EconometricsI

ProfessorWilliamGreene

Notes22.TwoStepMLandGMMEstimationTwostepestimationSetting,fittingamodelwhichcontainsparameterestimatesfromanothermodel.Typicalapplication,insertingapredictionfromonemodelintoanother.Procedures:Howit*sdone.Asymptoticresults:ConsistencyGettinganappropriateestimatoroftheasymptoticcovariancematrixTheMurphy-Topelresult(Nextpage)Application:Equation1:NumberofchildrenEquation2:LaborforceparticipationGMMestimation"MomentEquations11Examples:Fromfamiliarestimationsettings:LeastsquaresInstrumentalvariablesMaximumlikelihoodestimationFrombehavioralmodels:E[(ct-someexpectation)|somesetofvariablesinaninformationset]=0Whatdoesthisimply?TheMethodofMoments.SolvingthemomentequationsExactlyidentifiedcasesOveridentifiedcasesConsistency.Howdoweknowthemethodofmomentsisconsistent?Asymptoticcovariancematrix.Consistentvs.EfficientestimationAweightingmatrixTheminimumdistanceestimatorWhatistheefficientweightingmatrix?Estimatingtheweightingmatrix.TheGeneralizedmethodofmomentsestimator-howitiscomputed.ComputingtheappropriateasymptoticcovariancematrixTestingHypothesesintheGMMframework.Testinghypothesesabouttheparameters:WaldTestAcounterparttothelikelihoodratiotestTestingtheoveridentifyingrestrictionsApplication(Monday)?Datatransformations.Numberofkids,scaleincomevariables*Create;Kids=kl6+k618;income=faminc/10000;Wifeinc=ww*whrs/1000$?Equation1,numberofkids.StandardPoissonfertilitymodel.?Fitequation,collectparametersThetalandcovariancematrixVI?computefittedvalues.?Namelist;Z=one,wa,we,income,wifeinc$Poisson;Lhs=kids;Rhs=Z$Matrix;Thetal=b;VI=VARB$Create;ExpKids=Exp(Z1Thetal)$??Setupprobitlaborforceparticipationmodel?Computeprobitmodelandcollectresults.Gamma=coefficienton?fittednumberofkids.CollectTheta2andV2asabove.Namelist;X=one,wa,we,ha,he,income$Namelist;XF=X,ExpKids$Probit;Lhs=Ifp;Rhs=X,ExpKids$Calc;gamma=b(kreg)$Matrix;Theta2=b;V2=VARB$Create;bxf=Xf1Theta2$*?PoissonloglikelihoodZi(kidSiXZi10i-exp(Ziz0i)-logkidsi!)?Derivativeoftermwithrespectto0is(kidSi-exp(Zi^i))xzi*Create;gl=Kids-ExpKids$*?Probit,logL=2(lfp=0)log①(-P*-yexp(Zi'Oi))? +E(lfp=l)log①(+0'Xi+yexp(Zi。))?Obtainthiswithonetermbyusing2*lfp-1=-1or+1for0or1.?Thisisthederivativewithrespecttotheargumentof①.;g2=(2*lfp-l)*N01(bxf)/Phi((2*lfp-l)*bxf)?ThesearethetermsthatareusedtocomputeRandC.?**Typoinyourtext.Delete(1/n)inC-hatandR-hatonpage135.;vc=g2*g2*gamma*ExpKids;vr=g2*gl$?ComputematrixproductsandreportresultsMatrix;R=Xf1[vr]Z;C=Xf1[vc]Z;Q=C*Vl*Cf-R*Vl*Cf-C*Vl*Rf;V2s=V2+V2*Q*V2;Stat(Theta2fV2s)$+ IPoissonRegressionKIDS |ONE |250 |7 |-367.9654 |KIDS |ONE |250 |7 |-367.9654 |-432.0402 |128.1496 |4 |.0000000 |RsqP=.2720 |RsqD=.3108 | +IWeightingvariableINumberofobservationsIIterationscompletedILoglikelihoodfunctionIRestrictedloglikelihoodIChi-squaredIDegreesoffreedomISignificancelevelIChi-squared= 263.00713IG-squared= 284.19970+ TOC\o"1-5"\h\z+ + + + + + +IVariable | Coefficient | StandardError |b/St.Er.|P[|Z|>z] | Meanof X|+ + + + + + +Constant3.436586649.430037867.991.0000WA6765766674E-01.67769033E-02-9.984.0000WE1957100726E-01.27310844E-01-.717.4736INCOME .6461865674E-01.38825572E-011.664.0960WIFEINC4622239565E-01.14329665E-01-3.226.0013(Note:E+nnorE-nnmeansmultiplyby10to+or--rmpower.)42.92000012.3520002.30625403.1035352TOC\o"1-5"\h\z+ +IBinomialProbit Model |IMaximumLikelihood Estimates |I Dependentvariable LFP |I Weightingvariable ONE |I Numberofobservations 250 |I Iterationscompleted 6 II Loglikelihoodfunction -132.0586 |I Restrictedloglikelihood -168.2529 |I Chi-squared 72.38862 |I Degreesoffreedom 6 II Significancelevel .0000000 |+ ++ + + + + + +IVariable|Coefficient|StandardError|b/St.Er.|P[|Z|>z]|MeanofX|+ + + + + + +IndexfunctionforprobabilityConstant12.884250152.09230216.158.0000WA-.2114374146.37956821E-01-5.570.000042.920000WE.6868897361E-01.55472307E-011.238.215612.352000HA-.1926323530E-01.22665533E-01-.850.395445.024000HE-.1057757370E-01.37641934E-01-.281.778712.536000INCOME.1020796842.79918260E-011.277.20152.3062540EXPKIDS-2.197363852.31891089-6.890.00001.6000000-->Matrix;R=Xf?[vr]Z;C=Xf'[vc]Z;Q=C*V1*C!-R*V1*C'-C*V1*R*;V2s=V2+V2*Q*V2;Stat(Theta2zV2s)$Matrixstatisticalresults:Coefficients=THETA2 Variance=V2S+ + + + + +|Variable|Coefficient|StandardError|b/St.Er.|P[|Z|>z]|+ + + + + +THETA_112.884250152.81455324.578.0000T