Generalized Method of Moments.pdf

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1、GeneralizedMethodofMomentsPhDEconometricsWeiTanHanqingAdvancedInstituteofEconomicsandFinanceRenminUniversityofChinaWeiTan(RUC)GeneralizedMethodofMoments1/1TheMethodofMomentsIntroductionFrameworkforGMMConsistencyandAsymptoticPropertyofGMME¢cientEstimationWeiTan(RUC)General

2、izedMethodofMoments2/1TheMethodofMoments:ExampleofN(;2)yisarandomvariable,y~N(;2)Populationmoments:E(y)=;E(y2)=2+2Samplemoments:1Xn1Xny=^;y2=^2+^2iinnii=1Methodofmomentsestimators:1Xn1Xn1Xn^2=y=y;^2=y2(y2)=(yy)2iiinnnii=1i=1WeiTan(RUC)GeneralizedMethodofM

3、oments3/10TheMethodofMoments:Exampleofyi=xi+"iPopulationexpectation:E["ixik]=0.Samplemoments:Xn1(yixi1^1


xiK^K)xi1=0ni=1Xn1(yixi1^1


xiK^K)xi2=0ni=1


Xn1(yixi1^1


xiK^K)xiK=0ni=1Methodofmomentsestimators:^MM=^OLS.WeiTan(RUC)GeneralizedMethodofMoments4/1TheMe

4、thodofMomentsExample:LinearIVEstimationPopulationexpectation:E("iZik)=0forIVZ1;


;ZK.Samplemoment:Xn1(yixi1^1


xiK^K)Zi1=0ni=1Xn1(yixi1^1


xiK^K)Zi2=0ni=1


Xn1(yixi1^1


xiK^K)ZiK=0ni=1Methodofmomentsestimators:^MM=(Z0X)1(Z0y).WeiTan(RUC)GeneralizedMethodofM

5、oments5/1MaximumLikelihoodEstimationLog-likelihoodfunction:Xn1lnL=lnf(yijxi;1;


;K)ni=1Populationexpectations:@lnLE=0;k=1;


;K@kSamplemoments:Xn1@lnf(yijxi;1;


;K)=0n@Ki=1Solution:KnonlinearequationsinKunknowns.WeiTan(RUC)GeneralizedMethodofMoments6/1Identi…cat

6、ionExactlyidenti…edcase:KpopulationmomentequationsinKunknownparameters.Ourfamiliarcases,OLS,IVML,theMOMestimators.Overidenti…edcase.Instrumentalvariables.Thecovariancestructuremodel.Underidenti…edcaseMulticollinearity.Varianceparameterinaprobitmodel.WeiTan(RUC)Generalized

7、MethodofMoments7/1Identi…cation:Overidenti…cationPopulationexpectation:E("iZiM)=0forIVZ1;


;ZM,M>K.Samplemoment:thereareM>Kmomentsequations,morethannecessary.Xn1(yixi1^1


xiK^K)Zi1=0ni=1Xn1(yixi1^1


xiK^K)Zi2=0ni=1


Xn1(yixi1^1


xiK^K)ZiM=0ni=1Alinearsystemo

8、fMequationsinKunknowns.WeiTan(RUC)GeneralizedMethodofMoments8/1Identi…cation:Overidenti…cationTw