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AHeteroskedasticity-ConsistentCovarianceMatrixEstimatorandaDirectTestforHeteroskedasticityAuthor(s):HalbertWhiteSource:Econometrica,Vol.48,No.4(May,1980),pp.817-838Publishedby:TheEconometricSocietyStableURL::01/04/200907:46YouruseoftheJSTORarchiveindicatesyouracceptanceofJSTOR'sTermsandConditionsofUse,availableat://=econosoc.EachcopyofanypartofaJSTORtransmissionmustcontainthesamecopyrightnoticethatappearsonthescreenorprintedpageofsuchtransmission.JSTORisanot-for-profitorganizationfoundedin1995tobuildtrusteddigitalarchivesforscholarship.Weworkwiththescholarlycommunitytopreservetheirworkandthematerialstheyrelyupon,andtobuildacommonresearchplatformthatpromotesthediscoveryanduseoftheseresources.FormoreinformationaboutJSTOR,pleasecontactsupport@jstor.org.TheEconometricSocietyiscollaboratingwithJSTORtodigitize,preserveandextendaccesstoEconometrica.(e.g.,Griliches[10]),butstillwithoutthecertainknowledgethatanyofthemiscorrect.Inthissituationonecantesteachofthealternativetransformedmodelsforremainingheteroskedasticity(usinganyofseveralavailabletests),andeli-minatethosewhichfail.Butwhatisonetodoifallfailtheheteroskedasticitytest?Althoughtheinvestigatorwillhaveafairlygoodideaoftheparametervaluesofthelinearmodel,thereremainsaconsiderabledifficultyinassessingtheprecisionoftheparameterestimatesandtestinghypothesesduetothepossibleinconsis-tencyoftheusualcovariancematrixestimator.InthispaperIresolvethisdifficultybypresentingacovariancematrixestimatorwhichisconsistentinthepresenceofheteroskedasticity,butdoesnotrelyona(possiblyincorrect)specificformalmodelofthestructureoftheheteroskedasti-city.Thus,evenwhenheteroskedasticitycannotbecompletelyeliminated,properinferencescanbedrawn.Underappropriateconditions,anaturaltestforheteroskedasticitycanbeobtainedbycomparingtheconsistentestimatortotheusualcovariancematrixestimator;intheabsenceofheteroskedasticity,bothestimatorswillbeaboutthesame-otherwise,theywillgenerallydiverge.Thetestsharestheadvantageofthecovarianceestimator,inthatnoformalstructureonthenatureoftheheteroskedasticityisimposed,incontrasttothetestssuggested1IamgreatlyindebtedtoDavidTanny,JonWellner,DennisCarlton,thereferees,andtheparticipantsoftheUniversityofWesternOntarioandHarvard/MITeconometricsworkshopsforhelpfulcommentsandsuggestions.Anyerrorsareminealone.817818HALBERTWHITEbyGoldfeldandQuandt[8],RutemillerandBowers[20],Glejser[6],orHarvey[12].2.THEHETEROSKEDASTICITY-CONSISTENTCOVARIANCEESTIMATORTobegin,assumethatthemodelhasthefollowingstructure:ASSUMPTION1:ThemodelisknowntobeYi=X,lo+(il,.n)where(Xi,E,)isasequenceofindependentnot(necessarily)identicallydis-tributed(i.n.i.d.)randomvectors,suchthatXi(a1xKvector)andEi(ascalar)satisfyE(X!Ei)=0.EiisunobservablewhileYiandXiareobservable./3oisafiniteunknownKx1parametervectortobeestimated.Byassumingthattheelementsofthesequence(Xi,E,)arei.n.i.d.,thecaseoffixedregressorswith(possibly)heteroskedasticerrorsisautomaticallycovered.Alsocoveredbythisassumptionisthecaseinwhichobservationsareobtainednotfromacontrolledexperiment(asthefixedregressorassumptionrequires)butratherfroma(possibly)stratifiedcrosssection,acasefrequentlyencounteredinappliedmicroeconomics.NotethatbyassumingonlythatXiandEiareuncor-related,weautomaticallycoverthelessgeneralbutfrequentlyencounteredcasesinwhichE(EsjXj)=0orXiandEiareind