ln13 Generalized Method of Moments Estimation.pdf

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1、ApEc8212EconometricAnalysisII--Lecture#13GeneralizedMethodofMomentsEstimationReadings:Wooldridge,Chapter14(Sections1-3,6)Almostallestimationmethodsineconometricscanbereinterpretedas“generalizedmethodofmoments”(GMM)estimators.Sincethe1980s,econometricianshaveseenhowusin

2、gthisframeworkleadstonewkindsofestimators,andclarifiestherelationshipsbetweenexistingestimators.Thisisparticularlyusefulforappliedworkusingnonlinearmodelsandtimesseriesestimation.RecallthatGMMforalinearsystemofequationswasdiscussedinLecture7.I.MethodofMomentsEstimation

3、BeforelookatGMM,itisusefultoreviewwhatitgeneralizes,whichismethodofmomentsestimation.Supposewehavesomevariableywhosedistribution(density)isdeterminedbyKparameters,denotedby:f(y)=f(y

4、)=f(y

5、1,2,…K)Wewanttoestimatetheparametersbasedonasampleofnobservationsofy,wheree

6、achobservationfromthesampleisindependentof(and1identicallydistributedas)alloftheotherobservations(noserialcorrelationorheteroscedasticity).Averygeneralmethodforobtainingestimatesistocalculatek“moments”ofybasedonthesampledata.Thesampleestimateofthekthuncentered(“raw”)mo

7、mentis:1nkmk=yik=1,2,…Kni1Definekastheexpectationofthesampleestimate:kkE[mk]=E[yi]Usetodenotetheexpectationofthefirstuncenteredmoment:=1.Thevarianceofmkisk2Var[mk]=(1/n)Var[yi]=(1/n)[2k-(k)]Thefirstequalityfollowsfromtheindependenceofthesampleobserva

8、tions,andthesecondfollowsfromthedefinitionofvariance:foranyvariablex,Var[x]222≡E[(x-E[x])]=E[x]-(E[x]).Thesamplemomentmkisaconsistentestimateofk,anditisalsoasymptoticallynormallydistributed:kplim[mk]=k=E[yi]2d2n(mk-k)N[0,2k-(k)]Theimportantthingisthatweca

9、nusemomentstoestimateparametersofthedistributionofavariableifweassumeaspecificfunctionalformofthedistributionofthatvariable.Forexample,supposeyisnormallydistributed.Thenthedistributionofyiscompletelydeterminedbytwoparameters,and2.Supposewehaveestimatesofthe2uncentere

10、dmoments,m1andm2E[m1]=,soourestimateofk2ism1.SinceVar[yi]=[2k-(k)],wecan22estimateofasm2-(m1).Thisestima