EconometricsI

ProfessorWilliamGreene

Notes23.TheGeneralizedMethodofMomentsLEssentialResultsfortheMethodofMomentsEstimatorMomentEquation:Definesasamplestatisticthatmimicsapopulationexpectation:Thepeculationexpectation:E[mi(P)]=0.Subscriptiindicatesitdependsondatavectorindexedby'i'(ortforatimeseriessetting)Usenonlinearinstrumentalvariablesregressionasanexample:ThereareKparameters,0yi=f(xi,p)+&?ThereexistsasetofKinstrumentalvariables,ZisuchthatE[zj=0.Thesamplecounterpartisthemomentequation(l/n)SiZi8i=(l/n)ZiZi[yi-f(xi,p)]=(l/n)Zimi(p)=m(p)=0.Themethodofmomentsestimatoristhesolutiontothemomentequation(s).(Howthesolutionisobtainedisnotalwaysobvious,andvariesfromproblemtoproblem.ThisonecanbesolvedbysolvingtheoptimizationproblemP=thesolutiontoMinwrtp[m(P)]7m(p)]=(l/n2)[8,ZZr8]F.O.C.=(2/n2)G'ZZ*ewhereGisthenxKmatrixofderivativesoftheregressionfunctions.Notethatthesolutionhas2(l/n)G*Zx(l/n)Z,8whichiswhatwelookedfor.Example:Linearregression,f(xi,p)=x/p,thenG=X,asusual.Varianceofthemethodofmomentsestimator,basedontheSlutskyTheoremandtheDeltaMethod:LetV=Thecovariancematrixofm(P).Thisisasamplemean,sousually,V=(l/n)QforsomematrixQ.(Statedwithoutproof).ThecovariancematrixoftheestimatorisVar[bMM]=(G'1)V(G-1)'whereG=dm(p)/3prForthenonlinearleastsquares,IVproblem,supposeVarfgjJ=a2mi=Zi￡j,sothevarianceofm,=Sziz]Withindependentobservations,Var[m(P)]=(l/n2)Sin2ZiZir=o2/n2(ZrZ).Thederivativematrixisthesumoftheterms,G=(1/n)Si[-ZiXiO]=-(l/n)ZrX°.Combiningterms,then,(G-i)V(G-iy=[-(l/n)^0]-1{n2/n2(ZrZ)}[-(l/n)ZrX°]lr=o2(ZrX0)-1Z/Z(Z/X°)-p.Forthelinearregressionfunction,thisistheformulawehadbeforefortheIVestimator.Properties?Consistent,asymptoticallynormallydistributed,notnecessarilyefficient.WhatiftherightformulaforVisnotknown?UseS=(l/n)SimiForthenonlinearleastsquaresexample,mi=Ziei,sowewoulduseS=(1/n)Ziei2ZiZi'=(1/n)ZrE2ZjustlinetheWhiteestimator.Generalizingthemethodofmomentstooveridentifiedproblems:Whatiftherearemoremomentequationsthanparameters?A.istheeasierone.Theparametersareoveridentified:SupposethereareL>Kmomentsm(P)=0isL>KequationsinKunknowns.Assumingtheequationsareindependent,thereisnosolutionwhichsatisfiesallofthem,soweneedacriterion.Wecoulduseleastsquares,aswedidabove:p=thesolutiontoMinwrtpfin(p)]r[m(p)]=(l/n2)[8zZZz8]Intheexactlyidentifiedcase,theminimumvalueofthisfunctioniszero.Now,itispositive.SameF.O.C:(2/n)G^mCp)]=0.Momentsarenotexactlyequaltozero.Theyarenoworthogonaltothederivatives,G=6[m(0)]/60‘(LxKmatrix).NonlinearLeastsquaresexample.Supposetherearemoreinstrumentalvariablesthanparameters.FOCis-(2/n)(X°'Z)Z生=0.ThisisaKxLtimesanLx1.Thesolutioniszero,butZ#8isnotequalto0.Howtosolve?It'sanoptimizationproblem.Dependsonthemodel.Asymptoticcovariancematrix:(Again,withoutproof)Asy.Var[bMM]=[GrV-1G]-1.NoteifGissquare,thisistheresultabove.Here,Gisnotsquare.ItisLxKwhereL>K.HowtoestimateE?Sameasbefore.Aretherebetterwaystousethedata?Isleastsquaresthebestwaytosolvethemomentequations?Theorem:(Statedwithoutproof)—TheMinimumDistanceEstimatorsLetAbeanypositivedefinitematrix.ThesolutiontothecriterionfunctionMinimizewrt0[m(p)]rA[m(p)]isconsistentandasymptoticallynormallydistributed.ThisisaGMMestimatorTheorem:(Again,withoutproof)TheasymptoticcovariancematrixisAsy.Var[bGMM]=[G(對AG(p)]1[G(P)ZAVAG(P)][G(pyAG(P)]1AresomeA'sbetterthanothers?(Obviously)IsthereabestA?Yes,ifA=V-1,theprecedingisminimizedintheclassofMinimumDistanceestimators?ThisistheGMMestimatorA