Control-Function-Approach

Imbens/Wooldridge,CemmapLectureNotes14,June’09NewDevelopmentsinEconometricsCemmap,UCL,June2009Lecture14,Thursday,June18th,10:45-11;45ControlFunctionandRelatedMethodsThesenotesreviewthecontrolfunctionapproachtohandlingendogeneityinmodelslinearinparameters,anddrawscomparisonswithstandardmethodssuchas2SLSandmaximumlikelihoodmethods.CertainnonlinearmodelswithendogenousexplanatoryvariablesaremosteasilyestimatedusingtheCFmethod,andtherecentfocusonaveragemarginaleffectssuggestssomesimple,flexiblestrategies.Recentadvancesinsemiparametricandnonparametriccontrolfunctionmethodarecovered,andanexampleforhowonecanapplyCFmethodstononlinearpaneldatamodelsisprovided.1.Linear-in-ParametersModels:IVversusControlFunctionsMostmodelsthatarelinearinparametersareestimatedusingstandardinstrumentalvariablesmethods–eithertwostageleastsquares(2SLS)orgeneralizedmethodofmoments(GMM).Analternative,thecontrolfunction(CF)approach,reliesonthesamekindsofidentificationconditions.Inthestandardcasewhereaendogenousexplanatoryvariablesappearlinearly,theCFapproachleadstotheusual2SLSestimator.Buttherearedifferencesformodelsnonlinearinendogenousvariableseveniftheyarelinearinparameters.And,formodelsnonlinearinparameters,theCFapproachofferssomedistinctadvantages.ToillustratetheCFapproach,lety1denotetheresponsevariable,y2theendogenousexplanatoryvariable(ascalarforsimplicity),andzthe1Lvectorofexogenousvariables(whichincludesunityasitsfirstelement).Considerthemodely1z111y2u1,(1.1)1Imbens/Wooldridge,CemmapLectureNotes14,June’09wherez1isa1L1strictsubvectorofzthatalsoincludesaconstant.ThesenseinwhichzisexogenousisgivenbytheLorthogonality(zerocovariance)conditionsEz′u10.(1.2)Ofcourse,thisisthesameexogeneityconditionthatweuseforconsistencyofthe2SLSestimator,andwecanconsistentlyestimate1and1by2SLSunder(1.2)andtherankcondition,whichreducestorankEz′x1K1,wherex1z1,y2isa1K1vector.(WealsoneedtoassumeEz′zisnonsingular,butthisassumptionisrarelyaconcern.)Justaswith2SLS,thereducedformofy2–thatis,thelinearprojectionofy2ontotheexogenousvariables–playsacriticalrole.Writethereducedformwithanerrortermasy2z2v2Ez′v20(1.3)(1.4)where2isL1.Endogeneityofy2arisesifandonlyifu1iscorrelatedwithv2.Writethelinearprojectionofu1onv2,inerrorform,asu11v2e1,(1.5)where1Ev2u1/Ev22isthepopulationregressioncoefficient.Bydefinition,Ev2e10,andEz′e10becauseu1andv2arebothuncorrelatedwithz.Plugging(1.5)intoequation(1.1)givesy1z111y21v2e1,(1.6)wherewenowviewv2asanexplanatoryvariableintheequation.Asjustnoted,e1,isuncorrelatedwithv2andz.Plus,y2isalinearfunctionofzandv2,andsoe1isalsouncorrelatedwithy2.Becausee1isuncorrelatedwithz1,y2,andv2,(1.6)suggestsasimpleprocedurefor2Imbens/Wooldridge,CemmapLectureNotes14,June’09consistentlyestimating1and1(aswellas1):runtheOLSregressionofy1onz1,y2,andv2usingarandomsample.(Remember,OLSconsistentlyestimatestheparametersinanyequationwheretheerrortermisuncorrelatedwiththerighthandsidevariables.)Theonlyproblemwiththissuggestionisthatwedonotobservev2;itistheerrorinthereducedformequationfory2.Nevertheless,wecanwritev2y2−z2and,becausewecollectdataony2andz,wecanconsistentlyestimate2byOLS.Therefore,wecanreplacev2withv̂2,theOLSresidualsfromthefirst-stageregressionofy2onz.Simplesubstitutiongivesy1z111y21v̂2error,(1.7)where,foreachi,erroriei11zi̂2−2,whichdependsonthesamplingerrorin̂2unless10.Standardresultsontwo-stepestimationimplytheOLSestimatorsfrom(1.7)willbeconsistentfor1,1,and1.TheOLSestimatesfrom(1.7)arecontrolfunctionestimates.Theinclusionoftheresidualsv̂2“controls”fortheendogeneityofy2intheoriginalequation(althoughitdoessowithsamplingerrorbecause̂2≠2).ItisasimpleexerciseinthealgebraofleastsquarestoshowthattheOLSestimatesof1and1from(1.7)areidenticaltothe2SLSestimatesstartingfrom(1.1)andusingzasthevectorofinstruments.[Standarderrorsfrom(1.7)mustadjustforthegeneratedregressor.]Itistrivialtouse(1.7)totestH0:10,astheusualtstatisticisasymptoticallyvalidunderhomoskedasticityVaru1|z,y212underH0;orusetheheteroskedasticity-robustversion(whichdoesnotaccountforthefirst-stageestimationof2).AnestimatorthatcanbedifferentfromtheCFand2SLSestimatorsisthelimitedinformation(quasi-)maximumlikelihood(LIML)estimator.TheLIMLestimatorisobtainedfromequations(1.1)and(1.3)undertheassumptionthatu1,v2isindependentofzwitha3Imbens/Wooldridge,CemmapLectureNotes14,June’09mean-zerobivariatenormaldistribution.Infact,wecanworkoffof(1.3)and(1.6)andusetherelationshipfy1,y2|zfy1|y2,zfy2|z.If12Vare1and22Varv2,thequasi-log-likelihoodforobservationiis−log12/2−yi1−zi11−1yi2−1yi2−zi22/212−log22/2−yi2−zi22/222,(1.8)andallparametersareestimatedsimultaneously.When(1.1)isoveridentified,LIMLisgenerallydifferentfromCF(2SLS).And,astheweakinstrumentsnotesdocument,LIMLtypicallyhasbetterstatisticalpropertiesthan2SLSinsituationswithoveridentification.TheCFapproachcanbeseentobeatwo-stepversionofLIML,where2isobtainedinafirststepandthen1,1,and1areestimatedinasecondstep.(Thevarianceparameterscanb