mullevel models herarchcal models ad dvdual growh models. Joural of Educaoal ad Behavoral Sascs 7 33-355. Sger J. D. ad J. B. Wlle 003. Appled Logudal Daa Aalss: Modelg Chage ad Eve Occurrece. Oxford Uvers Press Oxford. Too T. J. 000. A Prmer Logudal Daa Aalss. Sage Publcaos Lodo. Webb N.L. W. H. Clue D. Bol A. Gamora R. H. Meer E. Oshoff ad C. Thor. 00. Models for aalss of NSF s ssemc ave programs The mpac of he urba ssem aves o sude acheveme Texas 994-000. Wscos Ceer for Educao Research Techcal Repor Jul. Avalable a hp://facsaff.wcer.wsc.edu/ormw/echcal_repors.hm. Wlle J. B. & Saer A. G. 994. Usg covarace srucure aalss o deec correlaes ad predcors of dvdual chage over me. Pschologcal Bulle 6 363-38. Oher Socal Scece Refereces Ashle T. Y. Lu ad S. Chag 999. Esmag e loer reveues for saes. Alac Ecoomcs Joural 7 70-78. Bale A. 950. Credbl procedures: LaPlace s geeralzao of Baes rule ad he combao of collaeral kowledge wh observed daa. Proceedgs of he Casual Acuaral Soce 37 7-3. Beck N. ad J. N. Kaz 995. Wha o do ad o o do wh me-seres cross-seco daa. Amerca Polcal Scece Revew 89 634-647. Beesock M. Dckso G. ad Khajura S. 988. The relaoshp bewee proper-labl surace premums ad come: A eraoal aalss. The Joural of Rsk ad Isurace 55 59-7. Brader T. ad J. A. Tucker 00. The emergece of mass parsashp Russa 993-996. Amerca Joural of Polcal Scece 45 69-83. Bühlma H. 967. Experece rag ad credbl. ASTIN Bulle 4: 99-07. Bühlma H. ad E. Sraub 970. Glaubwürdgke für Schadesäze. Meluge der Veregug Schwezerscher Verscherugsmahemaker 70: -33. Carroll A. M. 993. A emprcal vesgao of he srucure ad performace of he prvae workers compesao marke. The Joural of Rsk ad Isurace 60 85-. Daeburg D. R. R. Kaas ad M. J. Goovaers 996. Praccal Acuaral Credbl Models. Isue of Acuaral Scece ad Ecoomercs Uvers of Amserdam Amserdam The Neherlads. Delma T. E. 989. Pooled Cross-Secoal ad Tme Seres Daa Aalss. Marcel Dekker New York. Frees E. W. 99. Forecasg sae-o-sae mgrao raes. Joural of Busess ad Ecoomc Sascs 0 53-67. Frees E. W. 993. Shor-erm forecasg of eral mgrao. Evrome ad Plag A 5 593-606. Frees E. W. ad T. W. Mller 003. Sales forecasg usg logudal daa models. To appear he Ieraoal Joural of Forecasg. Frees E. W. Youg V. ad Y. Luo 999. A logudal daa aalss erpreao of credbl models. Isurace: Mahemacs ad Ecoomcs 4 9-47. Frees E. W. Youg V. ad Y. Luo 00. Case sudes usg pael daa models. Norh Amerca Acuaral Joural 4 No. 4 4-4.

A-34 / Appedces Frschma P. J. ad Frees E. W. 999. Demad for servces: Deermas of ax preparao fees. Joural of he Amerca Taxao Assocao Suppleme -3. Grabowsk H. Vscus W. K. Evas W.N. 989. Prce ad avalabl radeoffs of auomoble surace regulao. The Joural of Rsk ad Isurace 56 75-99. Gree R. K. ad S. Malpezz 003. A Prmer o U.S. Housg Markes ad Polc. The Urba Isue Press Washgo D.C. Haberma S. ad E. Pacco 999. Acuaral Models for Dsabl Isurace. Chapma ad Hall/CRC Boca Rao. Hachemeser C. A. 975. Credbl for regresso models wh applcaos o red. I Credbl: Theor ad Applcaos ed. P. M. Kah Academc Press New York 9-63. Hckma J.C. ad L. Heacox 999. Credbl heor: The corersoe of acuaral scece. Norh Amerca Acuaral Joural 3 No. - 8. Ja D. C. N. J. Vlcassm ad P. K. Chagua 994. A radom-coeffces log brad choce model appled o pael daa. Joural of Busess ad Ecoomc Sascs 37-38. Jewell W. S. 975. The use of collaeral daa credbl heor: A herarchcal model. Gorale dell Isuo Ialao degl Auar 38: -6. Also Research Memoradum 75-4 of he Ieraoal Isue for Appled Ssems Aalss Laxeburg Ausra. Kasprzk D. Duca G. Kalo G. ad Sgh M. P. edors 989 Pael Surves Wle New York. Km Y.-D. D. R. Aderso T. L. Amburge ad J. C. Hckma 995. The use of eve hsor aalss o exame surace solveces. Joural of Rsk ad Isurace 6 94-0. Klugma S. A. 99. Baesa Sascs Acuaral Scece: Wh Emphass o Credbl. Kluwer Academc Publshers Boso. Klugma S. H. Pajer ad G. Wllmo 998 Loss Models: From Daa o Decsos Wle New York. Kug Y. 996. Pael Daa wh Seral Correlao. Upublshed Ph.D. hess Uvers of Wscos Madso. Lazarsfeld P.F. ad M. Fske 938. The pael as a ew ool for measurg opo. Publc Opo Quarerl 596-6. Ledoler J. S. Klugma ad C.-S. Lee 99. Credbl models wh me-varg red compoes. ASTIN Bulle : 73-9. Lee H. D. 994. A Emprcal Sud of he Effecs of Tor Reforms o he Rae of Tor Flgs. upublshed Ph.D. hess Uvers of Wscos. Lee H. D. M. J. Browe ad J. T. Schm 994. How does jo ad several or reform affec he rae of or flgs? Evdece from he sae cours. Joural of Rsk ad Isurace 6 595-36. Ler J. 965. The valuao of rsk asses ad he seleco of rsk vesmes sock porfolos ad capal budges. Revew of Ecoomcs ad Sascs 3-37. Luo Y. V.R. Youg ad E.W. Frees 00. Credbl raemakg usg collaeral formao. Submed for publcao. Malpezz S. 996. Housg prces exerales ad regulao U.S. meropola areas. Joural of Housg Research 7 09-4. Markowz H. 95. Porfolo seleco. Joural of Face 7 77 9. Mowbra A.H. 94. How exesve a paroll exposure s ecessar o gve a depedable pure premum. Proceedgs of he Casual Acuaral Soce 4-30. Norberg R. 980. Emprcal Baes credbl. Scadava Acuaral Joural 980: 77-94. Norberg R. 986. Herarchcal credbl: Aalss of a radom effec lear model wh esed classfcao. Scadava Acuaral Joural 04-. Pque J. 997. Allowace for cos of clams bous-malus ssems. ASTIN Bulle 7 33-57.

Appedces / A-35 Rzzo J. A. 989. The mpac of medcal malpracce surace rae regulao. The Joural of Rsk ad Isurace 56 48-500. Sharpe W. 964. Capal asse prces: A heor of marke equlbrum uder rsk. Joural of Face 45-44. Shumwa T. 00. Forecasg bakrupc more accurael: A smple hazard model. Joural of Busess 74 0-4. Smso J. 985. Regresso space ad me: A sascal essa. Amerca Joural of Polcal Scece 9 94-47. Sohs M. H. ad D.C. Mauer 996. The deermas of corporae deb maur srucure. Joural of Busess 69 o. 3 79-3. Talor G. C. 977. Absrac credbl. Scadava Acuaral Joural 49-68. Thes C.F. ad Surrock T. 988. The peso-augmeed balace shee. The Joural of Rsk ad Isurace 55 467-480. Veer G. 996. Credbl. I Foudaos of Casual Acuaral Scece hrd edo edor I.K. Bass e al. Casual Acuaral Soce Arlgo Vrga. Vllas-Boas J. M. ad R. S. Wer 999. Edogee brad choce models. Maageme Scece 45 34-338. Wess M. 985. A mulvarae aalss of loss reservg esmaes proper-labl surers. The Joural of Rsk ad Isurace 5 99-. Zhou X. 000. Ecoomc rasformao ad come equal urba Cha: Evdece from pael daa. Amerca Joural of Socolog 05 35-74. Zor C. J. W. 00. Geeralzed esmag equao models for correlaed daa: A revew wh applcaos. Amerca Joural of Polcal Scece 45 470-490. Sascal Logudal Daa Refereces Baerjee M. ad Frees E. W. 997. Ifluece dagoscs for lear logudal models. Joural of he Amerca Sascal Assocao 9 999-005. Che Z. ad L. Kuo 00. A oe o he esmao of mulomal log model wh radom effecs. Amerca Sasca 55 89-95. Ch E. M. ad Resel G. C. 987. Models for logudal daa wh radom effecs ad AR errors. Joural of he Amerca Sascal Assocao 84 45-59. Coawa. M. R. 989. Aalss of repeaed caegorcal measuremes wh codoal lkelhood mehods. Joural of he Amerca Sascal Assocao 84 53-6. Corbel R. R. ad S. R. Searle 976a. Resrced maxmum lkelhood REML esmao of varace compoes he mxed model Techomercs 8: 3-38. Corbel R. R. ad S. R. Searle 976b A comparso of varace compoes esmaors Bomercs 3: 779-79. Crowder M. J. ad Had D. J. 990 Aalss of Repeaed Measures Chapma-Hall New York. Davda M. ad D. M. Gla 995. Nolear Models for Repeaed Measureme Daa. Chapma-Hall Lodo. Dggle P.J. P. Heagar K.-Y. Lag ad S. L. Zeger 00. Aalss of Logudal Daa. Secod Edo. Oxford Uvers Press. Fahrmer L. ad G. Tuz 00. Mulvarae Sascal Modellg Based o Geeralzed Lear Models Secod Edo. Sprger-Verlag New York. Fzmaurce G. M. Lard N. M. ad Rosk A. G. 993. Regresso models for dscree logudal resposes wh Dscusso. Sascal Scece 8 84-309. Frees E. W. 00. Omed varables pael daa models. Caada Joural of Sascs 9 4-3. Frees E. W. ad C. J 004. Emprcal sadard errors for logudal daa mxed lear models. Compuaoal Sascs o appear Ocober.

A-36 / Appedces Ghosh M. ad J. N. K. Rao 994. Small area esmao: a apprasal. Sascal Scece 9 55-93. Had D. J. ad Crowder M. J. 996. Praccal Logudal Daa Aalss. Chapma-Hall New York. Harve A. C. 989. Forecasg Srucural Tme Seres Models ad he Kalma Fler. Cambrdge Uvers Press Cambrdge. Harvlle D. A. 974 Baesa ferece for varace compoes usg ol error corass Bomerka 6 383-385. Harvlle D. 976. Exeso of he Gauss-Markov heorem o clude he esmao of radom effecs. Aals of Sascs. 384-395. Harvlle D. 977. Maxmum lkelhood esmao of varace compoes ad relaed problems. Joural of he Amerca Sascal Assocao 7 30-40. Harvlle D. ad J. R. Jeske 99. Mea square error of esmao or predco uder a geeral lear model. Joural of he Amerca Sascal Assocao 87 74-73. Herbach 959. Properes of model II pe aalss of varace ess A: opmum aure of he F-es for model II he balaced case. Aals of Mahemacal Sascs 30 939-959. Hldreh C. ad C. Houck 968. Some esmaors for a lear model wh radom coeffces. Joural of he Amerca Sascal Assocao 63 584-595. Joes R. H. 993. Logudal Daa wh Seral Correlao: A Sae-Space Approach. Chapma ad Hall Lodo. Jöreskog K. G. ad A. S. Goldberger 975. esmao of a model wh mulple dcaors ad mulple causes of a sgle lae varable. Joural of he Amerca Sascal Assocao 70 63-639. Kackar R. N. ad D. Harvlle 984. Approxmaos for sadard errors of esmaors of fxed ad radom effecs mxed lear models. Joural of he Amerca Sascal Assocao 79 853-86. L X. ad N. E. Breslow 996. Bas correco o geeralzed lear mxed models wh mulple compoes of dsperso. Joural of he Amerca Sascal Assocao 9 007-06. Ldsrom M. J. ad Baes D. M. 989. Newo-Raphso ad EM algorhms for lear mxed-effecs models for repeaed measures daa. Joural of he Amerca Sascal Assocao 84 04-. Lell R. C G. A. Mllke W. W. Sroup ad R. D. Wolfger 996. SAS Ssem for Mxed Models. SAS Isue Car Norh Carola. Parks R. 967. Effce esmao of a ssem of regresso equaos whe dsurbaces are boh serall ad coemporaeousl correlaed. Joural of he Amerca Sascal Assocao 6 500-509. Paerso H. D. ad R. Thompso 97. Recover of er-block formao whe block szes are uequal. Bomerka 58. 545-554. Phero J.C. ad D.M. Baes 000. Mxed-effecs Models S ad S-plus. Sprger New York. Rao C.R. 965. The heor of leas squares whe he parameers are sochasc ad s applcao o he aalss of growh curves. Bomerka 5 447-458. Rao C. R. 970 Esmao of varace ad covarace compoes lear models Joural of he Amerca Sascal Assocao 67: -5. Resel G. C. 98. Mulvarae repeaed-measureme or growh curve models wh mulvarae radom-effecs covarace srucure. Joural of he Amerca Sascal Assocao 77 90-95. Resel G. C. 984. Esmao ad predco a mulvarae radom effecs geeralzed lear model. Joural of he Amerca Sascal Assocao 79 406-4.

Appedces / A-37 Resel G. C. 985. Mea squared error properes of emprcal Baes esmaors a mulvarae radom effecs geeral lear model. Joural of he Amerca Sascal Assocao 79 406-4. Robso G. K. 99. The esmao of radom effecs. Sascal Scece 6 5-5. Searle S. R. G. Casella ad C. E. McCulloch 99. Varace Compoes. Joh Wle ad Sos New York. Self S. G. ad K.Y. Lag 987. Asmpoc properes of maxmum lkelhood esmaors ad lkelhood rao ess uder osadard codos. Joural of he Amerca Sascal Assocao 8 605-60. Sram D. O. ad J. W. Lee 994. Varace compoes esg he logudal mxed effecs model. Bomercs 50 7-77. Swallow W. H. ad S. R. Searle 978. Mmum varace quadrac ubased esmao MIVQUE of varace compoes. Techomercs 0 65-7. Tsmkas J. V. ad Ledoler J. 994. REML ad bes lear ubased predco sae space models. Commucaos Sascs: Theor ad Mehods 3 53-68. Tsmkas J. V. ad Ledoler J. 997. Mxed model represeaos of sae space models: New smoohg resuls ad her applcao o REML esmao. Sasca Sca 7 973-99. Tsmkas J. V. ad Ledoler J. 998. Aalss of mul-u varace compoes models wh sae space profles. Aals of he Isue of Sas. Mah 50 No. 47-64. Verbeke G. ad G. Molebergs 000. Lear Mxed Models for Logudal Daa. Sprger- Verlag New York. Voesh E. F. ad V.M. Chchll 997. Lear ad Nolear Models for he Aalss of Repeaed Measuremes. Marcel Dekker New York. Ware J. H. 985. Lear models for he aalss of logudal sudes. The Amerca Sasca 39 95-0. Wolfger R. R. Tobas ad J. Sall 994 Compug Gaussa lkelhoods ad her dervaves for geeral lear mxed models SIAM Joural of Scefc Compug 5 6: 94-30. Zeger S. L. ad M. R. Karm. Geeralzed lear models wh radom effecs: A Gbbs samplg approach. Joural of he Amerca Sascal Assocao 86 79-86. Geeral Sascs Refereces Agres A. 00. Caegorcal Daa Aalss. Joh Wle New York. Aderse E. B. 970. Asmpoc properes of codoal maxmum-lkelhood esmaors. Joural of he Roal Sascal Soce B 3 83-30. Agrs J. D G W. Imbes ad D. B. Rub 996. Idefcao of causal effecs usg srumeal varables. Joural of he Amerca Sascal Assocao 9 444-47. Aderso T. W. 958. A Iroduco o Mulvarae Sascal Aalss. Wle New York. Becker R. A. W. S. Clevelad ad M.-J. Shu 996. The vsual desg ad corol of rells graphcs dsplas. Joural of Compuaoal ad Graphcal Sascs 5 3-56. Bckel P. J. ad Doksum K. A. 977. Mahemacal Sascs. Holde-Da Sa Fracsco. Box G.E.P. 979. Robusess he sraeg of scefc model buldg. I Robusess Sascs ed. R. Lauer ad G. Wlderso 0-36. Academc Press New York. Box G. E. P. G. M. Jeks ad Resel G. 994 Tme Seres Aalss Prece Hall Eglewood Clffs NJ. Carroll R. J. ad Rupper D. 988 Trasformao ad Weghg Regresso Chapma-Hall. Clevelad W.S. 993. Vsualzg Daa. Summ N.J.: Hobar Press. Cogdo P. 003. Appled Baesa Modellg. Wle New York. Cook D. ad Wesberg S. 98. Resduals ad Ifluece Regresso. Chapma ad Hall Lodo.

A-38 / Appedces Draper N. ad Smh H. 98. Appled Regresso Aalss Secod Edo. Wle New York. Fredma M. 937. The use of raks o avod he assumpo of ormal mplc he aalss of varace. Joural of he Amerca Sascal Assocao 89 57-55. Fuller W. A. ad Baese G. E. 973. Trasformaos for esmao of lear models wh esed error srucure. Joural of he Amerca Sascal Assocao 68 66-3. Fuller W. A. ad Baese G. E. 974. Esmao of lear models wh cross-error srucure. Joural of Ecoomercs 67-78. Gelma A. J. B. Carl H. S. Ser ad D. B. Rub 004. Baesa Daa Aalss Secod Edo. Chapma & Hall New York. Gll J. 00. Baesa Mehods for he Socal ad Behavoral Sceces. Chapma & Hall New York. Godambe V. P. 960. A opmum proper of regular maxmum lkelhood esmao. Aals of Mahemacal Sascs 3 08-. Grabll F. A. 969. Marces wh Applcaos Sascs secod edo. Wadsworh Belmo CA. Hockg R. 985. The Aalss of Lear Models. Brooks/Cole:Wadsworh Moere Calfora. Hosmer D. W. ad S. Lemeshow 000. Appled Logsc Regresso. Joh Wle ad Sos New York. Hougaard P. 987. Modellg mulvarae survval. Scadava Joural of Sascs 4 9-304. Huber P. J. 967. The behavour of maxmum lkelhood esmaors uder o-sadard codos. Proceedgs of he Ffh Berkele Smposum o Mahemacal Sascs ad Probabl LeCam L. M. ad Nema J. edors Uvers of Calfora Press pp - 33. Huchso T.P. ad C. D. La 990. Couous Bvarae Dsrbuos Emphassg Applcaos. Adelade Souh Ausrala: Rumsb Scefc Publshg. Johso R. A. ad D. Wcher 999. Appled Mulvarae Sascal Aalss. Prece-Hall New Jerse. Kalma R. E. 960. A ew approach o lear flerg ad predco problems. Joural of Basc Egeerg 8 34-45. Laard M. W. 973. Robus large sample ess for homogee of varace. Joural of he Amerca Sascal Assocao 68 95. Lehma E. 99. Theor of Po Esmao. Wadsworh & Brooks/Cole Pacfc Grove CA. Lle R. J. 995. Modellg he drop-ou mechasm repeaed-measures sudes. Joural of he Amerca Sascal Assocao 90 -. Lle R. J. ad Rub D. B. 987. Sascal Aalss wh Mssg Daa. Joh Wle New York. McCullagh P. 983. Quas-lkelhood fucos. Aals of Sascs 59-67. McCullagh P. ad J. A. Nelder 989. Geeralzed Lear Models d ed. Chapma ad Hall Lodo. McCulloch C. E. ad S. R. Searle 00. Geeralzed Lear ad Mxed Models. Joh Wle ad Sos New York. Mller J.J. 977. Asmpoc properes of maxmum lkelhood esmaes he mxed model of aalss of varace. Aals of Sascs 5 746-76. Nelder J. A. ad R. W. Wedderbur 97. Geeralzed lear models. Joural of he Roal Sascal Soce Ser. A 35 370-84. Neer J. ad W. Wasserma 974. Appled Lear Sascal Models. Irw Homewood IL. Rub D. R. 976. Iferece ad mssg daa. Bomerka 63 58-59. Rub D. R. 978. Baesa ferece for causal effecs. The Aals of Sascs 6 34-58.

Appedces / A-39 Rub D. R. 990. Comme: Nema 93 ad causal ferece expermes ad observaoal sudes. Sascal Scece 5 47-480. Scheffé H. 959. The Aalss of Varace. Joh Wle ad Sos New York. Searle S. R. 97. Lear Models. Joh Wle ad Sos New York. Searle S. R. 987. Lear Models for Ubalaced Daa. Joh Wle ad Sos New York. Seber G. A. 977. Lear Regresso Aalss. Joh Wle New York. Serflg R. J. 980. Approxmao Theorems of Mahemacal Sascs. Joh Wle ad Sos New York. Sgler S. M. 986. The Hsor of Sascs: The Measureme of Ucera before 900. Harvard Uvers Press Cambrdge MA. Tufe E.R. 997. Vsual Explaaos. Cheshre Co.: Graphcs Press. Tuke J.W. 977. Exploraor Daa Aalss. Addso-Wesle Readg MA. Veables W. N. ad B. D. Rple 999. Moder Appled Sascs wh S-PLUS hrd edo. Sprger-Verlag New York. Wacher K. W. ad J. Trusell 98. Esmag hsorcal heghs. Joural of he Amerca Sascal Assocao 77 79-30. Wedderbur R. W. 974. Quas-lkelhood fucos geeralzed lear models ad he Gaussa mehod. Bomerka 6 439-47. Wog W. H. 986. Theor of paral lkelhood. Aals of Sascs 4 88-3.

Idex page umbers correspod o double-spaced verso aggregao bas 9 59 aalss of covarace model 6 34 363 arfcal regresso 9 55 aro v x x 6 3 6 5 70 94 300 459 Baesa 8 5 83 94 48 cojugae pror 67 40 4 emprcal Baes esmao 67 70 ferece v x 64 75 388 48 56 53 poseror dsrbuo 65 48 46 predcve dsrbuo 49 o-formave pror 40 pror dsrbuo 65 48 46 bes lear ubased esmaor BLUE 4 46 66 bes lear ubased predcor BLUP 45 66 0 496 BLUP forecas 53 79 BLUP predcor 46 5 7 76 79 80 04 343 346 347 496 BLUP resduals 45 5 79 0 BLUP varace 47 77 emprcal BLUPs 48 Hederso s mxed lear model equaos 8 bar depede varable xv 5 303 350 388 407 435 446 50 caocal lk 39 400 4 45 4 447 475 capal asse prcg model CAPM 335 caegorcal depede varable 43 causal effecs 8 cesorg 99 459 463 compuaoal mehods adapve Gaussa quadraure 366 38 Fsher scorg mehod 6 403 474 Newo-Raphso 6 384 403 4 474 codoal maxmum lkelhood esmaor 37 374 38 4 446 48 coemporaeous correlao 6 39 covarace srucure aalss 58 5 credbl credbl facor 89 90 43 44 69 7 73 credbl premum 4 credbl raemakg 70 7 credbl heor 4 70 73 75 5 srucure varable 69 cross-secoal correlao 47 48 49 309 30 3 cumulave hazard fuco 46 daa balaced daa 4 6 47 49 69 73 67 68 79 50 79 94 305 38 3 37 daa explorao 8 4 daa soopg 4 ubalaced daa 4 9 8 44 75 79 3 33 343 daa geerag process 4 34 dagosc sasc 45 6 45 dffereced daa 7 4 dsrbuos Beroull dsrbuo 388 39 407 4 430 exreme-value dsrbuo 44 443 expoeal faml 389 394 4 geeralzed exreme-value dsrbuo 44 443 mulomal dsrbuo 46 445 ormal dsrbuo 7 46 6 300 38 355 365 389 47 Posso dsrbuo v xv xv 389 394 397 408 409 45 44 430 445 damc model 46 309 EM algorhm 303 edogee 3 3 34 46 54 edogeous varable 3 3 55 63 exogee 36 43 68 8 309 exogeous varable 04 33 54 63 Grager-cause 7 predeermed 5 30 33 36 497 sequeall exogeous v 39 4 46 src exogee 5 33 34 39 76 srogl exogeous 7 weakl exogeous 6 error compoes model. See radom effecs. errors varables model 6 examples 7 capal srucure 39 charable corbuos 73 37 deal 96 0 67 group erm lfe surace 8

hospal coss 9 38 43 46 49 55 87 housg prce 77 38 8 come ax pames 73 8 9 38 77 85 9 98 350 359 366 379 454 457 loer sales 04 4 55 34 sude acheveme 86 90 7 50 or flgs 75 38 395 404 40 43 workers compesao 83 ogur 385 438 446 45 expoeal correlao model 36 facor aalss 59 6 facorzao heorem 45 45 464 falure rae 46 feasble geeralzed leas squares 74 75 30 feedback model 39 fxed effecs 83 87 90 98 04 8 38 4 44 63 89 90 38 40 4 70 73 75 76 77 79 80 85 86 87 88 30 30 39 3 338 350 363 370 37 388 407 4 43 48 454 48 487 495 basc fxed effecs model 5 59 64 67 68 69 83 89 6 74 fxed effecs esmaor 6 89 09 8 44 38 75 80 363 43 fxed effecs lear logudal daa model 56 fxed effecs model x 9 0 59 83 05 4 50 6 8 74 8 84 338 350 363 388 57 oe-wa fxed effecs model 6 75 77 79 80 8 74 30 wo-wa fxed effecs model 7 40 77 forecas 6 53 55 6 6 63 64 80 8 0 04 7 36 37 348 forward orhogoal devaos 43 Fuller-Baese rasform 304 Gaussa correlao model 36 Gaussa heor of errors 4 geeralzed esmag equao GEE xv 375 399 geeralzed lear model GLM 388 4 447 geeralzed mehod of momes GMM xv 375 40 growh curve model 6 95 67 Hausma es 70 75 hazard fuco 460 heerogee x 5 8 6 6 3 4 8 3 34 39 70 73 75 87 30 37 350 363 367 388 40 4 43 444 463 heerogee bas 9 heerogeeous model 9 66 subjec-specfc heerogee 6 34 es for heerogee 4 89. See also poolg es. heeroscedasc 50 5 55 57 58 98 3 4 79 87 30 33 35 404 homogeeous model xv 9 5 39 66 79 3 350 379 408 43 homoscedasc 50 5 9 45 35 defcao x 54 57 6 7 73 74 depedece of rreleva aleraves 438 44 44 drec leas squares 54 fe dmesoal usace parameers 4 fluece sasc 45 46 75 77 79 formao marx 6 359 365 393 473 474 al sae dsrbuo 448 450 460 srumeal varable 30 4 43 47 63 87 479 497 54 58 srumeal varable esmaor 30 44 48 479 497 ra-class correlao 85 eraed reweghed leas squares 377 39 475 Kalma fler xv 75 309 30 3 35 38 485 lagged depede varables 5 33 39 46 309 lagged durao depedece 458 Lagrage mulpler 5 9 48 lkelhood rao es 3 8 5 05 7 0 359 36 37 453 483 lear probabl model 35 35 lear projeco 5 34 43 69 497 lear red me model 3 lk fuco 35 390 39 4 log odds 356 log codoal log model 374 437 443 dsrbuo fuco 353 log fuco 355 358 365 50 mxed log model 438 mulomal log model v 437 44 445 463 esed log model 44 44 geeralzed log model 433 434 435 437 logudal ad pael daa sources Ceer for Research o Secur Prces CRSP 5 335 460 Compusa 5 460 Cosumer Prce Surve CPS 5 354

Naoal Assocao of Isurace Commssoers NAIC 5 50 Naoal Logudal Surve of Labor Marke Experece 5 Naoal Logudal Surve of Youh NLSY 7 507 Pael Sud of Icome Damcs PSID 3 7 503 scaer daa 5 386 438 margal dsrbuo 7 48 4 48 447 margal model xv 350 375 377 379 388 399 400 40 404 Markov model 447 456 margale dfferece 6 34 maxmum lkelhood esmao v 63 64 357 37 374 4 45 433 445 59 measureme equao 59 6 6 64 67 39 340 499 measureme error 59 66 330 MIMIC model 67 mssg gorable case 98 mssg a radom MAR 98 mssg compleel a radom MCAR 96 mssg daa v 5 68 94 95 96 97 300 304 39 53 o-gorable case 99 seleco model 97 98 30 selecv bas v x 94 uplaed orespose 96 mxed effecs lear mxed effecs model x 8 98 4 50 63 85 87 93 0 03 06 09 34 8 37 39 343 mxed effecs models 98 4 85 4 mxed lear model x 8 06 7 0 46 50 64 65 70 8 93 0 38 33 343 47 487 olear mxed effecs model 4 43 mome geerag fuco 44 mullevel cross-level effecs 90 cross-level eraco 88 daa 85 88 herarchcal model 64 88 44 50 5 hgh order mullevel models 08 09 mullevel model x 85 68 366 hree-level model 9 93 08 3 369 496 wo-level model 87 89 9 94 0 0 04 3 369 mulvarae regresso model 48 5 53 6 498 olear regresso model 4 359 37 388 4 osaoar model 34 usace parameers 5 88 43 observables represeao 386 99 37. See also margal model. observao equao 39 33 485 487 observaoal daa x 9 occurrece depedece 458 odds rao 356 357 386 434 438 omed varable x 0 40 3 70 76 8 86 88 model of correlaed effecs 8 83 84 umeasured cofouder 89 uobserved varable model 89 uobserved varables 4 30 90 30 overdsperso 378 394 398 408 409 paral lkelhood 450 46 53 pah dagram 6 plog mehods added varable plo 8 9 44 69 75 77 79 96 38 boxplo 8 95 9 mulple me seres plo 8 9 3 74 38 39 58 9 397 paral regresso plo 44 scaer plo wh smbols 3 rells plo 33 337 pooled cross-secoal model 96 59 6 pooled cross-secoal me seres 5 poolg es 4 43 69 9 9 96 3 6 populao parameer 5 6 45 83 88 06 74 populao-averaged model 86 00 poeal oucomes 9 predco x 4 4 46 00 34 346 4 485. See also forecasg. prob regresso model v 30 35 354 357 385 profle log-lkelhood 5 radom cesorg 459 radom effecs x 9 50 8 98 4 07 70 80 363 406 407 4 43 443 445 487 error compoes model 0 8 96 98 0 04 07 09 5 8 9 30 3 34 37 38 39 5 59 79 8 89 98 0 05 3 9

36 57 76 85 87 90 9 303 304 34 37 40 494 496 models wh radom effecs 8. See also radom effecs model. oe-wa radom effecs ANOVA model 04 43 48 67 7 radom coeffces model 0 03 0 9 33 5 7 79 90 0 86 radom effecs esmaors 38 75 79 radom effecs model x 363 406 443 radom erceps model 8 83. See also error compoes model. wo-wa error compoe 9 07 55 34 radom ul erpreao 385 44 radom walk model 34 repeaed measures desg 0 resdual sadard devao 36 37 roag pael v 94 95 samplg based model 4 5 87 8 30 score equaos 358 359 365 370 37 score vecor seemgl urelaed regresso 5 5 55 56 57 30 53 59 seral correlao x 6 5 98 53 54 95 50 309 3 30 33 408 auoregressve model 54 0 0 66 3 33 405 movg average model 33 seral correlao models x 3 seral covarace. See emporal varacecovarace marx. shrkage esmaor 4 43 44 495 496 smulaeous equaos model 498 small area esmao 4 75 spaal correlaos 37 39 sadard errors for regresso coeffce esmaors emprcal 378 379 404 405 model-based 39 379 404 405 pael-correced 3 robus 57 35 45 404 sae space models 38 39 58 srafed sample 3 56 85 9 0 srucural equao 9 3 59 6 6 64 65 66 67 68 499 srucural model 5 6 87 8 9 30 3 34 subjec-specfc parameers 9 5 4 45 5 55 8 66 7 88 389 subjec-specfc ercep 9 0 subjec-specfc slopes 55 7 suffcec 358 37 45 45 464 survval model 43 458 459 46 463 acceleraed falure me model 46 Cox proporoal hazard model 46 46 ssemac compoe 390 39 39 406 4 439 444 447 449 455 475 emporal varace-covarace marx 53 59 95 33 auoregressve 5308 0 54 0 65 66 33 405 exchageable correlao 378 40 compoud smmer 53 54 7 33 7 33 Toeplz marx 33 uform correlao 53 me-cosa varables 35 88 04 05 87 88 9 me seres aalss xv 7 5 54 33 38 483 me-seres cross-seco TSCS model 39 me-seres cross-seco daa 39 5 me-varg coeffces 3 338 339 340 50 raso equao 39 330 33 485 487 50 marx 448 model 43 448 449 455 457 458 459 probabl 456 wo-sage leas squares 3 54 55 57 58 wo-sage samplg scheme 84 64 363 u of aalss 55 93 94 00 9 varace compoe 53 8 89 99 08 4 48 59 0 04 73 33 33 334 344 40 403 40 47 479 varace compoes esmaors 4 0 55 mmum varace quadrac ubased esmaor MIVQUE 0 0 33 mome-based varace compoe esmaor 9 resdual maxmum lkelhood esmao 7 resrced maxmum lkelhood REML 6 43 334 345 esg varace compoes 04 Wald es 3 9 workg correlao 378 379 404 405