Stanislav Anatolyev. Intermediate and advanced econometrics: problems and solutions

Transcription

1 Staslav Aatolyev Itermedate ad advaced ecoometrcs: problems ad solutos Thrd edto KL/9/8 Moscow 9

2 Анатольев С.А. Задачи и решения по эконометрике. #KL/9/8. М.: Российская экономическая школа, 9 г. 78 с. (Англ.) Данное пособие сборник задач, которые использовались автором при преподавании эконометрики промежуточного и продвинутого уровней в Российской Экономической Школе в течение последних нескольких лет. Все задачи сопровождаются решениями. Ключевые слова: асимптотическая теория, бутстрап, линейная регрессия, метод наименьших квадратов, нелинейная регрессия, непараметрическая регрессия, экстремальное оценивание, метод наибольшего правдоподобия, инструментальные переменные, обобщенный метод моментов, эмпирическое правдоподобие, анализ панельных данных, условные ограничения на моменты, альтернативная асимптотика, асимптотика высокого порядка. Aatolyev, Staslav A. Itermedate ad advaced ecoometrcs: problems ad solutos. #KL 9/8 Moscow, New Ecoomc School, 9 78 pp. (Eg.) Ths maual s a collecto of problems that the author has bee usg teachg termedate ad advaced level ecoometrcs courses at the New Ecoomc School durg last several years. All problems are accompaed by sample solutos. Key words: asymptotc theory, bootstrap, lear regresso, ordary ad geeralzed least squares, olear regresso, oparametrc regresso, extremum estmato, maxmum lkelhood, strumetal varables, geeralzed method of momets, emprcal lkelhood, pael data aalyss, codtoal momet restrctos, alteratve asymptotcs, hgher-order asymptotcs ISBN Анатольев С.А., 9 г. Российская экономическая школа, 9 г.

3 CONTENTS I Problems 5 Asymptotc theory: geeral ad depedet data 7. Asymptotcs of trasformatos Asymptotcs of rotated logarthms Escapg probablty mass Asymptotcs of t-ratos Creepg bug o smplex Asymptotcs of sample varace Asymptotcs of roots Secod-order Delta-method Asymptotcs wth shrkg regressor Power treds Asymptotc theory: tme seres. Treded vs. d ereced regresso Log ru varace for AR() Asymptotcs of averages of AR() ad MA() Asymptotcs for mpulse respose fuctos Bootstrap 3 3. Bref ad exhaustve Bootstrappg t-rato Bootstrap bas correcto Bootstrap lear model Bootstrap for mpulse respose fuctos Regresso ad projecto 5 4. Regressg ad projectg dce Mxture of ormals Beroull regressor Best polyomal approxmato Hadlg codtoal expectatos Lear regresso ad OLS 7 5. Fxed ad radom regressors Cosstecy of OLS uder serally correlated errors Estmato of lear combato Icomplete regresso Geerated coe cet OLS olear model Log ad short regressos Rdge regresso Icosstecy uder alteratve Returs to schoolg CONTENTS 3

4 6 Heteroskedastcty ad GLS 3 6. Codtoal varace estmato Expoetal heteroskedastcty OLS ad GLS are detcal OLS ad GLS are equvalet Equcorrelated observatos Ubasedess of certa FGLS estmators Varace estmato Whte estmator HAC estmato uder homoskedastcty Expectatos of Whte ad Newey West estmators IID settg Nolear regresso Local ad global det cato Idet cato whe regressor s oradom Cobb Douglas producto fucto Expoetal regresso Power regresso Trasto regresso Nolear cosumpto fucto Extremum estmators Regresso o costat Quadratc regresso Nolearty at left had sde Least fourth powers Asymmetrc loss Maxmum lkelhood estmato 39. Normal dstrbuto Pareto dstrbuto Comparso of ML tests Ivarace of ML tests to reparameterzatos of ull Msspec ed maxmum lkelhood Idvdual e ects Irregular co dece terval Trval parameter space Nusace parameter desty MLE versus OLS MLE versus GLS MLE heteroskedastc tme seres regresso Does the lk matter? Maxmum lkelhood ad bary varables Maxmum lkelhood ad bary depedet varable Posso regresso Bootstrappg ML tests CONTENTS

5 Istrumetal varables 45. Ivald SLS Cosumpto fucto Optmal combato of strumets Trade ad growth Geeralzed method of momets 47. Nolear smultaeous equatos Improved GMM Mmum Dstace estmato Formato of momet codtos What CMM estmates Trty for GMM All about J Iterest rates ad future ato Spot ad forward exchage rates Returs from acal market Istrumetal varables ARMA models Hausma may ot work Testg momet codtos Bootstrappg OLS Bootstrappg DD Pael data Alteratg dvdual e ects Tme varat regressors Wth ad Betwee Paels ad strumets D erecg trasformatos Nolear pael data model Durb Watso statstc ad pael data Hgher-order dyamc pael Noparametrc estmato Noparametrc regresso wth dscrete regressor Noparametrc desty estmato Nadaraya Watso desty estmator Frst d erece trasformato ad oparametrc regresso Ubasedess of kerel estmates Shape restrcto Noparametrc hazard rate Noparametrcs ad perfect t Noparametrcs ad extreme observatos Codtoal momet restrctos 6 5. Usefuless of skedastc fucto Symmetrc regresso error Optmal strumetato of cosumpto fucto Optmal strumet AR-ARCH model Optmal strumet AR wth olear error Optmal IV estmato of a costat CONTENTS 5

6 5.7 Negatve bomal dstrbuto ad PML Nestg ad PML Msspec cato varace Mod ed Posso regresso ad PML estmators Optmal strumet ad regresso o costat Emprcal Lkelhood Commo mea Kullback Lebler Iformato Crtero Emprcal lkelhood as IV estmato Advaced asymptotc theory Maxmum lkelhood ad asymptotc bas Emprcal lkelhood ad asymptotc bas Asymptotcally rrelevat strumets Weakly edogeous regressors Weakly vald strumets II Solutos 69 Asymptotc theory: geeral ad depedet data 7. Asymptotcs of trasformatos Asymptotcs of rotated logarthms Escapg probablty mass Asymptotcs of t-ratos Creepg bug o smplex Asymptotcs of sample varace Asymptotcs of roots Secod-order Delta-method Asymptotcs wth shrkg regressor Power treds Asymptotc theory: tme seres 79. Treded vs. d ereced regresso Log ru varace for AR() Asymptotcs of averages of AR() ad MA() Asymptotcs for mpulse respose fuctos Bootstrap Bref ad exhaustve Bootstrappg t-rato Bootstrap bas correcto Bootstrap lear model Bootstrap for mpulse respose fuctos Regresso ad projecto Regressg ad projectg dce Mxture of ormals Beroull regressor Best polyomal approxmato Hadlg codtoal expectatos CONTENTS

7 5 Lear regresso ad OLS 9 5. Fxed ad radom regressors Cosstecy of OLS uder serally correlated errors Estmato of lear combato Icomplete regresso Geerated coe cet OLS olear model Log ad short regressos Rdge regresso Icosstecy uder alteratve Returs to schoolg Heteroskedastcty ad GLS Codtoal varace estmato Expoetal heteroskedastcty OLS ad GLS are detcal OLS ad GLS are equvalet Equcorrelated observatos Ubasedess of certa FGLS estmators Varace estmato 7. Whte estmator HAC estmato uder homoskedastcty Expectatos of Whte ad Newey West estmators IID settg Nolear regresso 3 8. Local ad global det cato Idet cato whe regressor s oradom Cobb Douglas producto fucto Expoetal regresso Power regresso Smple trasto regresso Nolear cosumpto fucto Extremum estmators 7 9. Regresso o costat Quadratc regresso Nolearty at left had sde Least fourth powers Asymmetrc loss Maxmum lkelhood estmato 3. Normal dstrbuto Pareto dstrbuto Comparso of ML tests Ivarace of ML tests to reparameterzatos of ull Msspec ed maxmum lkelhood Idvdual e ects Irregular co dece terval Trval parameter space Nusace parameter desty CONTENTS 7

8 .MLE versus OLS MLE versus GLS MLE heteroskedastc tme seres regresso does the lk matter? Maxmum lkelhood ad bary varables Maxmum lkelhood ad bary depedet varable Posso regresso Bootstrappg ML tests Istrumetal varables 9. Ivald SLS Cosumpto fucto Optmal combato of strumets Trade ad growth Geeralzed method of momets 33. Nolear smultaeous equatos Improved GMM Mmum Dstace estmato Formato of momet codtos What CMM estmates Trty for GMM All about J Iterest rates ad future ato Spot ad forward exchage rates Returs from acal market Istrumetal varables ARMA models Hausma may ot work Testg momet codtos Bootstrappg OLS Bootstrappg DD Pael data Alteratg dvdual e ects Tme varat regressors Wth ad Betwee Paels ad strumets D erecg trasformatos Nolear pael data model Durb Watso statstc ad pael data Hgher-order dyamc pael Noparametrc estmato 5 4. Noparametrc regresso wth dscrete regressor Noparametrc desty estmato Nadaraya Watso desty estmator Frst d erece trasformato ad oparametrc regresso Ubasedess of kerel estmates Shape restrcto Noparametrc hazard rate Noparametrcs ad perfect t CONTENTS

9 4.9 Noparametrcs ad extreme observatos Codtoal momet restrctos Usefuless of skedastc fucto Symmetrc regresso error Optmal strumetato of cosumpto fucto Optmal strumet AR-ARCH model Optmal strumet AR wth olear error Optmal IV estmato of a costat Negatve bomal dstrbuto ad PML Nestg ad PML Msspec cato varace Mod ed Posso regresso ad PML estmators Optmal strumet ad regresso o costat Emprcal Lkelhood Commo mea Kullback Lebler Iformato Crtero Emprcal lkelhood as IV estmato Advaced asymptotc theory Maxmum lkelhood ad asymptotc bas Emprcal lkelhood ad asymptotc bas Asymptotcally rrelevat strumets Weakly edogeous regressors Weakly vald strumets CONTENTS 9

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11 PREFACE Ths maual s a thrd edto of the collecto of problems that I have bee usg teachg termedate ad advaced level ecoometrcs courses at the New Ecoomc School (NES), Moscow, for already a decade. All problems are accompaed by sample solutos. Approxmately, chapters 8 ad 4 of the collecto belog to a course termedate level ecoometrcs ( Ecoometrcs III the NES teral course structure); chapters 9 3 to a course advaced level ecoometrcs ( Ecoometrcs IV, respectvely). The problems chapters 5 7 requre kowledge of more advaced ad specal materal. They have bee used the NES course Topcs Ecoometrcs. May of the problems are ot ew. Some are spred by my former teachers of ecoometrcs at PhD studes: Hyugtak Ah, Mahmoud El-Gamal, Bruce Hase, Yuch Ktamura, Charles Mask, Gautam Trpath, Keeth West. Some problems are borrowed from ther problem sets, as well as problem sets of other leadg ecoometrcs scholars or ther textbooks. Some orgate from the Problems ad Solutos secto of the joural Ecoometrc Theory, where the author has publshed several problems. The release of ths collecto would be hard wthout valuable help of my teachg assstats durg varous years: Adrey Vasev, Vktor Subbot, Semyo Polbekov, Alexader Vaschlko, Des Sokolov, Oleg Itskhok, Adrey Shabal, Staslav Kolekov, Aa Mkusheva, Dmtry Shak, Oleg Shbaov, Vadm Cherepaov, Pavel Stetseko, Iva Lazarev, Yula Shkurat, Dmtry Muravyev, Artem Shamguov, Dala Delya, Vktora Stepaova, Bors Gershma, Alexader Mgta, Iva Mrgorodsky, Roma Chkoller, Adrey Savochk, Alexader Kobel, Ekatera Lavreko, Yula Vakhrutdova, Elea Pkula, to whom go my deepest thaks. My thaks also go to my studets ad assstats who spotted errors ad typos that crept to the rst ad secod edtos of ths maual, especally Dmtry Shak, Des Sokolov, Pavel Stetseko, Georgy Kartashov, ad Roma Chkoller. Preparato of ths maual was supported part by the Swedsh Professorshp ( 3) from the Ecoomcs Educato ad Research Cosortum, wth fuds provded by the Govermet of Swede through the Eurasa Foudato, ad by the Access Idustres Professorshp (3 9) from Access Idustres. I wll be grateful to everyoe who ds errors, mstakes ad typos ths collecto ad reports them to saatoly@es.ru. CONTENTS

12 CONTENTS

13 NOTATION AND ABBREVIATIONS ID det cato FOC/SOC rst/secod order codto(s) CDF cumulatve dstrbuto fucto, typcally deoted as F PDF probablty desty fucto, typcally deoted as f LIME law of terated (mathematcal) expectatos LLN law of large umbers CLT cetral lmt theorem I fag dcator fucto equallg uty whe A holds ad zero otherwse Pr fag probablty of A E [yjx] mathematcal expectato (mea) of y codtoal o x V [yjx] varace of y codtoal o x C [x; y] covarace betwee x ad y BLP, BLP [yjx] best lear predctor I k k k detty matrx plm probablty lmt typcally meas dstrbuted as N ormal (Gaussa) dstrbuto k k ch-squared dstrbuto wth k degrees of freedom () o-cetral ch-squared dstrbuto wth k degrees of freedom ad o-cetralty parameter B (p) Beroull dstrbuto wth success probablty p IID depedetly ad detcally dstrbuted typcally sample sze cross-sectos T typcally sample sze tme seres k typcally umber of parameters parametrc models ` typcally umber of strumets or momet codtos X ; Y; Z; E; b E data matrces of regressors, depedet varables, strumets, errors, resduals L () (codtoal) lkelhood fucto ` () (codtoal) loglkelhood fucto s () (codtoal) score fucto m () momet fucto Q f typcally E [f] ; for example, Q xx = E [xx ] ; Q gge = E g g e ; = E [@m=@] ; etc. I Iformato matrx W Wald test statstc LR lkelhood rato test statstc LM Lagrage multpler (score) test statstc J Hase s J test statstc CONTENTS 3

14 4 CONTENTS

15 Part I Problems 5

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17 . ASYMPTOTIC THEORY: GENERAL AND INDEPENDENT DATA. Asymptotcs of trasformatos. Suppose that p T (^ ) d! N (; ). Fd the lmtg dstrbuto of T ( cos ^).. Suppose that T (^ ) d! N (; ). Fd the lmtg dstrbuto of T s ^. 3. Suppose that T ^! d. Fd the lmtg dstrbuto of T log ^.. Asymptotcs of rotated logarthms Let the postve radom vector (U ; V ) p U V u v be such that d! N!uu ;! uv! uv! vv as! : Fd the jot asymptotc dstrbuto of l U l V : l U + l V What s the codto uder whch l U l V ad l U + l V are asymptotcally depedet?.3 Escapg probablty mass Let X = fx ; : : : ; x g be a radom sample from some populato of x wth E [x] = ad V [x] =. Let A deote a evet such that P fa g = ; ad let the dstrbuto of A be depedet of the dstrbuto of x. Now costruct the followg radomzed estmator of : x ^ = f A happes, otherwse. () Fd the bas, varace, ad MSE of ^. Show how they behave as!. () Is ^ a cosstet estmator of? Fd the asymptotc dstrbuto of p (^ ): () Use ths dstrbuto to costruct a approxmately ( ) % co dece terval for. Compare ths CI wth the oe obtaed by usg x as a estmator of. ASYMPTOTIC THEORY: GENERAL AND INDEPENDENT DATA 7

18 .4 Asymptotcs of t-ratos Let fx g = be a radom sample of a scalar radom varable x wth E[x] = ; V[x] = ; E[(x ) 3 ] = ; E[(x ) 4 ] = ; where all parameters are te. (a) De e T x^ ; where x x ; = ^ (x x) : = Derve the lmtg dstrbuto of p T uder the assumpto =. (b) Now suppose t s ot assumed that =. Derve the lmtg dstrbuto of p T plm T :! Be sure your aswer reduces to the result of part (a) whe =. (c) De e R x ; where = x s the costraed estmator of uder the (possbly correct) assumpto =. Derve the lmtg dstrbuto of p R plm R! for arbtrary ad >. Uder what codtos o ad wll ths asymptotc dstrbuto be the same as part (b)?.5 Creepg bug o smplex Cosder a postve (x; y) orthat R + ad ts ut smplex,.e. the le segmet x + y = ; x ; y : Take a arbtrary atural umber k N: Image a bug startg creepg from the org (x; y) = (; ): Each secod the bug goes ether the postve x drecto wth probablty p; or the postve y drecto wth probablty p; each tme coverg dstace k : Evdetly, ths way the bug reaches the smplex k secods. Suppose t arrves there at pot (x k ; y k ): Now let k! ;.e. as f the bug shrks sze ad physcal abltes per secod. Determe (a) the probablty lmt of (x k ; y k ); (b) the rate of covergece; (c) the asymptotc dstrbuto of (x k ; y k ). 8 ASYMPTOTIC THEORY: GENERAL AND INDEPENDENT DATA

19 .6 Asymptotcs of sample varace Let x ; : : : ; x be a radom sample from a populato of x wth te fourth momets. Let x ad x be the sample averages of x ad x, respectvely. Fd costats a ad b ad fucto c() such that the vector sequece x a c() x b coverges to a otrval dstrbuto, ad determe ths lmtg dstrbuto. Derve the asymptotc dstrbuto of the sample varace x (x ) :.7 Asymptotcs of roots Suppose we are terested the ferece about the root of the olear system F (a; ) = ; where F : R p R k! R k ; ad a s a vector of costats. Let avalable be ^a; a cosstet ad asymptotcally ormal estmator of a: Assumg that s the uque soluto of the above system, ad ^ s the uque soluto of the system F (^a; ^) = ; derve the asymptotc dstrbuto of ^: Assume that all eeded smoothess codtos are sats ed..8 Secod-order Delta-method Let S = P = X ; where X ; = ; : : : ; ; s a radom sample of scalar radom varables wth E [X ] = ad V [X ] = : It s easy to show that p (S ) d! N (; 4 ) whe 6= : (a) Fd the asymptotc dstrbuto of S whe = ; by takg a square of the asymptotc dstrbuto of S. (b) Fd the asymptotc dstrbuto of cos(s ): Ht: appled to cos(s ). take a hgher-order Taylor expaso (c) Usg the techque of part (b), formulate ad prove a aalog of the Delta-method for the case whe the fucto s scalar-valued, has zero rst dervatve ad ozero secod dervatve (whe the dervatves are evaluated at the probablty lmt). For smplcty, let all volved radom varables be scalars..9 Asymptotcs wth shrkg regressor Suppose that y = + x + u ; ASYMPTOTICS OF SAMPLE VARIANCE 9

20 where fu g are IID wth E [u ] =, E u = ad E u 3 =, whle the regressor x s determstcally shrkg: x = wth (; ): Let the sample sze be : Dscuss as fully as you ca the asymptotc behavor of the OLS estmates (^; ^; ^ ) of (; ; ) as! :. Power treds Suppose that y = x + " ; = ; : : : ; ; where " IID (; ) whle x = for some kow ; ad = for some kow :. Uder what codtos o ad s the OLS estmator of cosstet? Derve ts asymptotc dstrbuto whe t s cosstet.. Uder what codtos o ad s the GLS estmator of cosstet? Derve ts asymptotc dstrbuto whe t s cosstet. ASYMPTOTIC THEORY: GENERAL AND INDEPENDENT DATA

21 . ASYMPTOTIC THEORY: TIME SERIES. Treded vs. d ereced regresso Cosder a lear model wth a learly tredg regressor: y t = + t + " t ; where the sequece " t s depedetly ad detcally dstrbuted accordg to some dstrbuto D wth mea zero ad varace : The object of terest s :. Wrte out the OLS estmator ^ of devatos form ad d ts asymptotc dstrbuto.. A researcher suggests removg the tredg regressor by takg d ereces to obta y t y t = + " t " t ad the estmatg by OLS. Wrte out the OLS estmator of ad d ts asymptotc dstrbuto. 3. Compare the estmators ^ ad terms of asymptotc e cecy.. Log ru varace for AR() Ofte oe eeds to estmate the log-ru varace V ze lm V p T! T T X t= z t e t! of a statoary sequece z t e t that sats es the restrcto E[e t jz t ] = : Derve a compact expresso for V ze the case whe e t ad z t follow depedet scalar AR() processes. For ths example, propose a method to cosstetly estmate V ze ; ad show your estmator s cosstecy..3 Asymptotcs of averages of AR() ad MA() Let x t be a martgale d erece sequece wth respect to ts ow past, ad let all codtos for the CLT be sats ed: p T x T = T = P T t= x d t! N (; ): Let ow y t = y t +x t ad z t = x t +x t ; where jj < ad jj < : Cosder tme averages y T = T P T t= y t ad z T = T P T t= z t:. Are y t ad z t martgale d erece sequeces relatve to ther ow past?. Fd the asymptotc dstrbutos of y T ad z T : ASYMPTOTIC THEORY: TIME SERIES

22 3. How would you estmate the asymptotc varaces of y T ad z T? 4. Repeat what you dd parts 3 whe x t s a k vector, ad we have p T x T = T = P T t= x t N (; ), y t = Py t +x t ; z t = x t +x t ; where P ad are k k matrces wth egevalues sde the ut crcle. d!.4 Asymptotcs for mpulse respose fuctos A statoary ad ergodc process z t that admts the represetato z t = + X j " t j ; j= where P j= j jj < ad " t s zero mea IID, s called lear. The fucto IRF (j) = j s called mpulse respose fucto of z t ; re ectg the fact that j t =@" t j ; a respose of z t to ts ut shock j perods ago.. Show that the strog zero mea AR() ad ARMA(,) processes ad y t = y t + " t ; jj < z t = z t + " t " t ; jj < ; jj < ; 6= ; are lear, ad derve ther mpulse respose fuctos.. Suppose a sample z ; : : : ; z T s gve. For the AR() process, costruct a estmator of the IRF o the bass of the OLS estmator of. Derve the asymptotc dstrbuto of your IRF estmator for xed horzo j as the sample sze T!. 3. Suppose that for the ARMA(,) process oe estmates from the sample z ; : : : ; z T by ^ = P T t=3 z tz t P T t=3 z t z t ; ad by a approprate root of the quadratc equato P ^ T + ^ = t= ^e t^e t P T ; ^e t = z t ^z t : t= ^e t O the bass of these estmates, costruct a estmator of the mpulse respose fucto you derved. Outle the steps (o eed to show all math) whch you would udertake order to derve ts asymptotc dstrbuto for xed j as T!. ASYMPTOTIC THEORY: TIME SERIES

23 3. BOOTSTRAP 3. Bref ad exhaustve Evaluate the followg clams.. The oly d erece betwee Mote Carlo ad the bootstrap s possblty ad mpossblty, respectvely, of samplg from the true populato.. Whe oe does bootstrap, there s o reaso to rase the umber of bootstrap repetto too hgh: there s a level whe makg t larger does ot yeld ay mprovemet precso. 3. The bootstrap estmator of the parameter of terest s preferable to the asymptotc oe, sce ts rate of covergece to the true parameter s ofte larger. 3. Bootstrappg t-rato Cosder the followg bootstrap procedure. Usg the oparametrc bootstrap, geerate bootstrap samples ad calculate ^ b ^ at each bootstrap repetto. Fd the quatles q= s(^) ad q = from ths bootstrap dstrbuto, ad costruct CI = [^ s(^)q = ; ^ s(^)q = ]: Show that CI s exactly the same as the percetle terval, ad ot the percetle-t terval. 3.3 Bootstrap bas correcto. Cosder a radom varable x wth mea : A radom sample fx g = s avalable. Oe estmates by x ad by x : Fd out what the bootstrap bas corrected estmators of ad are.. Suppose we have a sample of two depedet observatos z = ad z = 3 from the same dstrbuto. Let us be terested E[z ] ad (E[z]) whch are atural to estmate by z = (z + z ) ad z = 4 (z + z ) : Compute the bootstrap-bas-corrected estmates of the quattes of terest. BOOTSTRAP 3

24 3.4 Bootstrap lear model. Suppose oe has a radom sample of observatos from the lear regresso model y = x + e; E [ejx] = : Is the oparametrc bootstrap vald or vald the presece of heteroskedastcty? Expla.. Let the model be y = x + e; but E [ex] 6= ;.e. the regressors are edogeous. The OLS estmator ^ of the parameter s based. We kow that the bootstrap s a good way to estmate bas, so the dea s to estmate the bas of ^ ad costruct a bas-adjusted estmate of : Expla whether or ot the o-parametrc bootstrap ca be used to mplemet ths dea. 3. Take the lear regresso y = x + e; E [ejx] = : For a partcular value of x; the object of terest s the codtoal mea g(x) = E [yjx] : Descrbe how you would use the percetle-t bootstrap to costruct a co dece terval for g(x): 3.5 Bootstrap for mpulse respose fuctos Recall the formulato of Problem.4.. Descrbe detal how to costruct 95% error bads aroud the IRF estmates for the AR() process usg the bootstrap that attas asymptotc re emet.. It s well kow that spte of ther asymptotc ubasedess, usual estmates of mpulse respose fuctos are sg catly based samples typcally ecoutered practce. Propose a bootstrap algorthm to costruct a bas corrected mpulse respose fucto for the above ARMA(,) process. 4 BOOTSTRAP

25 4. REGRESSION AND PROJECTION 4. Regressg ad projectg dce Let y be a radom varable that deotes the umber of dots obtaed whe a far sx sded de s rolled. Let y f y s eve, x = otherwse. () Fd the jot dstrbuto of (x; y). () Fd the best predctor of y gve x. () Fd the best lear predctor, BLP [yjx], of y codtoal o x. (v) Calculate E U BP ad E U BLP, the mea square predcto errors for cases () ad () respectvely, ad show that E U BP E U BLP. 4. Mxture of ormals Suppose that pars (x ; y ); = ; : : : ; ; are depedetly draw from the followg mxture of ormals dstrbuto: 8 x >< N ; wth probablty p; 4 y 4 >: N ; wth probablty p; where < p < :. Derve the best lear predctor BLP [yjx] of y gve x.. Argue that the codtoal expectato fucto E [yjx] s olear. Provde a step-by-step algorthm allowg oe to derve E [yjx] ; ad derve t f you ca. 4.3 Beroull regressor Let x be dstrbuted Beroull, ad, codtoal o x; y be dstrbuted as N ; yjx ; x = ; N ; ; x = : Wrte out E [yjx] ad E y jx as lear fuctos of x: Why are these expectatos lear x? REGRESSION AND PROJECTION 5

26 4.4 Best polyomal approxmato Gve jotly dstrbuted radom varables x ad y; a best k th order polyomal approxmato BPA k [yjx] to E [yjx] ; the MSE sese, s a soluto to the problem m E E [yjx] x : : : k x k : ; ;:::; k Assumg that BPA k [yjx] exsts, d ts characterzato ad derve the propertes of the assocated predcto error U k = y BPA k [yjx] : 4.5 Hadlg codtoal expectatos. Cosder the followg stuato. The vector (y; x; z; w) s a radom quadruple. It s kow that E [yjx; z; w] = + x + z: It s also kow that C [x; z] = ad that C [w; z] > : The parameters ; ad are ot kow. A radom sample of observatos o (y; x; w) s avalable; z s ot observable. I ths settg, a researcher weghs two optos for estmatg : Oe s a lear least squares t of y o x: The other s a lear least squares t of y o (x; w): Compare these optos.. Let (x; y; z) be a radom trple. For a gve real costat ; a researcher wats to estmate E [yje [xjz] = ]. The researcher kows that E [xjz] ad E [yjz] are strctly creasg ad cotuous fuctos of z, ad s gve cosstet estmates of these fuctos. Show how the researcher ca use them to obta a cosstet estmate of the quatty of terest. 6 REGRESSION AND PROJECTION

27 5. LINEAR REGRESSION AND OLS 5. Fxed ad radom regressors. Commet o: Treatg regressors x a mea regresso as radom varables rather tha xed umbers smpl es further aalyss, sce the the observatos (x ; y ) may be treated as IID across.. A labor ecoomst argues: It s more plausble to thk of my regressors as radom rather tha xed. Look at educato, for example. A perso chooses her level of educato, thus t s radom. Age may be msreported, so t s radom too. Eve geder s radom, because oe ca get a sex chage operato doe. Commet o ths pearl. 3. Cosder a lear mea regresso y = x + e; E [ejx] = ; where x; stead of beg IID across ; depeds o through a ukow fucto ' as x = '() + u ; where u are IID depedet of e : Show that the OLS estmator of s stll ubased. 5. Cosstecy of OLS uder serally correlated errors Let fy t g + varace. t= be a strctly statoary ad ergodc stochastc process wth zero mea ad te () De e so that we ca wrte = C [y t; y t ] ; u t = y t y t ; V [y t ] y t = y t + u t : Show that the error u t sats es E [u t ] = ad C [u t ; y t ] = : () Show that the OLS estmator ^ from the regresso of y t o y t s cosstet for : () Show that, wthout further assumptos, u t s serally correlated. Costruct a example wth serally correlated u t. (v) A 994 paper the Joural of Ecoometrcs leads wth the statemet: It s well kow that lear regresso models wth lagged depedet varables, ordary least squares (OLS) estmators are cosstet f the errors are autocorrelated. Ths statemet, or a slght varato of t, appears vrtually all ecoometrcs textbooks. Recocle ths statemet wth your dgs from parts () ad (). Ths problem closely follows J.M. Wooldrdge (998) Cosstecy of OLS the Presece of Lagged Depedet Varable ad Serally Correlated Errors. Ecoometrc Theory 4, Problem LINEAR REGRESSION AND OLS 7

28 5.3 Estmato of lear combato Suppose oe has a radom sample of observatos from the lear regresso model y = + x + z + e; where e has mea zero ad varace ad s depedet of (x; z) :. What s the codtoal varace of the best lear codtoally (o the x ad z samples) ubased estmator ^ of where c x ad c z are some gve costats? = + c x + c z ;. Obta the lmtg dstrbuto of p ^ : Wrte your aswer as a fucto of the meas, varaces ad correlatos of x, z ad e ad of the costats ; ; ; c x ; c z ; assumg that all momets are te. 3. For whch value of the correlato coe cet betwee x ad z s the asymptotc varace mmzed for gve varaces of e ad x? 4. Dscuss the relatoshp of the result of part 3 wth the problem of multcollearty. 5.4 Icomplete regresso Cosder the lear regresso y = x + e; E [ejx] = ; E e jx = ; where x s k : Suppose that some compoet of the error e s observable, so that e = z + ; where z s a k vector of observables such that E [jz] = ad E [xz ] 6= : A researcher wats to estmate ad ad cosders two alteratves:. Ru the regresso of y o x ad z to d the OLS estmates ^ ad ^ of ad :. Ru the regresso of y o x to get the OLS estmate ^ of, compute the OLS resduals ^e = y x ^ ad ru the regresso of ^e o z to retreve the OLS estmate ^ of : Whch of the two methods would you recommed from the pot of vew of cosstecy of ^ ad ^? For the method(s) that yeld(s) cosstet estmates, d the lmtg dstrbuto of p (^ ) : 8 LINEAR REGRESSION AND OLS

29 5.5 Geerated coe cet Cosder the followg regresso model: y = x + z + u; where ad are scalar ukow parameters, u has zero mea ad ut varace, par (x; z) are depedet of u wth E x = x 6= ; E z = z 6= ; E [xz] = xz 6=. A collecto of trples f(x ; z ; y )g = s a radom sample. Suppose we are gve a estmator ^ of depedet of all u s, ad the lmtg dstrbuto of p (^ ) s N (; ) as! : De e the estmator ^ of as! ^ = x (y ^z ) : x = = Obta the asymptotc dstrbuto of ^ as! : 5.6 OLS olear model Cosder the equato y = ( + x)e, where y ad x are scalar observables, e s uobservable. Let E [ejx] = ad V [ejx] =. How would you estmate (; ) by OLS? How would you costruct stadard errors? 5.7 Log ad short regressos Take the true model y = x + x + e, E [ejx ; x ] = ; ad assume radom samplg. Suppose that s estmated by regressg y o x oly. Fd the probablty lmt of ths estmator. What are the codtos whe t s cosstet for? 5.8 Rdge regresso I the stadard lear mea regresso model, oe estmates k parameter by ~ = X X + I k X Y; where > s a xed scalar, I k s a k k detty matrx, X s k ad Y s matrces of data.. Fd E[ ~ jx ]. Is ~ codtoally ubased? Is t ubased?. Fd the probablty lmt of ~ as!. Is ~ cosstet? 3. Fd the asymptotc dstrbuto of ~. 4. From your vewpot, why may oe wat to use ~ stead of the OLS estmator ^? Gve codtos uder whch ~ s preferable to ^ accordg to your crtero, ad vce versa. GENERATED COEFFICIENT 9

30 5.9 Icosstecy uder alteratve Suppose that y = + x + u; where u s dstrbuted N (; ) depedetly of x: The varable x s uobserved. Istead we observe z = x + v; where v s dstrbuted N (; ) depedetly of x ad u: Gve a sample of sze ; t s proposed to ru the lear regresso of y o z ad use a covetoal t-test to test the ull hypothess = : Crtcally evaluate ths proposal. 5. Returs to schoolg A researcher presets hs research o returs to schoolg at a luchtme semar. He rus OLS, usg a radom sample of dvduals, o a Mcer-type lear regresso, where the left sde varable s a logarthm of hourly wage rate. The results are show the table. Regressor Pot estmate Stadard error t-statstc Costat :3 :6 :4 Male (g) :39 :7 5:54 Age (a) :4 : :6 Experece (e) :9 :36 :5 Completed schoolg (s) :7 :3 :5 Ablty (f) :8 :4 :65 Schoolg-ablty teracto (sf) : :4 :6. The preseter says: Our model yelds a :7 percetage pot retur per addtoal year of schoolg (I allow returs to schoolg to vary by ablty by troducg ablty-schoolg teracto, but the correspodg estmate s essetally zero). At the same tme, the estmated coe cet o ablty s 8: (although t s statstcally sg cat). Ths mples that oe would have to acqure three addtoal years of educato to compesate for oe stadard devato lower ate ablty terms of labor market returs. A perso from the audece argues: So you have just dvded oe sg cat estmate by aother sg cat estmate. Ths s lke dvdg zero by zero. You ca get ay aswer by dvdg zero by zero, so your umber 3 s as good as ay other umber. How would you professoally respod to ths argumet?. Aother perso from the audece argues: Your dummy varable Male eters the regresso oly as a separate varable, so the geder ueces oly the tercept. But the correspodg estmate s statstcally very sg cat ( fact, t s the oly sg cat varable your regresso). Ths makes me thk that t must eter the regresso also teractos wth the other varables. If I were you, I would ru two regressos, oe for males ad oe for females, ad test for d ereces coe cets across the two usg a sup-wald test. I ay case, I would compute bootstrap stadard errors to replace your asymptotc stadard errors hopg that most of parameters would become statstcally sg cat wth more precse stadard errors. How would you professoally respod to these argumets? 3 LINEAR REGRESSION AND OLS

31 6. HETEROSKEDASTICITY AND GLS 6. Codtoal varace estmato Ecoometrca A clams: I a IID cotext, to ru OLS ad GLS I do t eed to kow the skedastc fucto. See, I ca estmate the codtoal varace matrx of the error vector by ^ = dag ^e = ; where ^e for = ; : : : ; are OLS resduals. Whe I ru OLS, I ca estmate the varace matrx by (X X ) X ^X (X X ) ; whe I ru feasble GLS, I use the formula = (X ^ X ) X ^ Y: Ecoometca B argues: That a t rght. I both cases you are usg oly oe observato, ^e, to estmate the value of the skedastc fucto, (x ): Hece, your estmates wll be cosstet ad ferece wrog. Resolve ths dspute. 6. Expoetal heteroskedastcty Let y be scalar ad x be k vector radom varables. Observatos (y ; x ) are draw at radom from the populato of (y; x). You are told that E [yjx] = x ad that V [yjx] = exp(x + ), wth (; ) ukow. You are asked to estmate.. Propose a estmato method that s asymptotcally equvalet to GLS that would be computable were V [yjx] fully kow.. I what sese s the feasble GLS estmator of part e cet? I whch sese s t e cet? 6.3 OLS ad GLS are detcal Let Y = X ( + v) + U, where X s k, Y ad U are, ad ad v are k. The parameter of terest s. The propertes of (Y; X ; U; v) are: E [UjX ] =, E [vjx ] =, E [UU jx ] = I, E [vv jx ] =, E [Uv jx ] =. Y ad X are observable, whle U ad v are ot.. What are E [YjX ] ad V [YjX ]? Deote the latter by. Is the evromet homo- or heteroskedastc?. Wrte out the OLS ad GLS estmators ^ ad ~ of. Prove that ths model they are detcal. Ht: Frst prove that X b E =, where ^e s the vector of OLS resduals. Next prove that X b E =. The coclude. Alteratvely, use formulae for the verse of a sum of two matrces. The rst method s preferable, beg more ecoometrc. 3. Dscuss bee ts of usg both estmators ths model. HETEROSKEDASTICITY AND GLS 3

32 6.4 OLS ad GLS are equvalet Let us have a regresso wrtte a matrx form: Y = X + U, where X s k, Y ad U are, ad s k. The parameter of terest s. The propertes of u are: E [UjX ] =, E [UU jx ] =. Let t be also kow that X = X for some k k osgular matrx :. Prove that ths model the OLS ad GLS estmators ^ ad ~ of have the same te sample codtoal varace.. Apply ths result to the followg regresso o a costat: y = + u ; where the dsturbaces are equcorrelated, that s, E [u ] =, V [u ] = ad C [u ; u j ] = for 6= j: 6.5 Equcorrelated observatos Suppose x = + u ; where E [u ] = ad E [u u j ] = f = j f 6= j wth ; j = ; : : : ; : Is x = (x + : : : + x ) the best lear ubased estmator of? Ivestgate x for cosstecy. 6.6 Ubasedess of certa FGLS estmators Show that (a) for a radom varable z; f z ad z have the same dstrbuto, the E [z] = ; (b) for a radom vector " ad a vector fucto q (") of "; f " ad ad q ( ") = q (") for all ", the E [q (")] = : " have the same dstrbuto Cosder the lear regresso model wrtte matrx form: Y = X + E; E [EjX ] = ; E EE jx = : Let ^ be a estmate of whch s a fucto of products of least squares resduals,.e. ^ = F (MEE M) = H (EE ) for M = I X (X X ) X : Show that f E ad E have the same codtoal dstrbuto (e.g. f E s codtoally ormal), the the feasble GLS estmator s ubased. ~ F = X ^ X X ^ Y 3 HETEROSKEDASTICITY AND GLS

33 7. VARIANCE ESTIMATION 7. Whte estmator Evaluate the followg clams.. Whe oe suspects heteroskedastcty, oe should use the Whte formula Qxx Q xxe Qxx stead of good old Q xx, sce uder heteroskedastcty the latter does ot make sese, because s d eret for each observato.. Sce for the OLS estmator we have ad ^ = X X X Y E[^jX ] = V[^jX ] = X X X X X X ; we ca estmate the te sample varace by \ V[^jX ] = X X X = x x ^e X X (whch, apart from the factor ; s the same as the Whte estmator of the asymptotc varace) ad costruct t ad Wald statstcs usg t. Thus, we do ot eed asymptotc theory to do OLS estmato ad ferece. 7. HAC estmato uder homoskedastcty We look for a smpl cato of HAC varace estmators uder codtoal homoskedastcty. Suppose that the regressors x t ad left sde varable y t a lear tme seres regresso y t = x t + e t ; E [e t jx t ; x t ; : : :] = are jotly statoary ad ergodc. The error e t s serally correlated of ukow order, but let t be kow that t s codtoally homoskedastc,.e. E [e t e t j jx t ; x t ; : : :] = j s costat (.e. does ot deped o x t ; x t ; : : :) for all j : Develop a Newey West-type HAC estmator of the log-ru varace of x t e t that would take advatage of codtoal homoskedastcty. VARIANCE ESTIMATION 33

34 7.3 Expectatos of Whte ad Newey West estmators IID settg Suppose oe has a radom sample of observatos from the lear codtoally homoskedastc regresso model y = x + e ; E [e jx ] = ; E e jx = : Let ^ be the OLS estmator of, ad let ^V^ ad V^ be the Whte ad Newey West estmators of the asymptotc varace matrx of ^: Fd E[ ^V^jX ] ad E[ V^jX ]; where X s the matrx of stacked regressors for all observatos. 34 VARIANCE ESTIMATION

35 8. NONLINEAR REGRESSION 8. Local ad global det cato Cosder the olear regresso E [yjx] = + x; where 6= ad V [x] 6= : Whch det - cato codto for ( ; ) fals ad whch does ot? 8. Idet cato whe regressor s oradom Suppose we regress y o scalar x, but x s dstrbuted oly at oe pot (that s, Pr fx = ag = for some a). Whe does the det cato codto hold ad whe does t fal f the regresso s lear ad has o tercept? If the regresso s olear? Provde both algebrac ad tutve/graphcal explaatos. 8.3 Cobb Douglas producto fucto Suppose we have a radom sample of rms wth data o output Q; captal K ad labor L; ad wat to estmate the Cobb Douglas producto fucto Q = K L "; where " has the property E ["jk; L] = : Evaluate the followg suggestos of estmato of :. Ru a lear regresso of log Q log L o a costat ad log K log L. For varous values of o a grd, ru a lear regresso of Q o K L wthout a costat, ad select the value of that mmzes a sum of squared OLS errors. 8.4 Expoetal regresso Suppose you have the homoskedastc olear regresso y = exp ( + x) + e; E[ejx] = ; E[e jx] = ad radom sample f(x ; y )g = : Let the true be, ad x be dstrbuted as stadard ormal. Ivestgate the problem for local det ablty, ad derve the asymptotc dstrbuto of the NLLS estmator of (; ): Descrbe a cocetrato method algorthm gvg all formulas (cludg stadard errors that you would use practce) explct forms. NONLINEAR REGRESSION 35

36 8.5 Power regresso Suppose you have the olear regresso y = ( + x ) + e; E[ejx] = ad IID data f(x ; y )g = : How would you test H : = properly? 8.6 Trasto regresso Gve the radom sample f(x ; y )g = ; cosder the olear regresso y = + + e; E[ejx] = : + 3 x. Descrbe how to test usg the t-statstc f the margal uece of x o the codtoal mea of y, evaluated at x = ; equals.. Descrbe how to test usg the Wald statstc that the regresso fucto does ot deped o x. 8.7 Nolear cosumpto fucto Cosder the model E [c t jy t ; y t ; y t 3 ; : : :] = + I fy t > g + y t ; where c t s cosumpto at t ad y t s come at t: The par (c t ; y t ) s cotuously dstrbuted, statoary ad ergodc. The parameter represets a ormal come level, ad s kow. Suppose you are gve a log quarterly seres of legth T o c t ad y t.. Descrbe at least three d eret stuatos whe parameter det cato wll fal.. Descrbe detal how you wll ru the NLLS estmato employg the cocetrato method, cludg costructo of stadard errors for model parameters. 3. Descrbe how you wll test the hypothess H : = agast H a : < (a) by employg the asymptotc approach, (b) by employg the bootstrap approach wthout usg stadard errors. 36 NONLINEAR REGRESSION

37 9. EXTREMUM ESTIMATORS 9. Regresso o costat Cosder the followg model: y = + e; where all varables are scalars. Assume that fy g = s a radom sample, ad E[e] =, E[e ] =, E[e 3 ] = ad E[e 4 ] =. Cosder the followg three estmators of : ^ = arg m b ( ^ = y ; = log b + b ^ 3 = arg m b = y b ) (y b) ; Derve the asymptotc dstrbutos of these three estmators. Whch of them would you prefer most o the asymptotc bass? What s the dea behd each of the three estmators? = : 9. Quadratc regresso Cosder a olear regresso model where we assume: (A) The parameter space s B = ; +. y = ( + x) + u; (B) The error u has propertes E [u] =, V [u] =. (C) The regressor x has s dstrbuted uformly over [; ] depedetly of u. I partcular, ths mples E x = l ad E [x r ] = +r (r+ ) for teger r 6=. A radom sample f(x ; y )g = s avalable. De e two estmators of :. ^ mmzes S () = P = y ( + x ) over B.. ~ mmzes W () = P = y ( + x ) + l ( + x ) over B. For the case =, obta asymptotc dstrbutos of ^ ad ~. Whch oe of the two do you prefer o the asymptotc bass? EXTREMUM ESTIMATORS 37

38 9.3 Nolearty at left had sde A radom sample f(x ; y )g = s avalable for the olear model where the parameters ad are scalars. (y + ) = x + e; E[ejx] = ; E[e jx] = ;. Show that the NLLS estmator of ad ^^ = arg m a;b (y + a) bx = s geeral cosstet. What feature makes the model d er from a olear regresso where the NLLS estmator s cosstet?. Propose a cosstet CMM estmator of ad ad derve ts asymptotc dstrbuto. 9.4 Least fourth powers Suppose y = x + e; where all varables are scalars, x ad e are depedet, ad the dstrbuto of e s symmetrc aroud. For a radom sample f(x ; y )g = ; cosder the followg extremum estmator of : ^ = arg m (y bx ) 4 : b = Derve the asymptotc propertes of ^; payg specal atteto to the det cato codto. Compare ths estmator wth the OLS estmator terms of asymptotc e cecy for the case whe x ad e are ormally dstrbuted. 9.5 Asymmetrc loss Suppose that f(x ; y )g = s a radom sample from a populato satsfyg y = + x + e; where e s depedet of x; a k vector. Suppose also that all momets of x ad e are te ad that E [xx ] s osgular. Suppose that ^ ad ^ are de ed to be the values of ad that mmze y x over some set R k+ ; where for some < < u 3 f u ; (u) = ( )u 3 f u < : = Descrbe the asymptotc behavor of the estmators ^ ad ^ as! : If you eed to make addtoal assumptos be sure to specfy what these are ad why they are eeded. 38 EXTREMUM ESTIMATORS