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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+
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