计量经济学课件08

MultipleRegressionAnalysis

y=b0+b1x1+b2x2+...bkxk+u6.Heteroskedasticity1Economics20-Prof.AndersonWhatisHeteroskedasticity

Recalltheassumptionofhomoskedasticityimpliedthatconditionalontheexplanatoryvariables,thevarianceoftheunobservederror,u,wasconstantIfthisisnottrue,thatisifthevarianceofuisdifferentfordifferentvaluesofthex’s,thentheerrorsareheteroskedasticExample:estimatingreturnstoeducationandabilityisunobservable,andthinkthevarianceinabilitydiffersbyeducationalattainment2Economics20-Prof.Anderson.x

x1x2yf(y|x)ExampleofHeteroskedasticityx3..E(y|x)=b0+b1x3Economics20-Prof.AndersonWhyWorryAboutHeteroskedasticity?

OLSisstillunbiasedandconsistent,evenifwedonotassumehomoskedasticityThestandarderrorsoftheestimatesarebiasedifwehaveheteroskedasticityIfthestandarderrorsarebiased,wecannotusetheusualtstatisticsorFstatisticsorLMstatisticsfordrawinginferences4Economics20-Prof.AndersonVariancewithHeteroskedasticity5Economics20-Prof.AndersonVariancewithHeteroskedasticity6Economics20-Prof.AndersonRobustStandardErrors

Nowthatwehaveaconsistentestimateofthevariance,thesquarerootcanbeusedasastandarderrorforinferenceTypicallycalltheserobuststandarderrorsSometimestheestimatedvarianceiscorrectedfordegreesoffreedombymultiplyingbyn/(n–k–1)Asn→∞it’sallthesame,though7Economics20-Prof.AndersonRobustStandardErrors(cont)

Importanttorememberthattheserobuststandarderrorsonlyhaveasymptoticjustification–withsmallsamplesizeststatisticsformedwithrobuststandarderrorswillnothaveadistributionclosetothet,andinferenceswillnotbecorrectInStata,robuststandarderrorsareeasilyobtainedusingtherobustoptionofreg8Economics20-Prof.AndersonARobustLMStatistic

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

onř1ŭ,ř2ŭ,…,řq

ŭ,withnointerceptTheLMstatisticisn–SSR1,whereSSR1isthesumofsquaredresidualsfromthisfinalregression9Economics20-Prof.AndersonTestingforHeteroskedasticity

EssentiallywanttotestH0:Var(u|x1,x2,…,xk)=s2,whichisequivalenttoH0:E(u2|x1,x2,…,xk)=E(u2)=s2Ifassumetherelationshipbetweenu2andxjwillbelinear,cantestasalinearrestrictionSo,foru2=d0+d1x1+…+dk

xk+v)thismeanstestingH0:d1=d2=…=dk=010Economics20-Prof.AndersonTheBreusch-PaganTest

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

R2)/(n–k–1)],whichisdistributedFk,n–k-1TheLMstatisticisLM=nR2,whichisdistributedc2k11Economics20-Prof.AndersonTheWhiteTest

TheBreusch-PagantestwilldetectanylinearformsofheteroskedasticityTheWhitetestallowsfornonlinearitiesbyusingsquaresandcrossproductsofallthex’sStilljustusinganForLMtotestwhetherallthexj,xj2,andxjxharejointlysignificantThiscangettobeunwieldyprettyquickly12Economics20-Prof.AndersonAlternateformoftheWhitetest

ConsiderthatthefittedvaluesfromOLS,ŷ,areafunctionofallthex’sThus,ŷ2willbeafunctionofthesquaresandcrossproductsandŷandŷ2canproxyforallofthexj,xj2,andxjxh,soRegresstheresidualssquaredonŷandŷ2andusetheR2toformanForLMstatisticNoteonlytestingfor2restrictionsnow13Economics20-Prof.AndersonWeightedLeastSquares

Whileit’salwayspossibletoestimaterobuststandarderrorsforOLSestimates,ifweknowsomethingaboutthespecificformoftheheteroskedasticity,wecanobtainmoreefficientestimatesthanOLSThebasicideaisgoingtobetotransformthemodelintoonethathashomoskedasticerrors–calledweightedleastsquares14Economics20-Prof.AndersonCaseofformbeingknownuptoamultiplicativeconstant

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

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

15Economics20-Prof.AndersonGeneralizedLeastSquares

EstimatingthetransformedequationbyOLSisanexampleofgeneralizedleastsquares(GLS)GLSwillbeBLUEinthiscaseGLSisaweightedleastsquares(WLS)procedurewhereeachsquaredresidualisweightedbytheinverseofVar(ui|xi)16Economics20-Prof.AndersonWeightedLeastSquares

WhileitisintuitivetoseewhyperformingOLSonatransformedequationisappropriate,itcanbetedioustodothetransformationWeightedleastsquaresisawayofgettingthesamething,withoutthetransformationIdeaistominimizetheweightedsumofsquares(weightedby1/hi)17Economics20-Prof.AndersonMoreonWLS

WLSisgreatifweknowwhatVar(ui|xi)lookslikeInmostcases,won’tknowformofheteroskedasticityExamplewheredoisifdataisaggregated,butmodelisindividuallevelWanttoweighteachaggregateobservationbytheinverseofthenumberofindividuals18Economics20-Prof.AndersonFeasibleGLS

Moretypicalisthecasewhereyoudon’tknowtheformoftheheteroskedasticityInthiscase,youneedtoestimateh(xi)Typically,westartwiththeassumptionofafairlyflexiblemodel,suchas

Var(u|x)=s2exp(d0+d1x1+…+dkxk)Sincewedon’tknowthed,mustestimate19Economics20-Prof.AndersonFeasibleGLS(continued)

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

+d1x1+…+dkxk+eWhereE(e)=1andeisindependentofxNow,weknowthatûisanestimateofu,sowecanestimatethisbyOLS20Economics20-Prof.AndersonFeasibleGLS(continued)

Now,anestimateofhisobtainedasĥ