计量经济学导论电子ch08

IntroductoryEconometrics18.

MultipleRegressionAnalysis:Heteroskedasticity

y=b0+b1x1+b2x2+...bkxk+uIntroductoryEconometrics2WhatisHeteroskedasticity

Recalltheassumptionofhomoskedasticityimpliedthatconditionalontheexplanatoryvariables,thevarianceoftheunobservederror,u,wasconstantIfthisisnottrue,thatisifthevarianceofuisdifferentfordifferentvaluesofthex’s,thentheerrorsareheteroskedasticExample:estimatingreturnstoeducationandabilityisunobservable,andthinkthevarianceinabilitydiffersbyeducationalattainmentIntroductoryEconometrics3of75.Educationlevelprimarysecondaryf(y|x)IllustrationofHeteroskedasticity(wage2.dta)college..E(y|x)=b0+b1xwagehistogramsofwageratesforeacheducationdegree,IntroductoryEconometrics4CheckingtheExistenceofHSK:plottingtheresidualsagainstthefittedvaluesIntroductoryEconometrics5IntroductoryEconometrics6

Whenthereisheteroskedasticity…

OLSisstillunbiasedandconsistent.R-squaredoradjustedR-squaredarestillfinegoodness-of-fitmeasures.

IntroductoryEconometrics7

R-squaredoradjustedR-squaredTheyareestimatesofthepopulationR-squared,1

–[Var(u)/Var(y)],wherethevariancesaretheunconditionalvariancesinthepopulation.TheyconsistentlyestimatethepopulationR-squared,whetherornotVar(u|x)

=Var(y|x)dependsonx.

IntroductoryEconometrics8WhyWorryAboutHeteroskedasticity?ThestandarderrorsoftheestimatesarebiasedifwehaveheteroskedasticityIfthestandarderrorsarebiased,wecannotusetheusualtstatisticsorFstatisticsorLMstatisticsfordrawinginferencesIntroductoryEconometrics9

Whattodo?Econometricianshavelearnedhowtoadjuststandarderrors,t,F,andLMstatisticssothattheyarevalidinthepresenceofheteroskedasticityofunknownform.White(1980)showsthatthevariances,,canbeestimatedinthepresenceofheteroskedasticity.

IntroductoryEconometrics10VariancewithHeteroskedasticityIntroductoryEconometrics11VariancewithHeteroskedasticityIntroductoryEconometrics12of75VariancewithHeteroskedasticityThesquarerootofiscalled:

Heteroskedasticity-robuststandarderror,orWhitestandarderror,orHuberstandarderror,orEickerstandarderrors,orIntroductoryEconometrics13RobustStandardErrors

Nowthatwehaveaconsistentestimateofthevariance,thesquarerootcanbeusedasastandarderrorforinferenceTypicallycalltheserobuststandarderrorsSometimestheestimatedvarianceiscorrectedfordegreesoffreedombymultiplyingbyn/(n–k–1)Asn→∞it’sallthesame,thoughIntroductoryEconometrics14RobustStandardErrors(cont)

Importanttorememberthattheserobuststandarderrorsonlyhaveasymptoticjustification–withsmallsamplesizeststatisticsformedwithrobuststandarderrorswillnothaveadistributionclosetothet,andinferenceswillnotbecorrectInStata,robuststandarderrorsareeasilyobtainedusingtherobustoptionofregIntroductoryEconometrics15of75Example:robustseversususualse

(wage1.dta)IntroductoryEconometrics16IntroductoryEconometrics17IntroductoryEconometrics18

Example:robustseversususualseWhatdowelearn?Robuststandarderrorscanbeeitherlargerorsmallerthantheusualstandarderrors.Butempiricallytherobuststandarderrorsareoftenfoundtobelargerthanthestandarderrors.Ifthedifferencesbetweenthesetwoerrorsarelarge,thentheconclusionsforstatisticalinferencecanbeverydifferent.IntroductoryEconometrics19

Now,whycareabouttheusualse?Giventhatrobuststandarderrorsarevalidwhetherornotheteroskedasticityispresent,thenwhydowestillneedtheusualstandarderror?

NoticethatRobuststandarderrorsarejustifiedonlywhenthesamplesizeislarge.

IntroductoryEconometrics20RobustStandardErrorsWhenthesamplesizeissmallandthehomoskedasticyassumptionactuallyholds,theusualtstatisticshaveexacttdistribution,butthiswillnotbethecaseforrobuststandarderrors,henceinferencesmaynotbecorrect

Whenthesamplesizeislarge,reportingrobuststandarderrors(ortogetherwiththeusualstandarderrors)aremended,esp.inusingcross-sectionaldata.

IntroductoryEconometrics21

Heteroskedasticy(HSK)-robustInferenceafterOLSestimation（tstat.）LetrsedenoteHSK-robuststandarderrors

trse=(estimate-hypothesizedvalue)/(rse)

IntroductoryEconometrics22

Heteroskedasticy(HSK)-robustInferenceafterOLSestimation（Fstat.）TheHSK-RobustFstatisticWithHSKtheusualFstatisticisnolongerFdistributed.TheHSK-RobustFstatisticisalsocalledWaldstatistic HSKStataautomaticallycalculateitafterrobustregressionIntroductoryEconometrics23

Example:comparetheusualandrobustregressions:theusualregressions(birth.dta)IntroductoryEconometrics24

Example:usebirth.dta,FstatisticfortheusualregressionTotestwhetherthevariablemeasuringmother’seducation(motheduc)andwhetherlogfamilye(lfaminc)jointlyhavestatisticallysignificantimpacts,justtypeinSTATAExample:comparetheusualandrobustregressions:thetobustregressions(birth.dta)IntroductoryEconometrics25IntroductoryEconometrics26

Example:usebirth.dta,FstatisticfortherobustregressionFortherobustregression,theFstatisticisnowIntroductoryEconometrics27ARobustLMStatistic

RunOLSontherestrictedmodelandsavetheresidualsŭRegresseachoftheexcludedvariablesonalloftheincludedvariables(qdifferentregressions)andsaveeachsetofresidualsř1,ř2,…,řqRegressavariabledefinedtobe=1

onř1ŭ,ř2ŭ,…,řqŭ,withnointerceptTheLMstatisticisn–SSR1,whereSSR1isthesumofsquaredresidualsfromthisfinalregressionIntroductoryEconometrics28

Example:theLMfortheusualregression(1)

crime1.dtaH0:β2=β3=0H1:β2和β3至少有一个不为0Steps（i）对约束模型进行回归，得到残差（ii）用对无约束模型的所有解释变量进行回归，得到Ru2

IntroductoryEconometrics29IntroductoryEconometrics30IntroductoryEconometrics31of75Example:theLMfortheusualregression(2)

crime1.dta可知Ru2

=0.0013，进而有LM=nRu2=2725×0.0013=3.46Df=2，显著性水平为5%的

2

分布临界值为5.99，显然有LM<5.99，因此不能拒绝H0.IntroductoryEconometrics32

Example:theLMfortherobustregression(1)

crime1.dta从约束模型中得到残差将被排除的2个变量对所有未排除变量回归，保存残差，用r1和r2表示。分别求出与r1和r2的乘积，分别用x1和x2表示用1对x1和x2做不包括截距项的回归IntroductoryEconometrics33

Example:theLMfortherobustregression(2)

crime1.dtaIntroductoryEconometrics34IntroductoryEconometrics35of75Example:theLMfortherobustregression(3)

crime1.dta从而可得到LM统计量为3.997查自由度为2的

2分布5%的显著性水平下临界值为5.99.显然LM<5.99。因此不能拒绝零假设。注意：稳健回归和普通回归的LM检验结果一致IntroductoryEconometrics36

TestingforHSKThoughwehavemethodsofcomputingHSK-robustt,FandLMstatistics,therearestillreasonsforhavingsimpleteststhatcandetectthepresenceofheteroskedasticity.

IntroductoryEconometrics37

TestingforHSKReasonNo.1:WemayprefertoseetheusualOLSstandarderrorsandteststatisticsreportedunlessthereisevidenceofheteroskedasticity.

ReasonNo.2:Ifheteroskedasticityispresent,theOLSestimatorisnolongertheBLUE,thenitispossibletoobtainabetterestimatorthanOLS.IntroductoryEconometrics38TheBreuschnTestforHSK

EssentiallywanttotestH0:Var(u|x1,x2,…,xk)=s2,whichisequivalenttoH0:E(u2|x1,x2,…,xk)=E(u2)=s2Ifassumetherelationshipbetweenu2andxjwillbelinear,cantestasalinearrestrictionSo,foru2=d0+d1x1+…+dkxk+v)thismeanstestingH0:d1=d2=…=dk=0IntroductoryEconometrics39

TheBreuschnTestforHSKUnderthenullhypothesis,itisoftenreasonabletoassumethattheerrorvisindependentofx1,…,xk.TheneitherForLMstatisticsforoverallsignificanceoftheindependentvariablesinexplainingu2canbeusedtotestHSK.Theyareasymptoticallyvalidtestsinceu2isnotnormallydistributedinthesample.

IntroductoryEconometrics40TheBreusch-PaganTest

Don’tobservetheerror,butcanestimateitwiththeresidualsfromtheOLSregressionAfterregressingtheresidualssquaredonallofthex’s,canusetheR2toformanForLMtestTheFstatisticisjustthereportedFstatisticforoverallsignificanceoftheregression,F=[R2/k]/[(1–

R2)/(n–k–1)],whichisdistributedFk,n–k-1TheLMstatisticisLM=nR2,whichisdistributedc2kIntroductoryEconometrics41TheWhiteTest

TheBreusch-PagantestwilldetectanylinearformsofheteroskedasticityTheWhitetestallowsfornonlinearitiesbyusingsquaresandcrossproductsofallthex’sStilljustusinganForLMtotestwhetherallthexj,xj2,andxjxharejointlysignificantIntroductoryEconometrics42

TheWhiteTestforHSKThiscangettobeunwieldyprettyquickly.Forexample,ifwehavethreeexplanatoryvariables,x1,x2,and

x3thentheWhitetestwillhave9restrictions:3onlevels,3onsquares,and3oncross-products.Withsmallsamples,degreesoffreedomwillsoonberunoutwithmoreregressors.

IntroductoryEconometrics43AlternateformoftheWhitetest

ConsiderthatthefittedvaluesfromOLS,ŷ,areafunctionofallthex’sThus,ŷ2willbeafunctionofthesquaresandcrossproductsandŷandŷ2canproxyforallofthexj,xj2,andxjxh,soRegresstheresidualssquaredonŷandŷ2andusetheR2toformanForLMstatisticNoteonlytestingfor2restrictionsnowIntroductoryEconometrics44

B-P检验和White检验的stata命令regyx1x2…xkestathettest（B-P检验）estatimtest,white（white检验）Example8.4,8.5(hprice.dta)IntroductoryEconometrics45IntroductoryEconometrics46IntroductoryEconometrics47

FinalcommentsaboutHSKtestsItispossiblefortheHSKtesttorejectthenullwhenimportantvariablesareomitted,eventhoughthetruthisthereisnoHSK.

HSKcouldindicatemisspecification,therefore,whenpossible,thespecificationtestsshouldbecarriedoutearlierthantheHSKtest.IntroductoryEconometrics48WeightedLeastSquares

Whileit’salwayspossibletoestimaterobuststandarderrorsforOLSestimates,ifweknowsomethingaboutthespecificformoftheheteroskedasticity,wecanobtainmoreefficientestimatesthanOLSThebasicideaisgoingtobetotransformthemodelintoonethathashomoskedasticerrors–calledweightedleastsquaresIntroductoryEconometrics49Caseofformbeingknownuptoamultiplicativeconstant

SupposetheheteroskedasticitycanbemodeledasVar(u|x)=s2h(x),wherethetrickistofigureoutwhath(x)≡

hilookslikeE(ui/√hi|x)=0,becausehiisonlyafunctionofx,andVar(ui/√hi|x)=s2,becauseweknowVar(u|x)=s2hiSo,ifwedividedourwholeequationby√hiwewouldhaveamodelwheretheerrorishomoskedasticIntroductoryEconometrics50GeneralizedLeastSquares

EstimatingthetransformedequationbyOLSisanexampleofgeneralizedleastsquares(GLS)GLSwillbeBLUEinthiscaseGLSisaweightedleastsquares(WLS)procedurewhereeachsquaredresidualisweightedbytheinverseofVar(ui|xi)IntroductoryEconometrics51WeightedLeastSquares

WhileitisintuitivetoseewhyperformingOLSonatransformedequationisappropriate,itcanbetedioustodothetransformationWeightedleastsquaresisawayofgettingthesamething,withoutthetransformationIdeaistominimizetheweightedsumofsquares(weightedby1/hi)IntroductoryEconometrics52

WeightedLeastSquaresIntroductoryEconometrics53MoreonWLS

WLSisgreatifweknowwhatVar(ui|xi)lookslikeInmostcases,won’tknowformofheteroskedasticityExamplewheredoisifdataisaggregated,butmodelisindividuallevelWanttoweighteachaggregateobservationbytheinverseofthenumberofindividualsIntroductoryEconometrics54FeasibleGLS

Moretypicalisthecasewhereyoudon’tknowtheformoftheheteroskedasticityInthiscase,youneedtoestimateh(xi)Typically,westartwiththeassumptionofafairlyflexiblemodel,suchasVar(u|x)=s2exp(d0+d1x1+…+dkxk)Sincewedon’tknowthed,mustestimateIntroductoryEconometrics55FeasibleGLS(continued)

Ourassumptionimpliesthatu2=s2exp(d0+d1x1+…+dkxk)vWhereE(v|x)=1,thenifE(v)=1ln(u2)=a0

+d1x1+