洪永淼计量经济学讲义-ch08

CHAPTER8GENERALIZEDMETHODOFMOMENTSKeywords:CAPM,GMM,Momentmatching,Overidenti…cation,Rationalexpectations.Abstract:Manyeconomictheoriesandhypotheseshaveimplicationsonandonlyonamomentcondition.ApopularmethodtoestimatemodelparameterscontainedinthemomentconditionistheGeneralizedMethodofMoments(GMM).Inthischapter,we…rstprovidesomeeconomicexamplesforthemomentcondition,andde…netheGMMestimator.WethenestablishtheconsistencyandasymptoticnormalityoftheGMMestimator.SincetheasymptoticvarianceofaGMMestimatordependsonthechoiceofaweightingmatrix,weintroduceanasymptoticallyoptimaltwo-stageGMMestimatorwithasuitablechoiceofaweightingmatrix.Withtheconstructionofanasymptoticvarianceestimator,wethenproposeanasymptotically2Waldteststatisticforthehypothesisofinterest,andamodelspeci…cationtestforthemomentcondition.8.1IntroductionontheMethodofMomentsEstimation(MME)Tomotivatethegeneralizedmethodofmoments(GMM)estimation,we…rstconsideratraditionalmethodinstatisticswhichiscalledthemethodofmomentsestimation(MME).Procedures:Supposef(x;0)istheprobabilitydensityfunction(pdf)ortheprobabilitymassfunction(pmf)ofaunivariaterandomvariableXt.Question:Howtoestimatetheunknownparameter0usingarealizationoftherandomsamplefXtgnt=1?Answer:ThebasicideaofMMEistomatchthesamplemomentswiththepopulationmomentsobtainedundertheprobabilitydistributionalmodel.Speci…cally,MMEcanbeimplementedasfollows:Step1:ComputepopulationmomentskE(Xkt)underthemodeldensityf(x;0):Forexample,fork=1;2;wehaveE(Xt)=Z11xf(x;0)dx=(0)E(X2t)=Z11x2f(x;0)dx=2(0)+2(0):1Step2:ComputethesamplemomentsfromtherandomsampleXn=(X1;:::;Xn)0:Forexample,fork=1;2;wehave^m1=Xn!p(0)^m2=n1nXt=1X2t!pE(X2t)=2(0)+2(0);wheretheconvergencefollowsbytheWLLN.Step3Matchthesamplemomentswiththecorrespondingpopulationmomentsevaluatedatsomeparametervalue^:Forexample,fork=1;2;weset^m1=(^);^m2=2(^)+2(^):Step4:Solveforthesystemofequations.Thesolution^iscalledthemethodofmomentestimatorfor0:Remark:Ingeneral,ifisaK1vector,weneedKequationsofmatchingmoments.Question:IsMMEconsistentfor0?Because^mk!pk(0);weexpectthat^!p0asn!1:WenowillustrateMMEbytwosimpleexamples.Example1:SupposetherandomsamplefXtgnt=1i.i.d.EXP():Findanestimatorforusingthemethodofmomentestimation.Solution:Inourapplication,=:Becausetheexponentialpdff(x;)=exforx0;itcanbeshownthat()=E(Xt)=Z10xf(x;)dx=Z10xexdx=1:2Ontheotherhand,the…rstsamplemomentisthesamplemean:^m1=Xn:Matchingthesamplemeanwiththepopulationmeanevaluatedat^:^m1=(^)=1^;weobtainthemethodofmomentestimator^=1^m1=1Xn:Example2:SupposetherandomsamplefXtgnt=1i.i.d.N(;2):FindMMEfor0=(;2)0:Solution:The…rsttwopopulationmomentsareE(Xt)=;E(X2t)=2+2:The…rsttwosamplemomentsare^m1=Xn;^m2=1nnXt=1X2t:Matchingthe…rsttwomoments,wehaveXn=^;1nnXt=1X2t=^2+^2:ItfollowsthattheMME^=Xn;^2=1nnXt=1X2tX2n=1nnXt=1(XtXn)2:Itiswell-knownthat^!pand^2!p2asn!1:38.2GeneralizedMethodofMomentsSupposeisaK1unknownparametervector,andthereexistsal1momentfunctionmt()suchthatE[mt(0)]=0;wheresub-indextdenotesthatmt()isafunctionofbothandsomerandomvariablesrelatedtoindext.Forexample,wemayhavemt()=Xt(YtX0t)intheOLSestimation,ormt()=Zt(YtX0t)inthe2SLSestimation.Ifl=K;themodelE[mt(0)]=0iscalledexactlyidenti…ed.IflK;themodeliscalledoveridenti…ed.ThemomentconditionE[mt(0)]=0mayfollowfromeconomicand…nancialtheory(e.g.rationalexpectationsandcorrectdynamicassetpricing).Wenowillustratethisbythefollowingexample.Example[HansenandSingleton(1982,Econometrica)CapitalAssetPricingModel]:SupposearepresentativeeconomicagenthasaconstantrelativeriskaversionutilityoverhislifetimeU=nXt=0tu(Ct)=nXt=0tCt1;whereu(
)isthetime-invariantutilityfunctionoftheeconomicagentineachtimeperiod(hereweassumeu(c)=(c1)=),istheagent’stimediscountfactor,istheeconomicagent’sriskaversionparameter,andCtistheconsumptionduringperiodt:Lettheinformationavailabletotheagentattimet1berepresentedbythesigma-algebraIt1–inthesensethatanyvariablewhosevalueisknownattimet1ispresumedtobeIt1-measurable,andletRt=PtPt1=1+PtPt1Pt1bethegrossreturntoanassetacquiredattimet1atthepriceofPt1(weassumenodividendontheasset).Theagent’soptimizationproblemistomaxfCtgE(U)4subjecttotheintertemporalbudgetconstraintCt+Ptqt=Yt+Ptqt1;whereqtisthequantityoftheassetpurchasedattimetandWtistheagent’slaborincomeduringperiodt.De…nethemarginalrateofintertemporalsubstitutionMRSt=@u(Ct)@Ct@u(Ct1)@Ct1=CtCt11:The…rstorderconditionsoftheagentoptimizationproblemarecharacterizedbytheEulerequation:EMRSt(0)RtjIt1=1forsome0:Thatis,themarginalrateofintertemporalsubstitutiondiscountsgrossreturnstounity.Remark:Anydynamicassetpricingmodelisequivalenttoaspeci…cationofMRSt:WemaywritetheEulerequationasfollows:E[fMRStRt1gjIt1]=0:Thus,onemayviewthatfMRStRt1gisageneralizedmodelresidualwhichhastheMDSpropertywhenevaluatedatthetruestructralparameters0=(;)0:Question:Howtoestimatetheunknownparameterinanassetpricingmodel?Moregenerally,howtoestimatefromanylinearornonlineareconometricmodelwhichcanbeformulatedasasetofmomen