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

Transcription

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

2 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

3 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 Poolg es Added varable plos Ifluece dagoscs Cross-secoal correlao Heeroscedasc -8.5 Model exesos Seral correlao Subjec-specfc slopes 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 Models wh Radom Effecs 3. Error compoes / radom erceps model 3-3. Example: Icome ax pames Mxed effecs models Lear mxed effecs model Mxed lear model Iferece for regresso coeffces 3-6

4 3.5 Varace compoe esmao Maxmum lkelhood esmao Resrced maxmum lkelhood 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 Exercses ad Exesos Predco ad Baesa Iferece 4. Predco for oe-wa ANOVA models 4-4. Bes lear ubased predcors BLUP Mxed model predcors Lear mxed effecs model Lear combaos of global parameers ad subjecspecfc effecs BLUP resduals Predcg fuure observaos Example: Forecasg Wscos loer sales Sources ad characerscs of daa I-sample model specfcao Ou-of-sample model specfcao Forecass Baesa ferece Credbl heor Credbl heor models 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 Exercses ad Exesos Mullevel Models 5. Cross-secoal mullevel models Two-level models Mulple level models Mulple level modelg oher felds Logudal mullevel models Two-level models Mulple level models Predco Tesg varace compoes 5-3 Furher readg 5-5 Appedx 5A Hgh order mullevel models Exercses ad Exesos 5-9

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

6 8.4.4 Forecasg Kalma fler approach Traso equaos Observao se Measureme equaos Ial codos Kalma fler algorhm 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 Logsc ad prob regresso models Iferece for logsc ad prob regresso models Example: Icome ax pames ad ax preparers Radom effecs models Fxed effecs models 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 Geeralzed Lear Models 0. Homogeeous models Lear expoeal famles of dsrbuos Lk fucos Esmao Example: Tor flgs Margal models ad GEE Radom effecs models Fxed effecs models Maxmum lkelhood esmao for caocal lks Codoal maxmum lkelhood esmao for caocal lks Posso dsrbuo 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

7 0A.4 Margal Dsrbuos Exercses ad Exesos 0-7. Caegorcal Depede Varables ad Survval Models. Homogeeous models -.. Sascal ferece -.. Geeralzed log -..3 Mulomal codoal log Radom ul erpreao Nesed log 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

8 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

9 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

10 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.

11 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:// hp:// hp:// 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

12 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.

13 Chaper. Iroduco / 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

14 - / 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 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 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 Furher a plo wh boh he 965 ad 975 daa dsplas a egave relao bewee dvorce raes ad AFDC pames. AFDC

15 Chaper. Iroduco / -3 DIVORCE 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

16 -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.

17 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.

18 -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

19 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

20 -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

21 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

22 -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

23 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.

24 - / 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: N87 observaos. Bod Maur Face Bodma Subjecs are 38 frms over T0 ears: N380 observaos. Capal Srucure Charable Corbuos Face Capal Subjecs are 36 Japaese frms over T5 ears: N545 observaos. Accoug Char Subjecs are 47 axpaers over T0 ears; N470 observaos. Dvorce Socolog Dvorce Subjecs are 5 saes over T4 ears: 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: 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: N34 observaos. Isurace Pdemad Subjecs are coures over T7 ears: 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: N5 observaos. Tor Flgs Isurace Tflg Subjecs are 9 saes over T6 ears: 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.