CONTROLFUNCTIONAPPROACHESINLIMITEDDEPENDENTVARIABLEMODELSByWeiLinADISSERTATIONSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofEconomicsŒDoctorofPhilosophy2016ABSTRACTCONTROLFUNCTIONAPPROACHESINLIMITEDDEPENDENTVARIABLEMODELSByWeiLinThisdissertationconsistsofthreechaptersconcerningapplyingcontrolfunctionapproachestotakingcareofendogeneityofdifferentattributesinbinaryresponsemodels.Thechapterconsidersparametricforbinaryresponsemodelswhilethesecondchapterrelaxessomeofthedistributionalandfunctionalformassumptions.Thethirdchapterprovidesapracticalguidenceonchoosinganinterestingquantityofcausalinterestforbinaryresponsemodels.More,thechapterconsiderslatentvariablemodelsforbinaryresponseandfractionalresponsewithabinaryendogenousexplanatoryvariable(EEV)andpotentiallymanycontinuousendogenousexplanatoryvariables.Atwo-stepcontrolfunction(CF)approachispro-motedtoaccountforendogeneity.TheCFapproachenablesanuncoveringofpartialeffectsofcausalinterest.TheinferenceforthepartialeffectscanbeeasilyobtainedthroughbootstrappingbecauseofthecomputationsimplicityoftheCFapproach.Abasicprobitmodel,aendogenousswitchingprobitmodelandafractionalprobitmodelaredisucssedinthepaper.VariableadditiontestsonthegeneralizedresidualsareusedtodetectadditionalendogeneityfromthebinaryEEV.MonteCarloexperimentsshowthatpartialeffectsobtainedbyinsertinggeneralizedresidualsintobinaryresponsemodelsoutperformscoeffromlinearInfact,theyprovidefairlycloseapproximationstopartialeffectsfromjointestimations.Anempiricalillustrationofthedeterminationofthehousingbudegetshareshowsthat,inafractionalresponsemodel,usinggeneralizedresidualagainleadstoacloseapproximationtojointestimations.Thecoeffromlinearandpartialeffectsfromquasi-MLEarealsocloseinthiscase.Thesecondchapterproposesatwo-stepsieveM-estimationviaacontrolfunctionapproachforaspecialcaseoftriangularsystemsŒabinaryresponsemodelwithbothcontinuousanddummyendogenousexplanatoryvariables.Inastep,residualsareobtainedfrommethodofsievesestimationofreduced-formequationforcontinuousendogenousexplanatoryvariables.Inasec-ondstep,theresiduals(controlfunctions)arepluggedintoerrorsinthebinaryoutcomeequationandareduced-formequationforthedummyendogenousexplanatoryvariable.Functionalformsfortheresidualsenteringtheerrorsareassumedtobeunknown,andthetwoequationsarejointlyestimatedbysievemaximumlikelihoodestimation.Inordertoidentifycausaleffectsofinter-est,estimatorsfrombothstepsareemployedinathird-stepmethodofmomentsestimationoffunctionalscalledaveragepartialeffects(APEs),whichareasmarginaleffectsofanav-eragestructuralfunction.UndertheframeworkofHahn,LiaoandRidder(2015),IestablishpnasymptoticnormalityforAPEsandprovideconsistentestimatorsforasymptoticvariances.Duetotheirnumericalequivalenceresult,Ishowpracticalinferencefortheasymptoticvariance,usingaparametricmodelwiththenumberoftermsinbasisfunctionsedatagivensamplesize.Thus,theproposedtwo-stepsieveM-estimationwithcontrolfunctionapproachesarexible,robusttoeasytoimplement,computationallysimple,andfeasibletoconductpracticalinference.Asimulationstudydemonstratestheseadvantages.Thethirdchaptercomparethreedifferentapproachestoobtainingpartialeffectsinbinaryre-sponsemodels.Amongthethreeapproaches,wemaintainthattheaveragestructuralfunction(ASF)duetoBlundellandPowell(2003,2004)theqmarginaleffectofprimaryinterest,foritisbasedontheunconditionalmarginaldistributionofthestructuralerror.Analyticalexam-plesareprovidedtoshowthattheaverageindexfunction(AIF),proposedrecentlybyLewbel,Dong,andYang(2012),suffersfromessentiallythesameshortcomingsasthepropensityscoreasabasisforaveragepartialeffects.CopyrightbyWEILIN2016ACKNOWLEDGEMENTSIwouldliketoparticularlythankJeffreyM.Wooldridge,thechairofmydissertationcommitte,forhiseffectivementorship.Hiscontinuedadvicehasshapedmyviewofeconometricsandleadstothisdissertation.IalsoappreciatethesupportandguidiancefromKyooilKimandTimVogelsang,thedisser-tationcommitteemembers,thediscussionswithgraduatestudents,andtheadministrativesupportfromstuffmembersofEconomicsdepartment.Theusualdisclaimerapplies.vTABLEOFCONTENTSLISTOFTABLES.......................................viiiLISTOFFIGURES.......................................ixCHAPTER1BINARYANDFRACTIONALRESPONSEMODELSWITHCONTIN-UOUSANDBINARYENDOGENOUSEXPLANATORYVARIABLES(WITHJEFFREYM.WOOLDRIDGE)....................11.1Introduction.......................................11.2ModelandEstimationforBinaryResponse...............41.2.1ProbitModelswithOneBinaryEEVandManyContinuousEEVs.....41.2.2ProbitEndogenousSwitchingModelswithManyContinuousEEVs....91.3TestforEndogeneityfromaBinaryExplanatoryVariable...............141.4Quasi-LIMLandFractionalResponse.........................181.5MonteCarloSimulations................................201.6EmpiricalIllustration..................................261.7Conclusion.......................................30APPENDIX...........................................31BIBLIOGRAPHY........................................42CHAPTER2ACONTROLFUNCTIONAPPROACHINTWO-STEPSIEVEM-ESTIMATIONOFBINARYRESPONSEMODELSWITHENDOGENOUSEXPLANA-TORYVARIABLES..............................452.1Introduction.......................................452.2Model..................................482.2.1Theparametricmodel.............................482.2.2Thesemi-parametric......................512.3Estimation.......................................552.3.1Trueparameters................................552.3.2Sievespaces..................................562.3.3Theestimator.................................592.3.4Afunctionalofcausalinterest........................602.3.5Procedure...................................632.4AsymptoticProperties.................................632.4.1Consistency..................................632.4.2Asymptoticnormality............................652.4.3Consistentvarianceestimation........................682.5PracticalInference...................................702.6SimulationStudy....................................732.6.1Designs....................................732.6.2Results....................................782.7ConclusionandFutureWork..............................81viAPPENDIX...........................................84BIBLIOGRAPHY........................................110CHAPTER3ONDIFFERENTAPPROACHESTOOBTAININGPARTIALEFFECTSINBINARYRESPONSEMODELSWITHENDOGENOUSREGRES-SORS(WITHJEFFREYM.WOOLDRIDGE)................1143.1Introduction.......................................1143.2ALinearModel.....................................1173.3ABinaryResponseModel...............................1193.4Conclusion.......................................122APPENDIX...........................................123BIBLIOGRAPHY........................................129viiLISTOFTABLESTable1.1SimulationResultsforAPEofy2.........................36Table1.2SimulationResultsforAPEofy3.........................37Table1.3SummaryStatisticsoftheEstimationSample(N=2964).............38Table1.4TheFristStageReducedFormRegressionfortheEEVs.............39Table1.5Comparingmarginaleffectsinthestructuralequationofthehousingshare....40Table1.6VariableAdditionTestsforEndogeneity......................41Table2.1SimulationResultsforDesign1..........................107Table2.2SimulationResultsforDesign2..........................108Table2.3SimulationResultsforDesign3..........................109viiiLISTOFFIGURESFigure1.1EmpiricalDistributionofAPEsfory2fortheSampleSizeof1000underJointNormality..................................32Figure1.2EmpiricalDistributionofAPEsfory2fortheSampleSizeof1000underConditionalNormality...............................33Figure1.3EmpiricalDistributionofAPEsfory3fortheSampleSizeof1000underJointNormality..................................34Figure1.4EmpiricalDistributionofAPEsfory3fortheSampleSizeof1000underConditionalNormality...............................35Figure2.1EmpiricalDistributionForDesign1withz1˘Normal(0;9)...........102Figure2.2EmpiricalDistributionForDesign1withz1˘Normal(0;1)...........103Figure2.3EmpiricalDistributionForDesign2withz1˘Normal(0;9)...........104Figure2.4EmpiricalDistributionForDesign2withz1˘Normal(0;16)..........105Figure2.5EmpiricalDistributionForDesign3withz1˘Normal(0;9)...........106ixCHAPTER1BINARYANDFRACTIONALRESPONSEMODELSWITHCONTINUOUSANDBINARYENDOGENOUSEXPLANATORYVARIABLES(WITHJEFFREYM.WOOLDRIDGE)1.1IntroductionBinaryresponsemodelsplayaroleinmanyofempiricalstudies.Examplesincludeeconometricmodelsdeterminingtheprobabilityofmigrationinlaboreconomics(DongandLewbel,2015),thechanceofcollegeadmissionintheeconomicsofeducation(Conlinetal.,2013),andthelikelihoodoftakeoveractivityin(Edmansetal.,2012),tonamejustafew.Inpractice,linearprobabilitymodelsforbinaryresponseareoftenusedbecausetheyareeasytoestimate.However,alinearprojectiondisregardsthelimitednatureofthebinaryresponse,resultinginunrealisticpredictionsoftheresponseprobability.Therefore,coefofthelinearprobabilitymodelsonlyserve,atbest,asapproximationstomarginaleffectsofinterest(HorraceandOaxaca,2006).Latentvariablemodelsforbinaryresponse,ontheotherhand,arewellgroundedoneconomictheories.Thelatentthreshold-crossingstructurecapturesthetriggereffect,andthenonlineartrans-formationensurestheresponseprobabilityfallintotheunitinterval.Thereby,post-estimationquantitiesfromthelatentvariablemodels,suchasaveragepartialeffects(APEs),canbearacausalinterpretation,eveninthepresenceofendogenousexplanatoryvariables(LinandWooldridge,2015).Yetduetothenonlinearity,detectingandsolvingendogeneityissuesintheplaceislessstraightforward,especiallywhensomeofthesuspectedendogenousexplanatoryvariablesarealsodiscrete.ThispaperconsidersestimatingaspecialcaseoflatentvariablemodelsforbinaryresponsewithEEVsofdifferingattributes,whereacontrolfunctionapproachcanbeappliedtomakeiteasiertohandleEEVs.Namely,weallowforonebinaryEEVandpotentiallymanycontinuousEEVs.ThebinaryEEVcouldbeanindicatorofself-selectingintoatreatment,intoaregionorinto1asample,dependingonhowitappearsinthebinaryresponseequation.SomeempiricalexamplesforthebinaryEEVarewhethertoownahouse,whethertosubmitSATscoresinthecollegeapplication,orwhethertobeinthelaborforce.ThecontinuousEEVscouldbeyearsofeducation,familyincome,pricesofsubstitutes,orinputsfortheproduction.TheendogeneityhereismodeledtoariseasomittedvariableproblemsŠtheexistenceofcommonunobservablesaffectingboththebinaryoutcomeandtheEEVs.Toestimatethemodelabove,forsimplicity,wetakeasimpleparametricapproachbyassumingjointnormalityamongerrorterms,followingHeckman(1978)andAmemiya(1978)'streatmentofbinaryEEVandRiversandVuong(1988)'sofcontinuousEEV.Thisdistributionalassump-tion,besidesenablingcontrolfunctionapproaches,ineffectallowsforajointprocedure,suchasalimitedinformationmaximumlikelihood(LIML)estimation,oraquasi-limitedinformationmax-imumlikelihood(quasi-LIML)estimationifanyofthedistributionismisspHowever,thiskindofjointestimationsisrarelyconductedinpractice,duetothedifinsearchingforanu-mericalsolutionalongsomanydimensions(atleastthreeinthiscase).Onemightalsobetemptedtomimicthetwo-stageleastsquares,substitutingvaluesfromaestimationfortheEEVsinthebinaryresponseequation.Althoughindeedeasytoimplement,itleadstotheso-called"forbiddenregression"coinedbyHausman(1975),whichinturnyieldsinconsistentestimates.Alternatively,drawingonWooldridge(2014),thispaperpromotesatwo-stepcontrolfunction(CF)approachunderthequasi-LIMLframework,whichisnotonlycomputationalsimplebutalsodeliverssensibleestimatorsofAPEsinthepresenceofmultipleEEVsandvarioussourcesofheterogeneity.Tocarryoutthistwo-stepprocedure,residuals,insteadofvalues,fromtheestimationofreducedformsforthecontinuousEEVsarepluggedintothesecond-stagejointestimationofthebinaryoutcomeandbinaryEEV.Routinesincommonlyusedsoftwarecanbeexploited(orslightlytocarryoutthisprocedure.Mostimportantly,duetothecomputationsimplicityoftheCFapproach,bootstrappingcanbeeasilyappliedtoobtaininferenceforfunctionsoftheparameters,suchasAPEs,ratherthanusingcomplicateddeltamethod.Inaddition,asshowninWooldridge(2014),simplevariableadditiontests(VATs)forendo-2geneityareobtainedasby-productsoftheCFapproach.ThispaperextendstheVATsforasingleEEVtoVATsformultipleEEVs.TheVATsarebasedonstandardWaldtestsofthoseplugged-inresidualsobtainedfromthestageestimations.Inparticular,inthepresenceofabinaryEEV,testingongeneralizedresidualsenablesustodeterminewhetherwecanavoidajointestimationinthesecondstage.Further,sinceEEVsareoftencorrelatedwitheachother,conditioningonresid-ualsobtainedfromotherEEVshelpsreducethelikelihoodofdetectingadditionalendogeneityinthebinaryEEV.Anotherfeatureofthetwo-stepprocedure,whichstemsfromWhite(1982),isthatthequasi-LIMLframework.Theanalysisforbinaryresponsemodelsinthispapereasilycarrythroughtofractionalresponsemodels,asproposedinPapkeandWooldridge(1996).Solongasthecondi-tionalmeanofthefractionalresponseiscorrectlyweconsistentlyestimateparametersintheconditionalmeaneventhoughotherfeaturesofthedistributionareBesidesparametricapproachestoestimatingthistriangularmodelforbinaryresponse,semi-parametricandnonparametricapproachesarealsoavailable.Nevertheless,whilethoseapproachessensiblyrelaxparametricassumptionsinoneaspect,theyinevitablyimposerestrictionsinotherdirections.Forexample,BlundellandPowell(2003)advanceCFapproachestofullynonparamet-ricbinaryresponsemodelswithcontinuousEEVs.Unfortunately,theirassumptionofadditive,independenterrorsrulesoutdiscreteEEVs.DongandLewbel(2015)'sspecialregressormethodallowsforbothcontinuousandbinaryEEVsinsemiparametricbinaryresponsemodels.However,theirmethodrequiresaspecialregressortobeexcludedfromthereducedformsfortheEEVs,andthespecialregressorcannotappearinthestructuralequationinxiblefunctionalforms.Further,asdiscussedinLinandWooldridge(2015),theaverageindexfunctions(AIF),proposedasaba-sisformarginaleffectsforspecialregressormethod,lackacausalinterpretation.SomeotherexistingsemiparametricmethodstoestimatingthismodelarediscussedinmoredetailbyLin(2016).Therestofthepaperisorganizedasfollows.Section1.2startswithabasicmodelwithonebinaryandmanycontinuousEEVs.Thesameargumentsarethenextendedtoanendogenous3switchingmodelwheretheerrortermsandswitchingindicatorareallowedtointeract.Section1.3derivestheVATforendogeneityfromabinaryEEVgivenresidualsfromcontinuousEEVs.Section1.4showstheCFapproachcanbeappliedtofractionalresponsemodels.Section1.5presentsMonteCarlosimulationresultsofempiricaldistributionsofAPEsforbinaryresponsemodels.Section1.6illustratesbyrevisitingthestudyoftheeffectsofpriceandtotalexpenditureonhousingbudgetshareequation.Section1.7concludes.1.2ModelandEstimationforBinaryResponse1.2.1ProbitModelswithOneBinaryEEVandManyContinuousEEVsAsastartingpoint,weassumethattheonlycomplicationarisesfromEEVsofdifferingattributes,withnopresenceofheterogeneityoryet.More,considerasimplemodelforabinaryresponsey1withmanycontinuousEEVsy2andonebinaryEEVy3.Writethemodelrecursivelyinatriangularformasy1=1[x1b+u1>0];(1.1a)y2=zP+v2;(1.1b)y3=1[zd+u3>0]:(1.1c)(1.1a)isastructuralequationthatrepresentsacausalrelationship.(1.1b)and(1.1c)arereducedformsforthecontinuousEEVsy2ofdimension1GandthescalarbinaryEEVy3,respectively.1[]denotestheindicatorfunctionthattakesonavalueofonewhenthestatementinthebracketistrueandzerootherwise.x1isa1K1vectorwhereeachofitselementsisageneralfunctionof(z1;y2;y3),suchaspolynomials,interactions,logarithm,etc.,withx1=(z1;y2;y3)beingthelead-ingcase.z1isa1L1strictsubsetoftheentire1Lvectorofexogenousvariablesz(z1;z2),withLL1+L2andL2G+1.Sinceidentifyingoffnonlinearityoftentimesturnsoutpoorlyinpractice,soweneedatleastoneexcludedinstrumentforthebinaryEEV.Further,thesamerankconditionholdsasintwo-stageleastsquares:rankE(z0z)=L1+L2andrankE(z0x1)=K1.More-4over,letz1includeunityasitselement,whicheffectivelyforcestheerrorterms(u1;v2;v3)tohavezeromeans.PisaLGmatrixofparameters.Inthesimplestcase,whenG=1,y2isascalar.Thissystemofequationsdescribesendogeneityasanomittedvariableproblem.Thestructuralerroru1iscorrelatedwithexplanatoryvariablesy2andy3inthatitcontainsanunobservablethatalsoinerrortermsv2andu3.Writethelinearprojectionsof(u1;u3)onv2inerrorforms:u1=v2q+v1;(1.2a)u3=v2h+v3;(1.2b)whereqEv02v21Ev02u1andhEv02v21Ev02u3aretheG1vectorsofthepopu-lationregressioncoefAconvenientjointnormalityassumptionamong(u1;v2;u3)(Heckman,1978;Amemiya,1978;RiversandVuong,1988)leavesuswithabivariatenormallydistributedvectoroferrors(v1;v3)thatisindependentofv2,(byofalinearprojectionandbyapropertyofmultivariatenormality):D0B@v1v31CA=D0B@v1v3v21CA˘Normal2640B@001CA;0B@1rr11CA375;(1.3)whereD()denotesthedistribution,thevariancesofv1andv3arenormalizedtoone,andrCov(v1;v3)isthecovariance.Infact,ifwearewillingtoassumeastrongenoughexogeneityconditionfortheinstrumentsz,thebivariatedistribution(v1;v3)becomesindependentnotonlyofv2butalsoofzandthusofy2(becausey2isadeterministicfunctionofz;v2):D0B@v1v31CA=D0B@v1v3z;v21CA=D0B@v1v3y2;z;v21CA:(1.4)Giventhedistributionalassumption,wearriveatabivariateprobitmodelthataccountsfor5endogeneityissuesin(1.1)byaddingv2asextraexplanatoryvariables:y1=1[x1b+v2q+v10];(1.5a)y3=1[zd+v2h+v30]:(1.5b)Addingreducedformerrorsto"control"forendogeneityistheessenceofacontrolfunctionapproach.However,sincewecannotobservev2,tooperationalizeit,asimpletwo-stepprocedureproceedsasfollows:1.Estimate(1.1b),thereducedformsfory2,byordinaryleastsquares(OLS),equationbyequation,toobtaintheresidualsbv2=y2zbP.2.Estimate(1.5),thebivariateprobitmodelfory1andy3,jointlybymaximumlikelihoodestimation(MLE),replacingv2withbv2.Sincethereisnoone-to-onemappingbetweenthereducedformerrorv3andthebinaryEEVy3,wecannotobtainaproxyforv3andhencehavetorelyonajointestimationinthesecondstep.Bytheusualconsistencyargumentoftwo-stepM-estimations(see,forexample,Wooldridge,2010,section12.4.1),theresultingcontrolfunctionestimatorbP;bb;bd;bq;bhisconsistentforparametersbythefollowingpopulationproblems.Formally,P=Ez0z1Ez0y2;(1.6)and(b;d;q;h)istheuniquesolutiontomaxb2RK1;d2RL;r2RG;g2RG;r2RE[logP(y1;y3jy2;z;v2)]=Ey1y3logZ¥q3F(d)f(u3)du3+(1y1)y3logZ¥q3[1F(d)]f(u3)du3+y1(1y3)logZq3¥F(d)f(u3)du3+(1y1)(1y3)logZq3¥[1F(d)]f(u3)du3;(1.7)6wheredx1b+v2r+ru3p1r2;(1.8)q3zd+v2g:(1.9)However,asthemagnitudeofbdependsonthenormalizationoftheerrortermsandthusisonlyuptoscale,interpretingbisnotespeciallymeaningful.Instead,theprimarygoalinempiricalstudiesistoexplainmarginaleffectsofavariableofinterestonresponseprobabilities.InthepresenceofEEVs,P(y1=1jx1),theconditionalresponseprobabilityishardlyofanyinterest:itisaffectedbyy2andy3havingcorrelationswiththeomittedvariableintheunobservablesu1.Wemustusecareinconstructingainterestingresponsefunctionforderivingpartialeffects.For-tunately,BlundellandPowell(2003,2004)haveproposedtheaveragestructuralfunction(ASF),whichisintuitivelyappealingandcanbeobtainedviacounterfactualreasoning.IntheASFforthestructuralequation(1.1a),webreakthecorrelationsbyholdingtheobservablesx1asedargumentsandaveragingouttheunobservableui1withoutconditioningonx1:ASF(x1)=Eui1f1[x1b+ui1>0]g;(1.10)wherethesubscriptionui1emphasizesthatitisarandomvariable,andEui1fgistheexpectedvaluewithrespecttoui1.Inthetwo-stepCFprocedureabove,weidentifyparameterscorrespondtotheconditionalnormalityofu1givenv2,namely,u1jv2˘Normal(v2q;1):(1.11)Thus,bytheusuallawofiteratedexpectations,theASFin(1.10)canbeobtainedintwosteps.First,wetreatvi2ased,andthenaveragethemoutasrandomvariables:7ASF(x1)=Evi2nEui1jvi2f1[x1b+vi2q+vi1>0]jvi2go=Evi2fF(x1b+vi2q)g=Z¥¥F(x1b+ui2q)f(ui2)dui2;(1.12)wheref()isthedensityfunctionfortherandomvariablesvi2.Theaveragepartialeffects(APEs)foragivenx1arethenobtainedbytakingderivativesordifferencesof(1.12)APEy2(x1)=by2Z¥¥f(x1b+ui2q)f(ui2)dui2;(1.13a)APEy3(x1)=Z¥¥hFx(1)1b+vi2qFx(0)1b+vi2qf(ui2)dui2;(1.13b)whereby2isthecoefony2andx(1)1denotesexplanatoryvariablesataparticularedvaluewithy3=1andx(0)1denotesthesameedvalueoftheexplanatoryvariablesexceptthaty3=0.ThoseAPEscanbeconsistentlyestimatedbyusingsampleanalogueandinsertingconsistentestimatorsofbbandbqfromthetwo-stepCFapproach:dAPEy2(x1)=‹by2"N1Nåi=1fx1bb+bvi2bq#;(1.14a)dAPEy3(x1)=N1Nåi=1hFx(1)1bb+bvi2bqFx(0)1bb+bvi2bq:(1.14b)ToobtaininferencefortheestimatorsofAPEsasin(1.14a)and(1.14b),analyticalstandarderrorscanbederivedbythedeltamethodandbysettingthetwo-stepcontrolfunctionproblemasone-stepmethodofmomentsproblem.However,becuasealltheproceduresinvolvedintheestimationsarestandardroutines,bootstrapstandarderrorscanbeeasilyobtainedtoaccountforthesamplingerrors.Asshownin(1.13a)and(1.13b),APEsforthebinaryresponsemodelhavetheattractivefeatureofbuilt-inheterogeneityŠ-theydelivervaryingpartialeffectswhenevaluatedatdifferentvaluesof8x1.However,ifoneisinterestedinusingasinglesummarystatisticformarginaleffects,furtherav-eragingacrossx1shouldbeapplied.Ajointaveragingacrossx1;bv2(as"margins"commanddoesinSTATA)iscomputationallyeasierbutbearsadifferentcausalinterpretationfromsequentiallyaveragingoutbv2andx1(NamandWooldridge,2014).Althoughservingasastartingpoint,themodellingstrategyin(1.1)forCFapproachislimitedinseveralways.Onerestrictivefeatureisthatthereducedformerrorv2needstobeindependentoftheexogenousvariablesz.Thus,thelinearfunctionformforconditionalmeanofy2isunrealisticandcanberelaxedtobeanygenericfunctionp()forzasinBlundellandPowell(2003,2004).Moreimportantly,v2hereactsasasufstatistictocontrolforanyendogeneityfromy2inthestructuralerroru1:thatis,y2iscorrelatedwithu1onlythroughv2onitslevelform.However,asshowninMurtazashviliandWooldridge(2015),incaseofmoreheterogeneitysuchasrandomcoeftheunobservableu1cancontainfullinteractionsbetweenv2andz;x1.Besidesin-teractions,allowingforanunknownfunctionh()forv2asinLin(2016),eventhoughdoesnotcompletelymakethedependenceofu1onv2xible,neverthelessaddssomexiblitytothisrestrictiveassumption.1.2.2ProbitEndogenousSwitchingModelswithManyContinuousEEVsAsweareinterestedinmodelingsomeheterogeneitybesidesEEVs,weturntoaprobitswitchingregressionwithEEVs.ThebinaryEEVy3canbeviewedasaswitchingindicator.Inadditiontoshiftinginterceptswheny3appearingbyitselfinthelinearindex,theswitchingcanbemademoregeneral.Interactingy3withalltheobservablesallowsustoswitchingintoregimesofdifferingslopes.Theinteractionbetweeny3andtheunobservableindicatesthetworegimeshavedifferingunobservables.Theswitchingisendogenousbecausey3iscorrelatedwithunobservables.Inthetreatmenteffectframework,y3isthetreatmentindicatorandthetreatmenteffectisheterogenous.9Toseethis,writethemodelasfollows:y1=1[(1y3)x1b0+y3x1b1+(1y3)u0+y3u1>0](1.15a)y2=zP+v2(1.15b)y3=1[zd+u3>0];(1.15c)Underasimilarsetofnotationsandassumptionsasin(1.1),writethelinearprojectionofu1,u0andu3ontothereducedformerrorv2inerrorforms:u0=v2q0+v0(1.16a)u1=v2q1+v1(1.16b)u3=v2h+v3;(1.16c)whereq0Ev02v21Ev02u0,q1Ev02v21Ev02u1andhEv02v21Ev02u3.Then,wemaintainastrongexogeneityassumptionthattheremainingerrortermsv0andv1areindepen-dentofv2andaparametricassumptionthattheyhaveabivariatenormaldistributionwiththeremainingerrortermv3withcovariancer0andr1,respectively:D0B@v0v3v21CA˘Normal2640B@001CA;0B@1r0r011CA375;(1.17a)D0B@v1v3v21CA˘Normal2640B@001CA;0B@1r1r111CA375:(1.17b)Again,assuming(v1;v3)and(v0;v3)areindependentofzleadstoanindependencebetweeny2andthejointdistributionof(v0;v3)and(v1;v3)D0B@v0v31CA=D0B@v0v3z;v21CA=D0B@v0v3y2;z;v21CA;(1.18a)D0B@v1v31CA=D0B@v1v3z;v21CA=D0B@v1v3y2;z;v21CA:(1.18b)10Them,rewritemodel(1.15)inthetreatmentframeworky1=(1y3)y(0)1+y3y(1)1(1.19)y(0)1=1[x1b0+v2q0+v0>0];(1.20)y(1)1=1[x1b1+v2q1+v1>0];(1.21)y3=1[zd+v2h+v3>0];(1.22)wherey(0)1isthepotentialoutcomewhenthetreatmenty3equalszeroandy(1)1isthepotentialout-comewhenthetreatmentisone.Theself-selctionproblemisrepresentedbythenon-zerocorrela-tionbetweenthetreatmentindicatory3andtheunobservablesv0andv1inthepotentialoutcomes.Thosewhoself-selectintotreatmentinherentlyhaveadifferentdistributionofunobservablefromthosewhodonot.Toconsistentlyestimatetheparametersinthismodel,asimplethree-stepcontrolfunctionapproachsplitstheabovemodelintotwoHeckmansampleselectionmodelswithsub-samplesbythetreatmentstatus:1.Usingallobservation,estimate(1.15b),thereducedformsfory2,byordinaryleastsquares(OLS),equationbyequation,toobtaintheresidualsbv2=y2zbP.2.Sincey(1)1isobservedonlywheny3=1,jointlyestimate(1.21)and(1.22),thebinaryoutcomeequationfory(1)1andsampleselectionequationforindicatory3,bymaximumlikelihoodestimation(MLE),replacingv2withbv2,toobtainbb1andbq1.3.Sincey(0)1isobservedonlywheny3=0,jointlyestimate(1.20)and(1.22),thebinaryresponsemodelfory(0)1andsampleselectionequationforindicator1y3,bymaximumlikelihoodestimation(MLE),replacingv2withbv2,toobtainbb0andbq0.Theaboveprocedureisbysplittingtheobjectivefunctionforthesecond-stepestima-tionintotwoparts.11Namely,solvingmaxb02RK1;b12RK1;d2RL;r02RG;r12RG;g2RG;r02R;r12RE[logP(y1;y3jy2;z;v2)]=Ehy1y3logPy(1)1=1;y3=1jz;v2;y2+(1y1)y3logPy(1)1=0;y3=1jz;v2;y2+y1(1y3)logPy(0)1=1;y3=0jz;v2;y2+(1y1)(1y3)logPy(0)1=0;y3=0jz;v2;y2;(1.23)isequivalenttosolvingmaxb12RK1;d2RL;r12RG;g2RG;r12RE[logP(y1;y3jy2;z;v2)]=Ehy(1)1y3logPy(1)1=1;y3=1jz;v2;y2+1y(1)1y3logPy(1)1=0;y3=1jz;v2;y2+(1y3)logP(y3=0jz;v2;y2)];(1.24)andmaxb02RK1;d2RL;r02RG;g2RG;r02RE[logP(y1;y3jy2;z;v2)]=Ehy(0)1(1y3)logPy(0)1=1;y3=0jz;v2;y2+1y(0)1(1y3)logPy(0)1=0;y3=0jz;v2;y2+y3logP(y3=1jz;v2;y2)];(1.25)where12Py(1)1=1;y3=1jz;v2;y2=Z¥q3F(d1)f(u3)du3(1.26)Py(1)1=0;y3=1jz;v2;y2=Z¥q3[1F(d1)]f(u3)du3(1.27)Py(0)1=1;y3=0jz;v2;y2=Zq3¥F(d0)f(u3)du3(1.28)Py(0)1=0;y3=0jz;v2;y2=Zq3¥[1F(d0)]f(u3)du3(1.29)P(y3=1jz;v2;y2)=F(q3)(1.30)P(y3=0jz;v2;y2)=1F(q3)(1.31)d1x1b1+v2r1+r1u3q1r21(1.32)d0x1b0+v2r0+r0u3q1r20(1.33)q3zd+v2g:(1.34)Similarto(1.12),theASFfortheendogenousswitchingmodelisacombinationoftheASFsforthetworegimes:ASF(x1)=Z¥¥[y3F(x1b1+ui2q1)+(1y3)F(x1b0+ui2q0)]f(ui2)dui2:(1.35)APEsforacontinuousEEVy2andbinaryEEVy3areasfollowsrespectively:APEy2(x1)=Z¥¥"by(1)2y3f(x1b1+ui2q1)+by(0)2(1y3)f(x1b0+ui2q0)#f(ui2)dui2;(1.36a)APEy3(x1)=Z¥¥[F(x1b1+ui2q1)F(x1b0+ui2q0)]f(ui2)dui2;(1.36b)whereby(1)2isthecoeffory2in(1.21)andby(0)2isthecoefin(1.20).NoticethatAPEforabinaryexogenousvariablez1isnontrivailyasAPEz1(x1)=Z¥¥ny3hFx(1)1b1+ui2q1Fx(0)1b1+ui2q1+(1y3)hFx(1)1b0+ui2q0Fx(0)1b0+ui2q0;(1.37)13wherex(1)1denotesexplanatoryvariablesataparticularedvaluewithz1=1andx(0)1denotesthesameedvalueoftheexplanatoryvariablesexceptthatnowz1=0.Correspondingly,aconsistentestimatesoftheAPEsisasampleanalogof(1.36a)and(1.36b)withconsistentestimatesfortheparameterspluggedin:dAPEy2(x1)=N1Nåi=1"bby(1)2y3fx1bb1+bvi2bq1+bby(0)2(1y3)fx1bb0+bvi2bq0#;(1.38a)dAPEy3(x1)=N1Nåi=1hFx1bb1+bvi2bq1Fx1bb0+bvi2bq0:(1.38b)Asbefore,insteadofderivingcomplicatedanalyticalformulasforstandarderrorsforestimatesofAPEs,bootstrapstandarderrorcanbeeasilyappliedtoaccountforthesamplingvariationinthegeneratedregressorbv:Despitetheswitchingmodelbringsinadditionalxibilitybyallowingthestructuralerroru(1y3)u0+y3u1todependnotonlyonv2butalsooninteractionsbetweenv2andy3,as-sumingthereducedformsfory2remainunchangedacrosstworegimesisrestrictiveinempiricalapplications.1.3TestforEndogeneityfromaBinaryExplanatoryVariableThissectionfocusesonvariableadditiontestsforadditionalendogeneityfromabinaryexplana-toryvariable,conditioningonbv2,theresidualsfromreducedformsforcontinuousEEVs.Aswehaveseeninthe(1.1)and(1.15),theonlyconsistentapproachtodealwithabinaryEEVistomakedistributionalassumptionandconductajointestimation.Inrealapplication,wealwayswanttoavoidajointMLEestimationduetoitssensitivitytothedistributionalassumptionandcomputationaldifinarrivingatanumericalsolution.Avariableadditiontest(VAT),asproposedinWooldridge(2014),helpsusdeterminewhethersuchajointestimationisnecessarybytestingongeneralizedresidualsbeforeproceedingtoajointestimation.Especiallyifwehave14alreadycontrolledforendogeneityfromothercontinuousEEVsbyconditioningonbv2,thegen-eralizedresidualislesslikelytobecorrelatedwiththeremainingunobservable.ThefollowingshowsthattheVATonthegeneralizedresidualisavalidtestforendogeneityfromabinaryex-planatoryvariablebecauseitisasymptoticallyequivalenttoaLMtestunderthenullhypothesisofnoendogeneity.Moreformally,inthebasicmodel(1.1),weareinterestedintestingthefollowingnullhypoth-esis:H0:r=0.First,webeginbyshowinganinfeasibleLagrangemultiplier(score)testthathastheasymptoticdistributionofc21.Then,weshowtheVATtestofthegeneralizedresidualisasymptoticallyequivalenttotheinfeasibleLMtestandthushasthesameasymptoticc21distri-bution.Letg(b;q)andwi(xi1;vi2).Letedibediin(1.8)evaluatedatr=0andegbetheestimatesofgobtainedfromtherestrictedmodel.Therestrictedmodelisonewherer=0sowetreaty3asexogenousexplanatoryvariable.Letbq3ibeq3iin(1.9)evaluatedattheparametersbd;bhfromareduced-formprobitestimation.UsingthelikelihoodfunctionLiP(yi1;yi3jyi2;zi;vi2)foroneobservation,theLagrangemul-tiplier(LM)statisticplugstheestimatesfromtherestrictedmodelintothescorefromtheunre-strictedmodel:LM= Nåi=1eSi;r!0eA22heV22i1eA22 Nåi=1eSi;r!=N;(1.39)whereeSi;r¶lnLi¶rjg=eg;r=0=yi1FediFedi1Fedifedibgri3eA1N0BB@åNi=1E¶2lnLi¶g¶g0jyi3;yi2;zi;vi2jg=eg;r=0åNi=1E¶2lnLi¶r¶g0jyi3;yi2;zi;vi2jg=eg;r=0åNi=1E¶2lnLi¶g¶rjyi3;yi2;zi;vi2jg=eg;r=0åNi=1E¶2lnLi¶r¶rjyi3;yi2;zi;vi2jg=eg;r=01CCA=1N0BBBBB@åNi=1fedi2Fedi1Fediw0iwiåNi=1fedi2Fedi1Fediw0ibgri3åNi=1fedi2Fedi1Fedibgri3wiåNi=1fedi2Fedi1Fedibgr2i31CCCCCA15eA1=0B@eA11eA12eA21eA221CAeV=eA1eBeA1=0B@eV11eV12eV21eV221CAeB1NNåi=1eSi;reS0i;rbgri3yi3f(bq3i)F(bq3i)(1yi3)f(bq3i)F(bq3i)(1.40)MatrixeAaboveisanestimatoroftheexpectedvalueofthenegativeHessianmatrixthatusestheexpectedHessianform.TheouterproductofscoresorusualHessianformofthematrixcouldbeused.bgri3isaconsistentestimatorofgri3E(vi3jyi3;yi2;zi;vi2)AVATcanbecarriedoutbythefollowingprocedureoftestingongeneralizedresidual:1.UseOLStoestimatethereduced-formequationsforyi2(1.1b)toobtainbvi2:2.Useprobittoestimatetheaugmentedreduced-formforyi3(1.5b),constructbgri3accordingtotheformulain(1.40).3.Augmentthe(1.5a)bybgri3andestimatebyprobit.Usethet-testtoteststatisticalcanceofbgri3Underthenullhypothesisthecoefonbgri3iszero,andsoestimationoftheparametersinbgri3doesnotaffectthepN-asymptoticdistributionoftheteststatistic.Thereisnoneedtoaccountfortheestimationofbgri3whenperformingthetest.Also,asympototically,wehavebvi2p!vi2.ToshowthatthevariableadditiontestisasymptoticallyequivalenttotheLMtest,writethesecond-steploglikelihoodfunctionasLi=F(xi1b+bvi2q+tgri3)yi1[1F(x1b+bvi2q+tgri3)]1yi1:(1.41)Asmentionedabove,weignorethefactthatgr3isestimatedconsistentlyatthestep.The16scorevectorof(1.41)isSi=0B@¶lnLi¶g¶lnLi¶t1CA=yi1F(wig+tgr3)F(wig+tgr3)[1F(wig+tgr3)]f(wig+tgr3)0B@wigri31CA(1.42)Summingthescorevectoroveralliandusingamean-valueexpansionaboutthetrueparametervectorgivesN1=2Nåi=1bSi=N1=2Nåi=1SiApN0B@bggbtt1CA+op(1)=0(1.43)wherebSiisthescorevectorevaluatedattheestimatedparametersbg0;bt0,andAistheexpectedvalueoftheexpectedvalueofthenegativeHessianmatrix.pN0B@bggbtt1CA=A1"N1=2Nåi=1Si#+op(1)(1.44)WhentestingH0:t=0,therobustWaldteststatisticisgivenbyW=(btt)0bV22=N1(btt)=pN(btt)0bV122pN(btt)(1.45)wherebV=bA1bBbA1=0B@bV11bV12bV21bV221CA;(1.46)bB=1NNåi=1eSi;reS0i;r;(1.47)bA=1N0BB@åNi=1f(bpi)2F(bpi)[1F(bpi)]w0iwiåNi=1f(bpi)2F(bpi)[1F(bpi)]w0ibgri3åNi=1f(bpi)2F(bpi)[1F(bpi)]bgri3wiåNi=1f(bpi)2F(bpii)[1F(bpi)]bgr2i31CCA;(1.48)bpi=wibg+btbgr3;(1.49)bA1p!A1=0B@A11A12A21A221CA:(1.50)SotheWaldstatisticcanalsobewrittenasW= Nåi=1Si;t!0A22bV122A22 Nåi=1Si;t!/=N(1.51)17Underthenullofnoselectionbias(t=0;r=0),thescoreandHessianmatricesusedin1.39and1.51arethesamewhenevaluatedatthetrueparametervalues.Whenthenullistrue,btp!0,pN(bgg)andpN(egg)convergeindistribution.Therefore,LMWp!0,sothetestsareasymptoticallyequivalent.Inordertogetanasymptoticp-valueforthetestcanbeobtainedthroughbootstrapping.1.4Quasi-LIMLandFractionalResponseBasedontheliteratureofQuasi-MLE(White,1982),theabovecarrythroughiff1isafractionalresponsewithaconditionalmeanthathappenstohaveaprobitform.Thekeyinsightfromquasi-likelihoodestimationisthatwedonotneedtoknowthetruedistributionoftheentiremodeltoobtainconsistentparameterestimates.Thislikelihoodfunctioncouldalsobeappliedtothecasewherey1isfractionalresponse,aslongaswemodeltheconditionalmeanofy1tohaveaprobitform.WiththeBernoullidistributionbeinginthelinearexponentialfamily,quasi-LIMLwouldidentifyparametersinacorrectlyconditionalmeanregardlessofinotheraspectsofthedistribution.Namely,E(f1jx1;c1)=F(x1b+c1)(1.57a)y2=zP+v2(1.57b)y3=1[zd+u30];(1.57c)wherec1isanomittedvariablethoughttobecorrelatedwithy2andy3.Byassumingc1followsajointnormalitydistributionwithv2andu3,linearprojectionsofc1andu3ontov2havethefollowingerrorform:c1=v2q+a1(1.58a)u3=v2h+v3(1.58b)18whereqEv02v21Ev02c1andhEv02v21Ev02u3:Pluggingthelinearprojections(1.58a)and(1.58b)backto(1.57a)and(1.57c),wehaveaaugmentedequationsfortheconditionalmeanoffandthereducedformfory3:E(f1jx1;v2;a1)=F(x1b+v2q+a1)(1.59a)y3=1[zd+v2h+v30];(1.59b)wherea1istheremainingunobservablefactorthat,afterconditioningonv2,capturestheadditionalendogeneityfromy3throughv3.Again,assumeajointnormalityassumptionbetweena1andv3asD0B@a1v3v21CA˘Normal2640B@001CA;0B@s2arsarsa11CA375;(1.60)wheres2aVar(a1)andristhecovariance.Furtheraveragingouttheunobservablea1,thecondi-tionalmeanofthejointdistributionoff1andy3hastheexactsameformastheprobitmodelwithmanycontinuousEEVsandonebinaryEEVin(1.1).E(f1;y3=1jz;v2;y2)=E(y1;y3=1jz;v2;y2)=P(y1=1;y3=1jz;v2;y2)=Z¥q3F(d)f(u3)du3;(1.61a)E(f1;y3=0jz;v2;y2)=E(y1;y3=0jz;v2;y2)=P(y1=1;y3=0jz;v2;y2)=Zq3¥F(d)f(u3)du3;(1.61b)wheredx1b+v2r+ru3p1+(1r2)s2a;(1.62a)q3zd+v2g:(1.62b)BecausetheBernoulliloglikelihoodbelongstothelinearexponentialfamily,thesolutionfrom19thefollowingmaximizationproblem(b;q):maxb12RK1;d2RL;r12RG;g2RG;r12RE[logP(f1;y3jy2;z;v2)]=Ef1y3logZ¥q3F(d)f(u3)du3+(1f1)y3logZ¥q3[1F(d)]f(u3)du3+f1(1y3)logZq3¥F(d)f(u3)du3+(1f1)(1y3)logZq3¥[1F(d)]f(u3)du3:(1.63)1.5MonteCarloSimulationsInthissection,sixMonteCarloexperimentsareconductedtocomparethesamplebehaviorofdifferentestimatorsforbinaryresponsemodelwithbothcontinuousanddiscreteendogenousexplanatoryvariables(EEVs).ThesixMonteCarloexperimentsfallsintotwodesigns.Inthedesignerrorterms(u1;v2;u3)arejointlynormallydistributed.Intheseconddesign,conditionalonv2,u1andu3areassumedtohavebivariatenormaldistribution.Foreachdesign,threedatageneratingprocesses(DGPs)ŒŒincludingajustcase,anovercase,andaswitchingmodelwithtworegimesŒŒareconsidered.Nineestimatorsarecomparedineachcase,fourestimatorsassumingalinearprobabilitymodelforthebinaryoutcomeandtherestveesti-matorsacknowledgingthenonlinearfunctionalform.APEsaresimulatedforthoseestimatorsthatrespectthenonlinearfunctionalformandarecomparedwithcoeffromlinearestimators.More,inthedesignofjointnormality,theDGPfortheJustIDisy1=1[y2+y3+0:3z1+0:3z2+0:5v2+0:5v3+r1>0]y2=0:1z1+0:2z2+0:1z3+z4+v2(1.64)y3=1[0:2z1+0:1z2+z3+0:1z4+0:5v2+v3>0];20whereu1=0:5v2+v1(1.65)v1=0:5v3+r1(1.66)r1˘Normal(0;0:5)(1.67)sothat0BBBB@u1v2v31CCCCA˘Normal2666640BBBB@0001CCCCA;0BBBB@10:50:50:5100:5011CCCCA377775:(1.68)ThebinaryEEVy2andcontinuousEEVy3aregeneratedtohavecoefofoppositesignsinordertoshowhowbiasedestimatorsreacttosigndifference.Theexogenousvariablesaregeneratedas:z1˘Normal(0;1)e2˘Normal(0;1)z2=1[e2>0]z3˘Normal(0;1)e4˘Normal(0;1)z4=1[e3>0]:wherethecontinuousz3istheinstrumentmainlyforbinaryEEVy3andthebinaryz4istheinstrumentmainlyforcontinuousEEVy2:Tomakethemvalidinstruments,z3andz4areexcludedfromthestructuralequation.InthisDGP,thetrueASFisasASF(x1)=F(y2+y3+0:3z1+0:3z2):(1.69)21Thesecondcaseofoverhasthesameparametersexceptthatwehavetwoaddi-tionalinstrumentsz5andz6,wherez5˘Normal(0;1)e6˘Normal(0;1)z6=1[e6>0]:Continuousz5ismainlyforthecontinuesEEVy2andbinaryz6ismainlyforthebinaryEEVy3:ThetrueASFremainsthesameasin(1.69).Intheendogenousswitchingcase,toemphasizecoefonthecontinuousEEVy2andthecorrelationsbetweenthereducedformerrorsandthestructuralerroraredesignedtohaveoppositedirectionsacrossregimes,namelyy(1)1=1[y2+y3+0:3z1+0:3z2+0:5v2+v1>0]y(0)1=1[0:3y2+y30:5z1+0:1z20:5v2+v0>0](1.70)y2=0:1z1+0:2z2+0:1z3+z4+v2y3=1[0:2z1+0:1z2+z3+0:1z4+0:5v2+v3>0];whereu0=0:5v2+v0(1.71)v0=0:5v3+r1(1.72)0BBBB@u0v2v31CCCCA˘Normal2666640BBBB@0001CCCCA;0BBBB@10:50:50:5100:5011CCCCA377775:(1.73)ASFinthiscaseisASF(x1)=y3F(y2+y3+0:3z1+0:3z2)+(1y3)F(0:3y2+y30:5z1+0:1z2):(1.74)22Indesign2,parametrizationsarethesameasindesign1,butweassumev2followsademeanedc21distributionwithonedegreeoffreedomv2˘c211:(1.75)Inallexperiments,thenumberofreplicationsis1000,andtheresultsoftheexperimentsarepresentedforsamplesizesof1000,3000and5000.Table1.1andTable1.2reportbiasesandtherootmeansquarederrors(RMSEs)forestimatorsofAPEfory2andy3,respectively.Figure1.1andFigure1.2depicttheempiricaldistributionsofestimatorsofAPEfory2withsamplesizeof1000underdesign1anddesign2,respectively.Similarly,Figure1.3andFigure1.4depictthecounterpartfory3.Foreachoftheabovecase,coefoflinearprobabilitymodelsandAPEsofprobitmodelsarecompared.Further,forprobitmodels,jointestimationswiththebinaryEEVorallEEVsarecomparedwithtwo-stepsestimationswithcontrolfunctionterms(residualsorgeneralizedresiduals)pluggedin.Inaddition,aswitchingversionofeachmodelisconsideredtoaccountfortheendongenousswitchingDGPincase3.More,CFBiprobitisthecontrolfunctionapproachinserting-stageresidualfromreduced-formestimationofy2intothesecond-stepjointestimationbetweeny1andy3.CFBiprobitSwitchingperformsHeckmanprobitwithsampleselectionfory(1)1andy(0)1separatelyusingy3asasampleselectionindicator.CFProbitavoidsthejointestimationwithy3byinsertingageneralizedresidualfromy3asaproxyforendogeneitygivenresidualfromy2.CFProbitSwitchingperformstheCFProbitseparatelyforsub-samplesbyy3.CFLinearinsertsaresidualfromy2andageneralizedresidualfromy3intothelinearprobabilitymodelfory1.CFLinearSwitchingallowsforafullsetofinteractionsbetweeny3andotherobservablesandunobservableinthelinearprobabilitymodel.Usual2SLSuseslinearprobabilitymodelsforbothy1andy3andappliestheusualtwo-stepIVestimation.OptimalIVusespredictedvaluesfromreducedformsfory1andy3asinstrumentsforalinearprobabilitymodelofy1.y3ispredictedfromaprobitmodel.JointMLEisafulljointestimationofy1,y2andy3.23FortheAPEofy2underjointnormalityasinFigure1.1,CFBiprobitandJointMLEaretheconsistentestimatorsintheJustIDcaseandOverIDcasewhileCFBiprobitSwitchingistheconsistentestimatorintheSwitchingcase.TheirempiricaldistributionarecenteredaroundthetrueAPEdepictedbytheredverticalline.Besidesthoseconsistentestimators,approximationsprovidedbytheCFProbit(orCFProbitSwitchingintheSwitchingcase)outperform,toagreatextent,theapproximationsprovidedbythelinearprobabilityestimatorssuchasCFLinear(orCFLinearSwitchingintheSwitchingcase),Usual2SLSandOptimalIV.Infact,intheSwitchingcase,CFProbitSwitchingandCFBiprobitswitching(theconsistentestimatorinthiscase)seemtocompletelyoverlapwitheachother,suggestinganegligibleamountofbias.IntheJustIDcaseandOverIDcase,CFProbithasamildamountofupwardbiasandaslightlylowerpeakthanCFBiprobitandJointMLE.Incontrast,approximationsprovidedbylinearprobabilitymodelestima-tors(CFLinear,Usual2SLSOptimalIVandCFLinearSwitching)haveaamountofdownwardbiasinallcases.Thedifferencesinbiaswithinthelinearprobabilitymodelestimatorsarenotnoticeable:theyallseemtoclustertogether.IntheSwitchingcase,theyarejoinedbytheCFBiprobitandJointMLEwhohaveasimilaramountofdownwardbias.WhenCFBiprobitandJointMLEareconsistent,theystillcompletelyoverlapwitheachother.ThishappensnotonlyintheJustIDcasebutalsointheOverIDcase,suggestinganegligibleamountofefciencylossbycarryingoutatwo-stepprocedure.CFBiprobitSwitchingandCFProbitSwitching,however,sufferaslightlypeakcomparedtotheircounterpartnon-switchingestimators(CFBiprobitandCFProbit)intheJustIDcaseandOverIDcase,indicatinganefciencylossfromamorecomplexparametrization.Whentheerrortermsfollowaconditionalnormality,theestimatorsforAPEsfory2havefairlydifferentsamplebehaviorsfromthatunderjointnormality.AsinFigure1.2,JointMLElacksrobustnessandisnolongertheconsistentestimatorinanycase.Asbefore,CFBiprobitistheconsistentestimatorintheJustIDcaseandOverIDcasewhileCFBiprobitSwitchingistheconsistentestimatorintheSwitchingcase.ApproximationsprovidedbytheCFProbit(orCFProbitSwitchingintheSwitchingcase)isstillthebest:italmostoverlapwithCFBiprobit(orCF24BiprobitSwitchingintheSwitchingcase),theconsistentestimator.JointMLEisbiasedupwardstoanoticeabledegreeintheJustIDcaseandOverIDcase.IntheSwitchingcasewhereJointMLEisitisbiaseddownwardandjoinedbyotherinconsistentestimatorslikeCFBiprobitandlinearprobabilityestimators(CFLinear,Usual2SLSandOptimalIV).CFLinearSwitchingperformsmildlybetterthanotherlinearestimatorsintheSwitchingcasebutstillfaroffcomparedtotheconsistentestimator.Overall,linearprobabilityestimatorscontinuestoperformpoorlyinallcases:theyarefarbiaseddownwards.TheCFBiprobitSwitchingandCFProbitSwitchingstillleadstoefylossindicatedbypeaksintheJustIDcaseandOverIDcase.Underjointnormality,APEofy3followsasimilarpatternasthatofy2,withsomeminordifferences.AsinFigure1.3,CFBiprobitandJointMLEstilloverlapwitheachotherinallcases,whetherasconsistentestimatorsintheJustIDcaseandOverIDcase,orasestimatorsintheSwitchingcase.TheapproximationprovidedbyCFProbit(orCFProbitSwitchingintheSwitchingcase)isstillthebestbutwithapeakthanthoseinFigure1.1.Thelinearprobabilityestimatorsarebiasedupwards.ThedifferencesinempiricaldistributionsforthelinearestimatorsaremorepronouncedinthebinaryEEVy3thanforcontinuousEEVy2.Particularly,CFLinearusingthegeneralizedresidualisnolongclosetoUsual2SLSusinglinearprobabilitymodelfory3.IntheSwitchingcase,linearprobabilityestimatorsallliebetweentheCFBiprobitSwitchingandCFBiprobitwithvaryingdegreeofbiasandprecision.APEofy3underconditionalnormalityaresketchedinFigure1.4.LikeestimatorsinFigure1.3,theyhaveidenticalpatternswiththeircounterpartfory2.More,JointMLEisbiaseddownwardsinJustIDcaseandOverIDcasebutbiasedupwardsinSwitchingcase.Similarly,linearprobabilityestimatorsarebiasedupwardsratherthandownwards.CFProbit(orCFProbitSwitchingintheSwitchingcase)stillprovidesthebestapproximationsinallcases,betterthanlinearprobabilityestimators.Table1.1andTable1.2reportsthebiasandRMSEofy2andy3foralltheestimatorsinthesixcases,respective.Despitethedifferenceinsignandmagnitude,thepatternsofestimatorsfor25y2andy3aresimilar.MethodsusingCFapproachesarelistedfromColumn(1)to(6),followedbyconventionalmethodslikeIV2SLS,Opt.IV2SLSandJointMLEfromColumn(7)to(9).Withtheincreaseofsamplesizefrom1000to5000,thebiasofCFBiprobitinJustIDcaseandOverIDcase(orCFBiprobitSwitchingintheSwitchingcase)shrinksdrasticallytozero.TheirRMSEalsodecreasebyabouthalfatthesametime.ThebiasofCFProbitinJustIDcaseandOverIDcase(orCFProbitSwitchingintheSwitchingcase)issmallatsamplesizeof1000butshrinkbyalessamountassamplesizeincreasesto5000.TheRMSEofCFProbitorCFProbitSwitchingalsodecreasebyabouthalfasthesamplesizeincreases.Thebiasofthelinearprobabilityestimators(CF2SLS,CF2SLSSwitching,IV2SLSandOpt.IV2SLS)andJointMLE,however,ishugetostartwithanddoesnotshrinkorevenincreasesinsomecasesasthesamplesizeincrease.TheirRMSEsalsodoesnotdecreaseasmuch.Insummary,theMonteCarloresultsshowthatCFBiprobitdoesnotloseefycomparedtoJointMLEinthecorrectlycase.CFProbit(orCFProbitintheSwitchingcase)providesgoodapproximations,outperforminglinearestimatorsofanysorttoagreatextend.1.6EmpiricalIllustrationAsanempiricalillustration,werevisittheempiricalexampleofMurtazashviliandWooldridge(2015)underdifferentfunctionalformassumptionsandestimationmethods.MurtazashviliandWooldridge(2015)studythesensitivityofthebudgetshareofhousingexpendituretopriceandtotalexpenditureusingalinearprobabilitypaneldatamodelwithmanysourcesofheterogeneity.TotalexpenditureisconsideredtobethecontinuousEEVbecauseofitsjointdeterminationwiththebudgetshareonhousingexpenditure.HomeownershipdummyisconsideredtobethebinaryEEV.Itisalsoassumedtoplaytheroleofanendogenousswitchingindicatorisemployedforthebudgetshareofhousingexpenditureequation.Here,instead,weemployafractionalresponsemodelwithswitching,asin(1.76),thatacknowledgesthefractionalnatureofthebudgeshareand26thereforehasbuilt-inheterogeneity.E(HousingSharejx1;c1;c0)=F[b0+b1Log(Expend.)+b2Homeowner+z1b3+b4Log(Expend.)Homeowner+b5Homeownerz1+c0+Homeownerc1];(1.76a)Log(Expend.)=z0+zz1+v2;(1.76b)Homeowner=1[g0+zg1+u3>0]:(1.76c)Wealsousejustonecross-sectionalperiodfromtheirsample,whichturnsouttogivefairlycloseestimatesofmarginaleffectforthevariablesofinteresttothepanellinearmodelwithmanysourcesofheterogeneity.ThesummaryAPEsfromthenonlinearmodeliscomparedtothecoeffromthecoefinlinearprobabilitypaneldatamodels.Thesampleemployedintheestimationisthe2001waveofthePanelStudyofIncomeDynam-ics(PSID)thatconsistsof2355ownersand629renters.Sincewesuspecthomeownershipdummyindicatesswitchingintodifferingregions,wereportseparatesummarystatisticsfordifferenthomeownershipstatusasinTable1.3.Duetothewaythedependentvariablehousingbudgetshareisconstructedandtheincreaseinthepriceforhomes,84observationsinthesampleofhomeownersfaceanegativehousingbudgetshare.Asthedependentvariablehastobeintheunityintervalforafractionalresponse,thehousingbudgetforthoseobservationsaresettotheirlowerboundzero.Onaverage,ownersspendsmallersharesoftheirbudgetsonhousingthanrenters.Totalexpenditureandincomeofownersarelargerthanthoseofrenters.Logprice,ageofthehouseholdhead,maritalstatus,whetherrecentlymovedandtheracearetheexogenouscontrolvariables.Logincomeisconsideredtobetheinstrumentsmainlyforthelogexpenditurewhereyearsofeducationofthehouseholdheadandnumberofchildreninthehouseholdareinstrumentsmainlyforhomeownership.Table1.4reportsthereduced-formestimationforthetwoEEVs.Linearreducedformregressionsarereportedforlogtotalexpenditure,thecontinuousEEV,inColumn(1)and(2).Probitregressionarereportedforhomeownership,thebinaryEEV,inColumn(3)to(5).27Slightvariationsinthearereportedineachcase.Forexample,agesquaredareincludedinColumn(2)inadditiontoageinitslevelformasinColumn(1),whichturnsouttobebutpracticallyunimportant.Foranytheinstrumentsmentionedabovearestrongenough.TheprobitreducedformsforthehomeownershipdummyarereportedwithandwithoutthecontinuousEEV.PredictedvalueofhomeownershipfromColumn(3)containsonlyexogenousvariableandisusedasaninstrumentinRegression(3)inTable1.5.Columns(4)and(5)showthatincludingtheresidualfromthereducedformofthelogexpenditureissuftocontrolforalltheendogeneityfromthetotalexpenditureinhomeownership.Table1.5comparestheAPEsfromthefractionalresponsemodelstothecoefoflinearmodels.Columns(1)to(4)reportthecoeffromlinearmodelsforthehousingbudgetshare,andColumns(5)to(10)reporttheAPEsfromfractionalresponsemodels.TheestimatorsinthistableisnamedinthesimilarfashiontothatintheMonteCarlostudy,theonlydifferencebeingthatthedependentvariableisafractionalresponse,insteadofbinaryresponse.SoaFracisaddedtothenamestoindicatethataquasi-probitisassumedforhousingbudgetshare.WhenthehomeownershipisjointlyestimatedinthefractionalprobitasinColumn(9)andColumn(10),thebiprobitandheckprobitcommandinStataistoallowforafractionaldependentvariable.ThestandarderrorsfortheestimatesofAPEsarebootstrapped.AswecantellfromTable1.5,ofall,failingtoaccountforendogeneity(whetherasinLinearmodelrepresentedinColumn(1)orasinFracProbitmodelasinColumn(5))leadstofairlydifferentestimatesfromthosemethodsthattakecareofendogeneityusingthesamemodels.Amongthelinearprobabilitymodelsthathaveaccountedforendogeneity,theestimatesfromIV2SLSdifferfromthatOpt.IV2SLSandCF2SLS.BothOpt.IV2SLSandCF2SLSareclosetotheAPEsintheFracProbitmodelsthathaveaccountedforendogeneity.Thissuggeststhattherelationshipbetweenhousingbudgetshareandthecovariatesofinterestmightbeclosetolinearintheunitintervalsothattwo-stageleastsquaresprovidesagoodapproximation.AmongtheFracProbitmodels,theestimatesfromdifferentmethodsofaccountingforendogeneityarefairlycloseacrossboard.Thedifferencebetweenconductingjointestimationswiththehome28ownershipasinColumn(9)andColumn(10)andplugginginthegeneralizedresidualfromhomeownershipasinColumn(7)andColumn(8)issmall.Particularly,ifweusehomeownershipastheswitchingindicator,theestimatesandstandarderrorfromColumn(10)andColumn(8)arethesameatleasttothethirddecimalplace.ThedifferencebetweenColumn(7)andColumn(9)arealsonegligiblysmall.Table1.6reportstheteststatisticsfortestsofendogeneityandtheirp-values.Thep-valuesareobtainedfrombootstrappingtheteststatistics.OnlyestimatorsthatemployCFapproachesareconsideredinthistable.AlltheteststatisticsreportedareWaldtestsforThenamesrefertotheestimatorsasin1.5.Column(1)to(4)reportthevariableadditiontests(VATs)oncon-trolfunctiontermsonly.Column(5)and(6)alsoreportWaldtestsonthecorrelationparameterr(orr0andr1forthetwoswitchingregimes)representingtheendogeneityfromhomeownershipgiventhecontrolfunctiontermfromlogexpenditure.Inanycase,theevidenceofendogeneityfromlogexpenditureisstrong:p-valuesareidenticallyzeroinanytestontheofbv2,thecontrolfunctiontermfromlogexpenditure.Notestbasedonthefractionalresponsemodeltheendogeneityfromhomeownship,whetheritisaVATtestonthegeneralizedresidualoronthecorrelationparameter,althoughtestonthegeneralizedresidualinthelinearmodelCF2SLSturnsouttobeTheteststatisticsandp-valuefromVATtestongeneralizedresid-ualinColumn(3)isquitesimilartotheWaldtestononthecorrelationparameterrinColumn(5),suggestingthevalidityofusingVATongeneralizedresidualtodetecttheadditionalendo-geneityfromhomeownership.LMorLRtestonrcanalsobeperformed,butpropermethodofbootstrappingforp-valueshastobeout.AnotheradvantageofperformingaVATtestongeneralizedresidualisitsrobustnesstomodelInthecaseofswitching,ajointtestconcerningtworegimesneedstobeconductedtodetectendogeneity.Thiscanbeeasilydonebyperformingajointtestoninteractiontermsbetweenthecontrolfunctiontermsandtheswitchingindicator.Findingawaytocombiningcorrelationparametersr0andr1obtainedfromdifferentregimes,however,isnoeasyjob.291.7ConclusionThispaperhaveshowncontrolfunctionapproachesappliedtoaccountforonebinaryEEVandmanycontinuousEEVsinbinaryandfractionalresponsemodels.Thecontrolfunctionapproachiscomputationalsimpleandallowsforaxibleincorporationofheterogeneityasinanendoge-nousswitchingmodel.PartialeffectsbasedontheASFareofcausalinterpretationandcanbeeasilybootstrappedtoobtaininferenceduetothecomputationsimplicityofthecontrolfunctionapproach.AVATtestbasedonthegeneralizedresidualisshowedtobeavalidtestfordetectingadditionalendogeneityfromthebinaryEEVconditioningontheresidualsfromthecontinuousEEVs.Simulationstudyshowsthatusinggeneralizedresidualstoaccountforendogeneitypro-videsafairlygoodapproximationtothetrueAPE,betterthanapproximationspro-videdbylinearprobabilitymodels.ApplyingtheCFapproachtoanempiricalillustrationusingfractionalresponsemodelforthehousingbudgetshare,weshowthatthehomeownershipisnotendogenousafterconditioningonthetotalexpenditure,thecontinuousEEV.ThisisfoundoutbyperformingVATsonthegeneralizedresidualsandcrossvalidatedbyaWaldtestonthecorrela-tionparameter.Theseresultsimpliesthatpluggingingeneralizedresidualintoabinaryresponsemodel(orafractionalresponsemodel)actsasabetterapproximationtothecausalmarginaleffectsofinterestthanotherconventionalmethods.Itisalsocomputationallysimplerandenablesaeasierdetectionofendogeneity.30APPENDIX31APPENDIXFORCHAPTER1A.1FiguresandTablesforSection1.5Figure1.1EmpiricalDistributionofAPEsfory2fortheSampleSizeof1000underJointNormality32Figure1.2EmpiricalDistributionofAPEsfory2fortheSampleSizeof1000underConditionalNormality33Figure1.3EmpiricalDistributionofAPEsfory3fortheSampleSizeof1000underJointNormality34Figure1.4EmpiricalDistributionofAPEsfory3fortheSampleSizeof1000underConditionalNormality35Table1.1SimulationResultsforAPEofy2(1)(2)(3)(4)(5)(6)(7)(8)(9)(1)(2)(3)(4)(5)(6)(7)(8)(9)Design1:JointNormalityDesign2:ConditionalNormalityCFCFCFCFCFCFIVOpt.IVMLECFCFCFCFCFCFIVOpt.IVMLEBiprobitBiprobitProbitProbit2SLS2SLS2SLS2SLSBiprobitBiprobitProbitProbit2SLS2SLS2SLS2SLSSwitchingSwitchingSwitchingSwitchingSwitchingSwitchingCase1:JustID,OneRegion,APEy2=-.2650JustID,OneRegion,APEy2=-.2426N=1000Bias.0011.0020.0029.0041-.0566-.0568-.0541-.0542.0011.0009.0018.0024.0028-.0463-.0465-.0474-.0476.0121RMSE.0102.0105.0112.0117.0634.0635.0613.0614.0102.0133.0147.0138.0151.0560.0562.0570.0571.0160N=3000Bias.0003.0005.0018.0022-.0571-.0572-.0548-.0549.0003.0002.0004.0016.0013-.0458-.0459-.0468-.0469.0114RMSE.0054.0055.0060.0061.0593.0593.0571.0572.0054.0075.0081.0078.0084.0490.0490.0498.0500.0128N=5000Bias.0002.0004.0018.0020-.0573-.0574-.0550-.0551.0002-.0001.0001.0013.0009-.0461-.0461-.0472-.0472.0112RMSE.0044.0045.0050.0051.0587.0587.0564.0565.0044.0058.0062.0061.0064.0480.0480.0490.0490.0121Case2:OverID,OneRegion,APEy2=-.2279OverID,OneRegion,APEy2=-.2123N=1000Bias.0000.0003.0008.0011-.0342-.0342-.0322-.0325.0000.0005.0006.0015.0015-.0269-.0273-.0291-.0294.0059RMSE.0068.0069.0072.0074.0359.0358.0341.0343.0067.0072.0074.0078.0080.0298.0301.0319.0321.0092N=3000Bias.0001.0002.0008.0009-.0335-.0334-.0315-.0317.0001.0000.0000.0011.0009-.0272-.0277-.0292-.0294.0057RMSE.0039.0039.0041.0041.0341.0339.0321.0323.0039.0041.0042.0045.0046.0281.0286.0300.0302.0070N=5000Bias.0000.0000.0007.0007-.0338-.0339-.0317-.03200000.0000.0001.0010.0009-.0271-.0276-.0290-.0293.0057RMSE.0029.0029.0031.0032.0341.0635.0321.0323.0029.0032.0033.0035.0036.0277.0281.0295.0298.0065Case3:JustID,TwoRegions,APEy2=-.0902JustID,TwoRegions,APEy2=-.0624N=1000Bias-.0248-.0001-.0237.0010-.0274-.0261-.0252-.0255-.0248-.0239-.0001-.0230.0001-.0176-.0158-.0176-.0182-.0241RMSE.0345.0171.0337.0171.0382.0365.0368.0369.0346.0363.0197.0357.0196.0336.0317.0336.0339.0363N=3000Bias-.0252-.0003-.0251-.0003-.0269-.0165-.0249-.0192-.0252.-.0259-.0004-.0251-.0003-.0187-.0166-.0186-.0192-.0261RMSE.0289.0098.0295.0108.0311.0225.0293.0251.0289.0302.0108.0295.0108.0247.0225.0246.0251.0303N=5000Bias-.0256.0001-.0245.0009-.0271-.0260-.0250-.0253-.0256-.0251.0000-.0243.0001-.0175-.0159-.0175-.0181-.0253RMSE.0280.0076.0269.0077.0298.0286.0279.0282.0280.0278.0083.0270.0083.0213.0197.0214.0218.0279aSequentialaveragingofthecontrolfunctiontermv2andxisappliedtocomputeestimatesofAPEs.bThebiasisasthedifferencebetweenthetrueAPEsandtheestimates.RMSEistherootmeansquarederror.cEstimator(1)istheCFapproachinsertingtheresidualbv2toasecond-stagejointbiprobitbetweeny1,y3.Estimator(2)istheCFapproachinsertingtheresidualbv2toasecond-stagejointbiprobitbetweeny(1)1,y3andy(0)1,y3.Estimator(3)istheCFapproachinsertingresidualbv2andbgr3intotheprobitmodelfory1.Estimator(4)istheCFapproachinsertingresidualbv2andbgr3intotheprobitmodelfory1separatelyforsub-samplesbyy3.Estimator(5)istheCFapproachappliedtolinearprobabilitymodelfory1byinsertingresidualbv2andbgr3.Estimator(6)istheCFapproachappliedtolinearprobabilitymodelfory1byinsertingresidualbv2andbgr3forsub-samplesbyy3.Estimator(7)isthe2SLSIVapproachforalinearprobabilitymodelofy1.Estimator(8)isthe2SLSIVapproachusingpredictedvaluesfromthereducedformsfory2andy3asinstruments.y2ispredictedusingalinearmodelandy3ispredictedusingprobitmodel.Estimator(9)isthejointestimationofy1,y2andy3bymaximumlikelihood.36Table1.2SimulationResultsforAPEofy3(1)(2)(3)(4)(5)(6)(7)(8)(9)(1)(2)(3)(4)(5)(6)(7)(8)(9)Design1:JointNormalityDesign2:ConditionalNormalityCFCFCFCFCFCFIVOpt.IVMLECFCFCFCFCFCFIVOpt.IVMLEBiprobitBiprobitProbitProbit2SLS2SLS2SLS2SLSBiprobitBiprobitProbitProbit2SLS2SLS2SLS2SLSSwitchingSwitchingSwitchingSwitchingSwitchingSwitchingCase1JustID,OneRegion,APEy3=.2573JustID,OneRegion,APEy3=.2385N=1000Bias-.0013-.0012-.0116-.0119.1010.1010.0809.0823-.0014-.0003.0013-.0087-.0075.0667.0656.0738.0752-.0119RMSE.0380.0382.0440.0442.1117.1119.0966.0972.0380.0422.0438.0472.0484.0866.0869.0956.0961.0432N=3000Bias.0001.0002-.0090-.0091.1032.1033.0852.0857.0000-.0001.0001-.0084-.0083.0675.0663.0744.0757-.0118RMSE.0228.0229.0267.0268.1069.1070.0904.0906.0228.0236.0241.0276.0281.0742.0735.0819.0830.0259N=5000Bias-.0004-.0003-.0096-.0096.1024.1026.0840.0849-.0004.0008.0009-.0073-.0071.0683.0671.0768.0815-.0112RMSE.0172.0172.0213.0214.1047.1048.0873.0880.0171.0193.0198.0224.0228.0727.0717.0816.0815.0221Case2OverID,OneRegion,APEy3=.2132OverID,OneRegion,APEy3=.2026N=1000Bias.0002-.0008-.0003-.0006.0672.0653.0512.0534.0001.0009.0009-.0012.0003.0353.0493.0492.0513-.0044RMSE.0323.0349.0350.0367.0813.0811.0744.0729.0322.0394.0394.0400.0411.0622.0736.0772.0762.0372N=3000Bias.0002.0004-.0003.0003.0663.0647.0501.0521.0001.0004.0008-.0017-.0006.0330.0339.0494.0510-.0041RMSE.0188.0197.0198.0207.0715.0705.0587.0595.0188.0217.0229.0237.0234.0444.0455.0608.0608.0226N=5000Bias-.0001-.0002-.0012-.0007.0655.0635.0491.0513-.0001.0003.0006-.0016-.0005.0338.0478.0485.0506-.0041RMSE.0147.0152.0158.0162.0687.0670.0545.0559.0146.0173.0180.0185.0189.0414.0540.0557.0569.0174Case3JustID,TwoRegions,APEy3=.0622JustID,TwoRegions,APEy3=.0864N=1000Bias.0436.0005.0339.0065.0292.0327.0128.0153.0437.0535.0007.0462.0028.0257-.0045.0257.0301.0549RMSE.0721.03920585.0444.0576.0575.0559.0546.0722.0766.0409.0669.0455.0555.0500.0597.0601.0779N=3000Bias.0437.0001.0467.0023.0292-.0051.0129.0294.0438.0536-.0003.0467.0023.0254-.0051.0246.0294.0546RMSE.0555.0236.0547.0268.0413.0294.0351.0428.0555.0624.0238.0547.0268.0386.0294.0408.0428.0634N=5000Bias.0447.0010.0353.0076.0300.0335.0130.0159.0447.0536-.0003.0466.0023.0245-.0057.0247.0291.0545RMSE.0516.0180.0413.0215.0372.0395.0273.0280.0516.0592.0185.0516.0208.0333.0237.0350.0378.0601aSequentialaveragingofthecontrolfunctiontermv2andxisappliedtocomputeestimatesofAPEs.bThebiasisasthedifferencebetweenthetrueAPEsandtheestimates.RMSEistherootmeansquarederror.cEstimator(1)istheCFapproachinsertingtheresidualbv2toasecond-stagejointbiprobitbetweeny1,y3.Estimator(2)istheCFapproachinsertingtheresidualbv2toasecond-stagejointbiprobitbetweeny(1)1,y3andy(0)1,y3.Estimator(3)istheCFapproachinsertingresidualbv2andbgr3intotheprobitmodelfory1.Estimator(4)istheCFapproachinsertingresidualbv2andbgr3intotheprobitmodelfory1separatelyforsub-samplesbyy3.Estimator(5)istheCFapproachappliedtolinearprobabilitymodelfory1byinsertingresidualbv2andbgr3.Estimator(6)istheCFapproachappliedtolinearprobabilitymodelfory1byinsertingresidualbv2andbgr3forsub-samplesbyy3.Estimator(7)isthe2SLSIVapproachforalinearprobabilitymodelofy1.Estimator(8)isthe2SLSIVapproachusingpredictedvaluesfromthereducedformsfory2andy3asinstruments.y2ispredictedusingalinearmodelandy3ispredictedusingprobitmodel.Estimator(9)isthejointestimationofy1,y2andy3bymaximumlikelihood.37A.2TablesforSection1.6Table1.3SummaryStatisticsoftheEstimationSample(N=2964)VariableOwnerRenterBudgetShareonHousing.20.41(.16)(.16)Ln(Expenditure)10.359.82(.59)(.59)Ln(Income)10.9410.29(.75)(.73)Ln(Price)8.558.93(.21)(.12)Age49.8844.45(12.97)(13.69)Married.79.35(.40)(.48)Moved.10.33(.30)(.47)Black.21.46(.41)(.50)Yearsofeducation13.2712.12(2.73)(2.90)NumberofChildren.941.05(1.14)(1.24)Obs.2355629aThesampleisbasedonthe2001wavesofthePanelStudyofIncomeDynamics(PSID).Allmonetaryvariableswereconvertedto1998dollarsbeforetheywerelogged.bSamplestandarddeviationsareinparenthesesbelowthesamplemeans.38Table1.4TheFristStageReducedFormRegressionfortheEEVs(1)(2)(3)(4)(5)EstimationMethodOLSOLSProbitProbitProbitDependentVariableLn(Expenditure)Ln(Expenditure)OwnerOwnerOwnerbv2.058(.009)Ln(Expenditure).058(.009)Ln(Income).410.385.055.057.033(.013)(.013)(.006)(.006)(.007)Education.025.024.005.004.002(.0031)(.003)(.001)(.0015)(.0015)Children.056.062.012.011.008(.0077)(.008)(.004)(.003)(.004)Age-.0025.029.003.003.003(.0007)(.004)(.0003)(.0003)(.0003)Age2-.0002(.00004)Ln(Price)-.068-.067-.646-.624-.620(.0318)(.0314)(.017)(.016)(.016)Married.27.26.054.053.037(.021)(.021)(.010)(.009)(.010)Moved.012.037-.065-.063-.064(.022)(.022)(.009)(.009)(.009)Black-.058-.079-.064-.060-.057(.019)(.0189)(.0087)(.009)(.009)abv2denotestheresidualfromRegression(1),thereducedformforlogtotalexpenditure.bRegression(1)and(2)areregressionsforlogtotalexpenditure,thecontinuousEEV.Regression(3)-(5)areregressionsforhomeownership,thebinaryEEV.c*p-value<10%**p-value<5%***p-value<1%39Table1.5Comparingmarginaleffectsinthestructuralequationofthehousingshare(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)EstimationMethodIVOpt.IVCFCFCFCFCFCFFunctionalformfory1Linear2SLS2SLS2SLSFracProbitFracProbitFracProbitFracProbitFracBiprobitFracBiprobitNoEEVsNoEEVsOneEEVTwoEEVsSwitchingSwitchingMarginalEffectsCoeffCoeffCoeffCoeffAPEAPEAPEAPEAPEAPELn(Expenditure)-.109-.176-.056-.061-.106-.064-.061-.059-.062-.059(.005)(.084)(.01)(.01)(.005)(.009)(.010)(.009)(.01)(.009)Owner-.094.336-.14-.124-.074-.088-.101-.124-.098-.124(.009)(.330)(.016)(.015)(.0082)(.012)(.020)(.038)(.020)(.038)Age.002.0003.003.003.002.0025.0026.0026.0025.0026(.0002)(.001)(.0002)(.0002)(.0002)(.0002)(.0002)(.0002)(.0002)(.0002)Ln(Price).147.54.109.124.151.147.136.138.139.138(.013)(.29)(.017)(.017)(.013)(.016)(.021)(.022)(.021)(.022)Married-.005-.06-.027-.028-.0026-.027-.027-.027-.027-.027(.007)(.025)(.008)(.009)(.006)(.008)(.008)(.008)(.008)(.008)Moved.011.071.005.007.009.009.007.005.007.005(.007)(.047)(.008)(.007)(.007)(.007)(.008)(.009)(.007)(.009)Black-.012.041-.010-.009-.012-.006-.008-.009-.007-.009(.006)(.036)(.006)(.006)(.006)(.006)(.006)(.007)(.006)(.007)aThedependentvariableistheexpenditureshareonhousing.bStandarderrorsfortheestimatedAPEswerebootstrapstandarderrorswith200replications.cRegression(1)istheOLSforlinearprobabilitymodelthatassumsnoEEVs.Regression(2)isthe2SLSIVestimatorforlinearprobabilitymodelthatusesalinearprobabilitymodelforthereducedformofhomeownership.Regression(3)isthe2SLSIVestimatorforlinearprobabilitymodelthatusesthepredictedvaluesfromthestageregressions(1)and(3)inTable1.4asIV.Regression(4)istheCFapproachforlinearprobabilitymodelusingbv2andbgr3.Regression(5)isthefractionalresponsemodelthatassumesnoEEVs.Regression(6)istheCFapproachforthefractionalresponsemodelbyinsertingonlybv2.Regression(7)istheCFapproachforthefractionalresponsemodelbyinsertingbothbv2andbgr3.Regression(8)istheCFapproachforthefractionalresponsemodelbyinsertingbothbv2andbgr3appliedseparatelytotwosub-samplesofhouseholdsbytheirhomeownershipstatus.Regression(9)istheCFapproachtoajointestimationbetweenfractionalresponseandhomeownershipinthesecondstep.Regression(10)istheCFapproachappliedseparatelytotwo-samplesofhouseholdsbytheirhomeownershipstatusandajointestimationbetweenfractionalresponseandhomeownershipinthesecondstep.d*p-value<10%**p-value<5%***p-value<1%40Table1.6VariableAdditionTestsforEndogeneity(1)(2)(3)(4)(5)(6)EstimationMethodCFCFCFCFCFCFFunctionalformfory12SLSFracProbitFracProbitFracProbitFracBiprobitFracBiprobitOneEEVTWoEEVsSwitchingofbv241.7938.4338.4276.91(.000)(.000)(.000)(.000)ofbgr34.070.71(.036)(.405)Jointofbv2;bgr321.2538.67(.000)(.000)Jointofbgr(1)3;bgr(0)32.87(0.238)ofbgr(1)30.61(.431)ofbgr(0)32.26(.154)Jointofbv(1)2;bv(0)244.73(0.000)ofbv(1)220.5441.44(.000)(.000)ofbv(0)224.1841.36(.000)(.000)ofr0.73(.407)ofr1.064(.410)ofr0.207(.131)Jointofbgr(1)3;bgr(0)3;bv(1)2;bv(0)246.56(.000)aTheWaldteststatisticsfornullhypothesesofnoendogeneityarereportedforcontrolfunctiontermsandthecorrelationparameters.bBootstrapp-valueisreportedintheparenthesesundertheWaldteststatistic.cbv2istheresidualfromreduced-formregressionoflogexpenditureinColumn(1)of1.4.bgr3isthegeneralizedresidualfromreduced-formregressionofhomeownershipinColumn(4)ofTable1.4.Anysuperscript(1)indicatesthatthistermappearsintheregimewithhomeowernship=1inaswitchingregression.Anysuperscript(0)indicatesthatthistermappearsintheregimewithhomeowernship=0inaswitchingregression.risthecorrelationfromthejointestimationwithhomeownshipgiventhecontrolfunctiontermbv2hasbeenpluggedin.Anysubscript1indicatesitisthecorrelationwithhousingbudgetshareinregime1.Anysubscript0indicatesitisthecorrelationwithhousingbudgetshareinregime0.41BIBLIOGRAPHY42BIBLIOGRAPHYAmemiya,T.,1978.Theestimationofasimultaneousequationgeneralizedprobitmodel.Econo-metrica46,1193Œ1205.Blundell,R.W.,Powell,J.L.,2003.Endogeneityinnonparametricandsemiparametricregressionmodels.EconometricSocietyMonographs36,312Œ357.Blundell,R.W.,Powell,J.L.,2004.EndogeneityinSemiparametricBinaryResponseModels.ReviewofEconomicStudies71,655Œ679.Conlin,M.,Dickert-Conlin,S.,Chapman,G.,2013.Voluntarydisclosureandthestrate-gicbehaviorofcolleges.JournalofEconomicBehavior&Organization96,48Œ64.doi:10.1016/j.jebo.2013.09.007Dong,Y.,Lewbel,A.,2015.ASimpleEstimatorforBinaryChoiceModelswithEndogenousRegressors.EconometricReviews34,82Œ105.doi:10.1080/07474938.2014.944470Edmans,A.,Goldstein,I.,Jiang,W.,2012.TheRealEffectsofFinancialMarkets:TheImpactofPricesonTakeovers.TheJournalofFinance67,933Œ971.doi:10.1111/j.1540-6261.2012.01738.xHausman,J.A.,1975.Aninstrumentalvariableapproachtofullinformationestimatorsforlinearandcertainnonlineareconometricmodels.Econometrica43,727Œ38.Heckman,J.,1978.Dummyendogenousvariablesinasimultaneousequationsystem.Economet-rica46,931Œ959.Horrace,W.C.,Oaxaca,R.L.,2006.Resultsonthebiasandinconsistencyofordinaryleastsquaresforthelinearprobabilitymodel.EconomicsLetters90,321Œ327.Lin,W.,Wooldridge,J.M.,2015b.Ondifferentapproachestoobtainingpartialeffectsinbinaryresponsemodelswithendogenousregressors.EconomicsLetters134,58Œ61.doi:10.1016/j.econlet.2015.05.019Lin,W.,2016.AControlFunctionApproachinTwo-StepSieveM-EstimationofBinaryResponseModelswithEndogenousExplanatoryVariables.Mimeo.Murtazashvili,I.,Wooldridge,J.M.,2015.Acontrolfunctionapproachtoestimatingswitchingregressionmodelswithendogenousexplanatoryvariablesandendogenousswitching.JournalofEconometrics.doi:10.1016/j.jeconom.2015.06.014Papke,L.E.,Wooldridge,J.M.,1996.EconometricMethodsforFractionalResponseVariableswithanApplicationto401(k)PlanParticipationRates:Summary.JournalofAppliedEcono-metrics(1986-1998)11,619.Rivers,D.,Vuong,Q.H.,1988.Limitedinformationestimatorsandexogeneitytestsforsimulta-neousprobitmodels.Journalofeconometrics39,347Œ366.43Roodman,D.,2015.CMP:Statamoduletoimplementconditional(recursive)mixedprocessesti-mator,StatisticalSoftwareComponents.BostonCollegeDepartmentofEconomics.Semykina,A.,Wooldridge,J.M.,2015.BinaryResponsePanelDataModelswithSampleSelec-tionandSelfSelection(WorkingPaperNo.wp2015_05_01).DepartmentofEconomics,FloridaStateUniversity.Nam,S.,Wooldridge,J.M.,2014.OnComputingAveragePartialEffectsinModelswithEndo-geneityorHeterogeneity.Mimeo.DepartmentofEconomics,MichiganStateUniversity.White,H.,1982.Maximumlikelihoodestimationofmodels.Econometrica50,1Œ25.Wooldridge,J.M.,2005.Unobservedheterogeneityandestimationofaveragepartialeffects.Iden-andInferenceforEconometricModels:EssaysinHonorofThomasRothenberg27Œ55.Wooldridge,J.M.,2010.Econometricanalysisofcrosssectionandpaneldata.MITpress.Wooldridge,J.M.,2014.Quasi-maximumlikelihoodestimationandtestingfornonlinearmodelswithendogenousexplanatoryvariables.JournalofEconometrics,Causality,Predic-tion,andSpeciAnalysis:RecentAdvancesandFutureDirections182,226Œ234.doi:10.1016/j.jeconom.2014.04.02044CHAPTER2ACONTROLFUNCTIONAPPROACHINTWO-STEPSIEVEM-ESTIMATIONOFBINARYRESPONSEMODELSWITHENDOGENOUSEXPLANATORYVARIABLES2.1IntroductionItisnotablychallengingtodealwithendogeneityinlimiteddependentvariablemodels,especiallywhentheendogenousexplanatoryvariables(EEVs,forshort)canhavedifferingattributes,withsomebeingcontinuousandsomebeingdiscrete.Inprinciple,onecanmakeenoughdistributionalassumptionssothatmaximumlikelihoodestimation(MLE)ispossible.Dependingonhowcom-plicatedthemodelis,MLEcanbecomputationallydemanding(althoughsimulationmethodsofestimationcanmakecomplicatedmodelsfeasible).Inanycase,theestimatesofparametersandotherquantitiesofinterest,suchasmarginaleffects,canbesensitivetoparametricassumptions.Thispaperfocusesonasemiparametricandestimationofaspecialcaseofatrian-gularsystem.Namely,theresponsevariableisbinary,thereisasinglebinaryEEV,andpotentiallymanycontinuousEEVs.ThemodelcanbeappliedtoendogenousswitchingwithabinaryresponseandmultiplecontinuousEEVsŒascouldhappen,forexample,inaproductionenvironmentwithtwoproductionregimesandseveralcontinuousinputs.Or,themultiplecontinuousEEVscanbeprices,asinPetrinandTrain(2010).Becauseofthesemiparametricsetting,theapproachprovidesamorexibleandrobustwaytoconductestimationandinferenceforcausalmarginaleffects.Thispaperextendsthetwo-stepparametricapproachinLinandWooldridge(2015a).There,jointnormalityandlinearityassumptionsareusedtoobtainasimpletwo-stepcontrolfunctionestimator,wherereducedformsforallcontinuousEEVsareestimatedtoobtainreducedformresiduals(controlfunctions).Inasecondstep,abivariateprobitmodelisusedwhichincludesthecontrolfunctionsandanyfunctionsoftheEEVs.Themethodiscomputationallysimpleandpn-asymptoticallynormal,providedtheparametricassumptionsareInaddition,averagemarginaleffectsareandeasilyestimated.Bootstrappingthefullprocedureprovides45straightforwardinference;or,thedeltamethodcanbeusedtoobtainproperstandarderrors.OnecanextendtheLinandWooldridge(2015a)inseveraldirectionsbecausetherearemultipleunobservablesandseverallinearfunctionalforms.ItisessentiallyimpossibletoallowafullynonparametricanalysisbecauseoneloseswithdiscreteEEVs(Chesher,2003,forexample).ThispaperrelaxesassumptionsonthereducedformsforthecontinuousEEVsandtherelationshipsbetweenthestructuralerrorsandthecontrolfunctions.Imaintainlinearityinthestructuralequation,andIalsomaintainaconditionalbivariatenormalityassumptionontheerrorsinthebinaryresponseequations.Whileonecandebatethesechoices,ataminimumtheyextendtheanalysisinLinandWooldridge(2015a)toallowforthekindofnonparametricreducedformsforthecontinuousEEVsinBlundellandPowell(2004).UnliketheBlundellandPowellframework,theanalysishereallowsforabinaryEEV.UnlikeBlundellandPowell(2004),whichusesamatchingapproachinthesecondstepesti-mation,andRothe(2009),whichappliestheKleinandSpady(1993)semiparametricmaximumlikelihoodestimator,hereIusesieveestimationforthenonparametriccomponentsinbothsteps.Themethodofsieves,formallyintroducedbyGrenander(1981),allowsforunknownfunctionsthatlieinparameterspacestobeapproximatedbydimensionalspacesofgrowingdimensions.Thegrowthinthedimensionalityislinkedtoincreaseinsamplesizetoguaranteeconsistency.Asapracticalmatter,sieveestimationisappealingbecause,foragivensamplesize,itisxibleparametricestimation.Thedifliesintheoreticalderivationsoftheasymptoticpropertiesofestimatorswithagrowingparameterset.Tothisend,IapplygeneralresultsinChen(2007)andChen,LintonandVanKeilegom(2003)forconsistencyandratesofconvergencefortwo-stepsieveestimation,wherethestepconsistsofsieveleastsquaresandthesecondstepsieveMLE.Athirdstepestimationisusedtoestimatetheaveragepartialeffects(APEs)basedontheaveragestructuralfunction(ASF)ofBlundellandPowell(2004).DuetothenumericalequivalenceresultshowninHahn,LiaoandRidder(2015),wecantreatthetwo-stepsieveM-estimationasifitisastandardtwo-stepparametricproblem,andthuseasilyconductprac-ticalinferenceonAPEsusingdeltamethods.Itseemsverylikelythatbootstrappingcanalsobe46Inadditiontofullparametricapproachesandthesemiparametricsieveapproachproposedhere,thereareothermodelingstrategiesandestimationmethodsthathavebeenproposedŒsomeonlyforthespecialcaseofasinglebinaryEEVandothersthatalsoallowdiscreteEEVs.Recently,HanandVytlacil(2015)advanceanalysesforunknownmarginaldistributionsinabivariateresponsemodelusingthecopulaapproach.ArestrictivefeatureŒotherthanthatitappliestothespecialcaseofasinglebinaryEEVthatappearsadditivelyinthestructuralequationŒisthatthecopulafunctionisrequiredtobeparametric.Here,Irestrictthemarginalsbut,ineffect,allowgeneralcorrelationamongthereducedformerrorsofthecontinuousandbinaryEEVsandthestructuralerror.ForhandlingdummyEEVs,Terza,Basu,andRathouz(2008)andWooldridge(2014)argueforplugginginaresidualorgeneralizedresidualobtainedfromareducedformprobit.Whilesimple,theseCFmethodsarecontroversialbecausetheyrelyonnonstandardparametricassumptionstoachieveandthereforeconsistency1.TheCFapproachismucheasiertojustifyforthecontinuousEEVs,andthatismotivationforthecurrentpaper.Lewbel(2000)proposesadifferentsemiparametricapproachforestimatingtheparametersofabinaryresponse,andheallowsbothcontinuousanddiscreteEEVs.Unfortunately,Lewbel'sapproachislimitedbyitsrequirementofafispecialregressor,"whichisanobservedcovariateassumedtoappearinthestructuralequationbutnotinthereducedformsfortheEEVs.Exceptwhenthespecialregressorisrandomized,thisisastrongrequirement.Moreover,asarguedinLinandWooldridge(2015b),Lewbel'sspecialregressorapproachcurrentlydoesnotallowonetouncovermeaningfulpartialeffects.Tosummarize,theapproachherebuildsofftheparametricapproachinLinandWooldridge(2015a),relaxlinearityassumptionsinbothreducedformandcontrolfunctionrelationships.ItallowsthecontinuousandbinaryEEVstoappearincomplicatednonlinearwaysinalinearindex1AspointedoutinWooldridge(2014),itisnaturaltouseplugginginofgeneralizedresidualundernonstandardassumptionasameanstotestforendogeneityofdummyvariableinbianryoutcomeequation,ratherthantoestimatethecoef47Œforexample,squaresandinteractionsareallowed.Ialsoshowhowtoestimateandperforminfer-enceonaveragepartialeffects,somethingthatisoftenignoredintheliteratureonsemiparametricanalysis.Theremainderofthepaperisorganizedasfollows.Section2.2presentsthemodelscationsandsufconditions,startingwiththeparametriccasetoprovideback-ground.Section2.3elaboratesonthetwo-stepsieveM-estimationproblem.Section2.4presentsasymptoticspropertiesofthetwo-stepsieveestimator,includingconsistencyandasymptoticnor-malityofnonlinearfunctionals.Section2.5showsapracticalinferencefortheAPEsresultingfromthesemiparametrictwo-stepprocedure.Insection2.6,IundertakeaMonteCarlostudyshowinghowthesieveapproximationworkscomparedwithotherchoices.Section2.7concludesandproposesdirectionsforfutureresearch.2.2Model2.2.1TheparametricmodelAsastartingpoint,considerasimpleparametrictriangularsystemforabinaryresponsey1withacontinuousEEVy2andadummyEEVy3ofthefollowingform,chap1conclusiony1=1[x1bo+u10];(2.1a)y2=zdo+v2;(2.1b)y3=1[zgo+u30];(2.1c)wherex1f1(z1;y2;y3)isa1K1vectoroffunctionsthatcouldincludeinteractions,squaresandlogarithmsofitsarguments,suchasz1,y2z1,andlog(y2).Theleadingcaseisx1(1;z1;y2;y3),wheretheelementofx1isnormalizedat1.Sameforzf(z1;z2),whichisa1Lvectoroffunctionsoffullsetofexogenousvariables,withz(1;z1;z2)beingtheleadingcase.z1aretheincludedexogenousvariablesandz2theexcludedexogenousvariables.Onereasonthatwe48writevectorx1andzthiswayisbecausecontrolfunctionapproachcouldeasilyaccommodateforxiblefunctionsinx1andz,withoutchangingtheestimationprocedure.Foritissuftomakeajointnormalityassumptionfor(u1;v2;u3),alongwithanexogeneityassumptionontheexogenousvariableszbeingindependentof(u1;v2;u3),fol-lowingclassicalliterature(BlundellandSmith,1994;Heckman,1978;RiversandVuong,1988,forexample).Becauseofthenonlinearity,herewerequireastrongerexogeneityconditiononzthananusualuncorrelationconditioninlinearcase.Additionally,weassumeherethattheendogeneitycomesfromomittedvariableproblemsothatcovariancesoftheerrortermsareassumedtobenonzero.Formally,adistributionalandexogeneityassumptionforinthisparametricsettingisstatedinAssumption2.1.Assumption2.1D0BBBB@u1v2u3z1CCCCA=D0BBBB@u1v2u31CCCCA˘N2666640BBBB@0001CCCCA;0BBBB@1rotoros2vwotowo11CCCCA377775;wherero;woandtoarecovariances,ands2visthevarianceforv2.Thevariancesforu1;u3areassumedtobeatunityforsimplicity.Asv2isaproxyforendogeneityfromy2,bythetrivariatenormalityinAssumption2.1,projectu1andu3ontov2,u1=rosvv2+v1;u3=wosvv2+v3:Itfollowsthatthe(v1;v3)arebivariatenormalandindependentof(v2;z),D0B@v1v3z;v21CA=D0B@v1v31CA˘N2640B@001CA;0B@1r2otorowotorowo1w2o1CA375:49Usingv2asanadditionalcontrolvariable,(2.1)canbereducedtoy1=1x1bo+rosvv2+v10;(2.2a)y3=1zgo+wosvv2+v30:(2.2b)Sincewehavetwononlinearequations(2.2a)and(2.2b),alongwithbivariatenormalerrorterms(v1;v3),jointMLEshouldbeemployedtoestimatetheparameters.AnorderconditionLK1isrequiredbecauseidentifyingoffnonlinearityresultsinpoorstatisticalpropertiesandsometimesevendifinconvergence.Further,weshouldhaveatleastoneinstrumentmainlyforeachreducedformequationtoensureenoughvariationinthestrengthoftheinstruments.Procedure2.1UnderthejointnormalityandexogeneityrequirementinAssumption2.1,andtheorderconditionstatedabove,thefollowingparametrictwo-stepbiprobitprocedureisconsistentforparametersofinterestboandrosvinstructuralequation(2.2a):(a)RuntheOLSfory2onztoobtainthereducedformresidualbv2:(b)Runabiprobittojointlyestimateequations(2.2a)and(2.2b)byMLE,usingbv2astheproxyforv2.UnderthesufconditionsinAssumption2.1,consistencyresultingfromProcedure2.1isstraightforward.Inaddition,itisfairlyeasytoimplementProcedure2.1,foritonlyinvolvesroutinescommonlyusedinstandardsoftware.Anotherkeyfeatureofcontrolfunctionapproach,whichisnotenoughillustratedinthesystem(2.1),isthatitcouldeasilyac-commodateforavectorofcontinuousEEVsy2,withoutincreasingthedimensioninthesecondstepjointlikelihoodfunction.However,assumingtrivariatejointnormalityisrestrictiveandun-necessary.Fortunately,duetotherobustnatureofcontrolfunctionapproach,Procedure2.1cangetthroughunderaweakersetofconditions.First,wedropthenormalityassumptiononv2andinsteadassumethatconditionaldistributiondistributionof(u1;u3)givenv2followabivariatenormality,asformallystatedinAssumption2.10.50Assumption2.10(a)(v1;v3)areindependentofzandv2andfollowbivariatenormaldistributionwithvariancesoneandcovariancero,D0B@v1v3z;v21CA=D0B@v1v31CA˘N2640B@001CA;0B@1roro11CA375:(b)v2isconditionallymeanindependentofz.(c)Theconditionalmeansof(u1;u3)givenv2arelinearwithadditiveerrors,u1=lov2+v1;u3=hov2+v3:AlthoughweakerthanAssumption2.1,Assumption2.10isstillrestrictivealongtwodimen-sions.First,linearconditionalmeanof(u1;u3)givenv2ishardtomaintain,especiallywhenweassumetheremainingadditiveerrorsareindependentofv2.Second,theindependencebetweenzand(u1;u3)requirestheconditionalmeanfory2tobelinearinz,whichalsolacksxibility.Afterall,sinceweareinherentlyignorantaboutthedistributionfromeconomictheory,ideally,wewouldliketoimposeaslittleandharmlessassumptionsonfunctionalformanderrordistri-butionaspossible.SothesemiparametricdrawsontheweakerconditionalbivariatenormalityinAssumption2.10.Itaddsxibilitybyintroducingarbitraryunknownfunctionsalongtheabovetwodimensions.2.2.2Thesemi-parametricThesemi-parametriccationrelaxestheabovetworestrictions.First,followingBlundellandPowell(2004),IallowthereducedformforendogenousEEVy2tobefullynonparametric,composedofho(z)andanadditiveerrorv2:y2=ho(z)+v2;(2.3)51whereho(z)E(y2jz)istheconditionalmeanofy2.Ialsoletv2enterthestructuralerrorsu1andu3inapartiallinearfashion:u1=mo(v2)+v1;(2.4)u3=qo(v2)+v3;(2.5)wheremo()andqo()areunknowncontrolfunctionswithzeromeans.Justlikeintheparametriccase,thesetwononparametricfunctionsofv2areassumedto"partialout"fromu1alltheendo-geneityinthecontinuousEEVy2.Equation(2.4)and(2.5)arepartiallinearinthesensethattheerrors(v1;v1)arestillassumedtofollowtheparametricbivaraitenormaldistribution.Decomposingu1andu3inthispartiallinearfashionismorethanharmless.Oneofassumingaconditionalnormalityisthatwearenotimposinganyrestrictiononthedistributionoftheunconditionaldistributionoferrorterms(u1;u3):Also,conditionsforsinglein-dexbinaryresponseisautomaticallybymaintainingtheaboveassumptions.v1arev3arebivariatenormalandthushasalargesupport.Wehaveacontinuouslydistributedcomponenty2inx1.x1isnormalizedtohaveaninterceptsothatu1ismeanzero.Moreover,forpracticalpropose,itiseasiesttoconductajointMLEtohandledummyEEVinbinaryoutcomemodel,soweremainajointnormalityforeasyofestimation.Additionally,assumingbivariatenormalityenablesustomakeextrapolationtothecounterfactualoutcome,therebyrecoveringacounterfactualmarginalef-fectofinterest.Investigationonrelaxingdistributionalassumptionalongthisdimensionismainlytheoretical,andisleftfordiscussiononadifferentpaper.Afterplugginginequations(2.4)and(2.5)foru1andu3,alongwithreplacingequation(2.1b)withequation(2.3),weobtainasemiparametricestimationmodelasfollows:y1=1[x1bo+mo(v2)+v10];(2.6a)y2=ho(z)+v2;(2.6b)y3=1[zgo+qo(v2)+v30]:(2.6c)Tosimplifynotation,Ireservesubscriptoonlytosignalthatatrueparameterisanunknownfunctionofdimensions,butomitthesubscriptoifaparameterisinespace.Other52notationremainsthesameasintheparametriccase.ThedistributionalassumptionisstatedinAssumption2.2below.Assumption2.2(a)(v1;v3)isajointbivariatenormalwithvariancesatunityandcovariancer,andisindependentofzandv2;D0B@v1v3z;v21CA=D0B@v1v31CA˘N2640B@001CA;0B@1roro11CA375:(b)v2isarbitrarymeanzerodistribution,(c)u1andu3haveadditiveindependenterrors,namely,u1=mo(v2)+v1;u3=qo(v2)+v3:Unlikemakinganunitvariancesassumptionforunconditionaldistributionof(u1;u3)inAs-sumption2.1,asubtlechangehereisnowtheconditionalvariancesareassumedatunity,forsimplicityofconductingestimationinthesieveM-estimationframeworklateron.Notethatwhiletheunconditionaldistributionof(u1;u3)isrestrictivetobemeanzerobynormalization,theconditionalmeanofu1givenarealizationofv2canbeanynumber.Sincethefunctionalformforv2isarbitrary,andallthefunctionsofv2correlatedwithu1andu3havebeentakencareof,itismoresensibletohaveinAssumption2.2(c)thattherestoftheerrorsv1?v2andv3?v2,where?denotesindependence.Nowthatwehavedimensionalparameters(mo();ho();qo()),assumingsinglein-dexessuchasx1binmodel(2.6)playsamoreroleindimensionreduction,achievingagreaterestimationprecisionthanfullynonparametricmodel.Sinceitcontainsbothparam-eters(b;g;r)anddimensionalparameters(mo();ho();qo()),(2.6)isasemiparametrictriangularsystem.Tocarryoutestimationonthissystem,onewouldhavethoughtofseparat-ingouttheandpartsthroughatransformation.Lookssimilartopartiallinearmodels53astheequations(2.6a)and(2.6c)are,thenon-invertibilityoftheindicatorfunction1[]effectivelypreventsusfromseparatelyestimatingtheparametricandnonparametricpart.Amoreplausibleprocedureistofollowatwo-stepcontrolfunctionmethodtomimicthepara-metrictwo-stepbiprobitProcedure2.1.Optimizingacriterionoveracompactparame-terspace(b;r)oftenbringsusdesirablestatisticalproperties.Bycontrast,unknownfunction(mo();ho();qo())livesinparameterspacethatis"unbounded".Itisoftenproblematictotrytosolveasampleoptimizationproblemovere"unbounded"parameterspace.Thingsareorunclearatthe,orthe"boundary".Sowecouldnoteasilypassthroughthepropertyofthecriterionfunctionattheinteriorofthespacetothe"boundary"point.Togetaroundthis,methodofsievesoptimizesthecriterionoveradimensionalspacecalledsieves.Thedimensionofthesievespacesgrowswithsamplesizessothatinthelimitittendsinthetruespace.Procedure2.2Heuristically,atwo-stepsieveM-estimationgoesasfollows,(a)Conductsieveleastsquaresestimationofconditionalmeanho(z)fory2toobtainthere-ducedformresidualbv2:(b)Plugbv2intothesecondstageunknowncontrolfunctions(mo(bv2);qo(bv2)).Thedimen-sionalparameters(b;g;r)anddimensionalparameters(mo();qo())inthesecondsteparethenestimatedsimultaneouslybysievemaximumlikelihoodestimation.Parsimonisityentitledbycontrolfunctionapproachisstilltrueinthissemiparametrictwo-stepProcedure2.2.ItcouldeasilyaccommodateforavectorofcontinuousEEVsy2,withoutchangingthevariance-covariancestructureinconditionalbivariatenormalityinAssumption2.2(a).Procedure2.2isheuristicinthesensethatwehavenotyetthespecoptimizationprobleminthisnonstandardprocedure,neitherareweclearabouthowtoimplementtheestimationinpractice.Further,unlikeProcedure2.1intheparametriccase,theconsistencyandasymptoticsforthisProcedure2.2isnotstandard.Showingthemisnoeasywork.Weturntonextsectionsto54establishtheseresultsformally.2.3EstimationThissectionaimstoformallystudyandestimationforthesemiparametricmodel(2.6)intheframeworkoftwo-stepsieveM-estimation.Somewhattechnicalandabstractasitmightbe,itisanimportantbuildingblockforderivingasymptoticpropertiesinthenextsection.2.3.1Trueparametersthetrueparameterashoho(),andtheparameterspaceinthestepasH,whereho()2H.Inaddition,lethdenoteagenericelementinH,formallyh2H.Assumption3.2formallytheofhoinpopulation.Assumption3.1Letrandomvectorwi1=(yi2;zi);i=1;:::;ndenotethedataintheststep,andrandomvectorwi2=(yi1;yi2;yi3;zi);i=1;:::;ndenotethedatainthesecondstep.Thedataisassumedtobei:i:d:Assumption3.2Thetrueparameterhointheststepisasauniquesolutiontomaxi-mizeapopulationleastsquaresproblemoverthedimensionalparameterspaceH:ho=argsuph2HEh(yi2h(zi))2i:thetruesecond-stepparametersasgo(qo;mo();qo()).SimilarlytoH,theparameterspaceinthesecondstepasG,wherego2G.LetgdenoteagenericelementinG:Forthedimensionalcomponentqo,theparameterspaceasQ,whereqo2Qandqo(bo;ro;go).Fornotationalclarity,werestorethesubscriptoforqotodenotethetruecomponentinthesecondstep,butletq(b;r;g)denoteagenericpointintheparameterspaceQ.55Forthecomponentsmo()andqo(),theparameterspaceasMandQ,wheremo()2Mandqo()2Q,respectively.LetmdenoteagenericelementinMandqagenericelementinQ.ItiseasytonotethattheparameterspaceGisatensorproductofaparameterspaceQandtwofunctionalspacesMandQ.Assumption3.3theofthesecond-steptrueparametergoinpopulation.Assumption3.3Havingpluggedinthetrueparameterhointheststep,thetrueparametergointhesecondstepisasauniquesolutiontoapopulationmaximumlikelihoodestimationproblemoverthedimensionalparameterspaceG:go=argsupg2QMQE[logF2(di1[xi1b+m(yi2ho(zi))];di2[zig+q(yi2ho(zi))];di1di2r)];whereF2(;;)denotesastandardbivariatenormalCDFforthejointdistributionof(v1;v3)withvariancesatunity.Ittakesthevaluesofthetwoindexesandthecovariancerasitsarguments.denotestensorproduct.di12yi11,di22yi21:Here,di1anddi2aretransformationsofthebinaryoutcomeyi1andbinarydummyvariableyi2soastoindicatethedirectionofintegrationforv1andv3respectively.2.3.2SievespacesSincewehavethetrueparameterho()inthestep,andthetrueparameter(mo();qo())inthesecondstep,atheoreticaldifofestimatingsuchamodeloftenliesinthatoptimizingoveranfunctionalspacecouldresultinanill-posedproblem2.ThisisevenmoretrueinthisparticularProcedure2.2,becauseunknowncontrolfunctions(mo();qo())inthesecondsteptakesthenonparametricestimationofv2=y2ho(z)asinputs.2Theofill-posedproblemandasurveyonitisprovidedinChen(2007).Heuristi-cally,iteithermeansthereisnouniquemaximizerofthecriterionfunctionatall.Orevenwhensuchauniquemaximizerexists,wehavetroublebackingitout,becuasethereareotherconfound-ingsolutionsatwhichthevalueofcriteronfunctionissufclosetoitsmaximum.56Tomitigatetheill-posedproblemincarryingoutthistwo-stepProcedure2.2toestimatethissemiparametricsystemofequations(2.6),sieveapproximationsforunknownfunctions(mo();ho();qo())canbeemployed.Thus,onecouldoptimizeoverasequenceofdimen-sionalsievespacesthatbecomesdense3inthetruefunctionalspaceinthelimitassamplesizeincreases.Therearethreemajorreasonsthatsieveapproximationischosenoverothercommonnonpara-metrictechniques,suchaskernelestimation,tocarryoutthesemi-parametrictwo-stepcontrolfunctionprocedure.First,methodofsievesinherentlyembracesahigherdegreeofxibility.Unlikemethodofkernel,itiseasytoimposestructuresinthefunctionalforms,orinformalterminology,shaperestrictionsinthemodelThankstothisfeatureofsieves,weareabletodecomposetheerrortermsinmodel(2.6)inapartiallinearfashion.Second,sieveapproximationiseasytoimplement.Instandardsoftware,empiricalresearcherscansimplyaddbyhandsomesievesatasmallscaleasadditionalregressors,wheneverfunctionalformassumptionsforacontrolvariablesaredubious.Toestimatethesemiparametricmodel(2.6)inparticular,onecouldcarryoutthemethodofsievebysimplyaddingsomesievefunctionstotheparametrictwo-stepbiprobit.Lastly,unlikemethodofkernel,whereasymptoticsneedstobederivedonacase-by-casebasis,methodofsievehasastreamofmoreliteratureonasymptotictheoryforaestimationframework.Sinceourmodelintothetwo-stepsieveM-estimationframeworkbyHahn,LiaoandRidder(2015),wemaptheirresultbyverifyingconditions.sievespaceHnforthetrueparameterhointhestepatsamplesizenasHnnh()=pk1(n)1()0dh:dh2Rk1(n)o;3Densemeansforeveryelementinthetrueparameterspace,thereisanelementinthelimitofthesequenceofthesievespacethatisarbitrarilyclosetothatelementinthetrueparameterspace.57whereh()denotesagenericelementinHn,whichisareal-valuedfunction.pk1(n)1()(p1;1();:::;p1;k1(n)())0(2.7)isasak1(n)1vectorofbasisfunctions.Thenumberofbasisfunctionscharacterizesthecomplexityofthesievespace.Thefollowingformallystatesthenotionofapproximationassamplesizeincreases,inwhichasequenceofpsudo-trueparametersconvergestothetrueparameter.Assumption3.4AnondecreasingsequenceofsievespacesapproximateHasngetslargerHnHn+1:::H;andthereexistsasequenceph;nhosuchthatdh(ho;pnho)!0asn!¥,wheredh(;)denotesa(pseudo)metriconH,ph;nisaprojectionmappingfromHtoHn.LetGndenotethesievespaceatsamplesizenforthetrueparameter(qo;mo();qo())inthesecondstep,whichisatensorproductofparameterspaceQandsievespaceMnandQn,GnQMnQnf(b;r;g)gfm()=pl(n)2()0dm:dm2Rl(n)gfq()=pl(n)2()0dq:dq2Rl(n)gwherem()denotesagenericelementinMnandq()denotesagenericelementinQn:pl(n)2()(p2;1();:::;p2;l(n)())0isasal(n)1vectorofbasisfunctions.Letdqdenotethedimensionofthecomponentqinsecondstep.Sotheoveralldimensioninthesecond-stepisk2(n)=dim(Gn)=dq+2l(n).AnondecreasingsequenceofsievespacesapproximatesGasngetslarger.Formally,58Assumption3.5AnondecreasingsequenceofsievespacesapproximateGasngetslargerGnGn+1:::Gandthereexistsasequencepg;ngosuchthatdggo;pg;ngo!0asn!¥,wheredg(;)denotesa(pseudo)metriconG,pg;nisaprojectionmappingfromGtoGn.2.3.3TheestimatorFormally,thefollowingestimationproblems(2.8)and(2.3.3)thesieveM-estimatorsinProcedure2.2.ThetrueparameterhointhestepisestimatedbymaximizinganaturalsampleanalogofthesieveleastsquarescriterionasinAssumption3.2overtheapproximatingsievespacesHn.bhn=argmaxh2Hn1nnåi=1(yi2h(zi))2:(2.8)Thissieveleastsquaresestimatorbhn()inthestephasaclosed-formsolutionbhn()=pk1(n)1()0(P01P1)nåi=1pk1(n)1(zi)yi2;whereP1pk1(n)1(z1);:::;pk1(n)1(zn)0denotesnk1(n)datamatrixinthewholesample,eachrowconsistingofbasisfunctionsforeachobservationatlengthofk1(n);and(P01P1)denotestheMoore-Penrosegeneralizedinverse.Theresidualinthestepbvi2=yi2bhn(zi)=yi2pk1(n)1(zi)0(P01P1)nåi=1pk1(n)1(zi)yi2isobtainedasaninputfortwounknownfunctionsm()andq()insievespacesMnandQn,wherem(bvi2)=P2;l(n)(bvi2)0dmandq(bvi2)=P2;l(n)(bvi2)0dq.Similarly,thetrueparameterestimatorgointhesecondstepisestimatedbymaximizinganaturalsampleanalogofthesievemaximumlikelihoodproblemininAssumption3.3over59theapproximatingsievespacesGn,where(q;m();q())isjointlyestimatedbyMLE,assumingbivariatenormaldistribution.bgn=argmaxg2QMnQn1nnåi=1logF2di1hxi1b+m(yi2bhn(zi))i;di2hzig+q(yi2bhn(zi))i;di1di2rUnfortunately,thissievemaximumlikelihoodestimatorbgninthesecondstepdoesnothaveaclosed-formsolution,butweknowithasthreecomponents,bgn()=bq;bmn();bqn():2.3.4AfunctionalofcausalinterestAlthoughwehavepreviouslyintheAssumptionin3.2andAssumption3.3andestimationinequations(2.8)and(2.3.3)forparameters(ho;go)inthetwo-stepprocedure,itdoesnotdeliveracausaleffectofinterestdirectly.Ithasbeenwidelyrecognizedthattheco-efofthisnonlinearmodelaregenerallyoflimitedinteresttoempiricalresearchersŒthecoefthedirectionoftheimpactbutnotthemagnitude.Parameters(ho;go)insemi-parametricbinaryresponsemodel(2.6)havethesamedrawback.Bycontrast,theaveragepartialeffect(APE)basedontheaveragestructuralfunction(ASF)givesthemagnitudeoftheimpact,fortheexogenousshiftintheregressors,onthebinaryresponse.Thismeasureisguaranteedtoprovideasummarynumberthatiscounterfactualinnatureandhasthesamesignasthecoefofthevariableofinterest.,theASFforthestructuralequation(2.1a)inthistriangularbinaryresponsemodel(2.6)isafunctionofx1.Itiscounterfactuallybyholdingx1edandtakingexpectationwithrespecttounobservableui1ASFx1Eui1(1[x1b+ui10]);wheresubscriptidenotesarandomvariable.Operationally,thisisequivalenttoplugx1bintheCDFofui1,ASFx1Fui(x1b):60AsinAssumption2.2(c),structuralerrorui1=mo(vi2)+vi1,wherevi1isindepen-dentofvi2.Bylawofiteratedexpectation,takeexpectationwithrespecttothestructuralerrorui1isequivalenttotakeaconditionalexpectationofui1givenvi2,whichisvi1,andthentakeexpectationwithrespecttovi2,ASFx1Evi2;vi1(1[x1b+mo(vi2)+vi10])=Evi2hEvi1(1[x1b+mo(vi2)+v1i0])i=Evi2(F[x1b+mo(vi2)]):(2.9)Sincevi2isobtainedasarandomfunctionalofthedatawi1,andfunctionho,namely,vi2=yi2ho(zi)(2.10)Takingexpectationwithrespecttovi2asinequation(2.9)isequivalenttoplugginginequation(2.10)andtakingexpectationwithrespecttowi1asfollows,ASFx1Ewi1[F(x1b+mo[yi2ho(zi)])]:(2.11)AstherandomnessinASFonlyarisesfromtheunobservable,andithasbeenremovedbytakingexpectation,ASFisataedvalueofx1.APEsareasdifferentiatedASFwithrespecttothevariableofinterest.Inthistriangularbinaryresponsemodel(2.6),APEsarebasedonASFinequation(2.11).Letry2(ho;go)inequation(2.12a)denotetheAPEforcontinuousEEVy2,whichisaderivativeofASFwithrespecttoy2.Letry3(ho;go)inequation(2.12b)denotetheAPEforadummyEEVy3,whichisadifferenceofASFatdifferentoutcomesofy3.ry2(ho;go)b2Ewi1hF(1)ii;(2.12a)ry3(ho;go)Ewi1[Fi1Fi0];(2.12b)61whereF(1)i¶F(t)¶tt=x1b+mo[yi2ho(zi)]=f(x1b+mo[yi2ho(zi)]);Fi1Fx1(3)b(3)+b3+mo[yi2ho(zi)];Fi0Fx1(3)b(3)+mo[yi2ho(zi)]:Fornotationalclarity,b2isthecoeffory2andb3isthecoeffory3,embeddedinthelinearindexx1bstructuralequation(2.1a).(3)insubscriptofx1(3)b(3)denotesalinearindexx1bwithy3excluded.Duetothenatureofthenonlinearstructuralequation(2.1a),bothry2(;)andry3(;)arenonlinearfunctionalsthattaketheparameters(ho;b;mo)ˆ(ho;go)frombothstepsasarguments.x1isheldasaedargumentandisabsorbedintothefunctionalformassumption.However,sinceweareignorantaboutthedistributionofdatainstagewi1,thereisnowayforustosolvetheexpectationanalytically,buthavetoestimatethenonlinearfunctionalbymethodofmoments.ThisistomakesampleanalogsofthetrueAPEsinequations(2.12a)andequation(2.12b),withthetwo-stepestimatorsbhn;bb;bmnpluggedin.Formally,bry2bhn;bgn=bb2"n1nåi=1bF(1)i#;(2.14a)bry3bhn;bgn=n1nåi=1bFi1bFi0;(2.14b)wherebF(1)ifhx1bb+bmn(yi2bhn(zi))i;bFi1Fx1(3)bb(3)+bb3+bmn(yi2bhn(zi));bFi0Fx1(3)bb(3)+bmn(yi2bhn(zi));f()isthenormaldensityandF()isthenormalCDF,bhn;bb;bmnareobtainedfromequations(2.8)and(2.3.3).622.3.5ProcedureAfterpresentingallthetechnicaldetailofestimation,sievespacesandafunctionalofempiricalinterest,thissectionconcludeswithaProcedure3.1thatalsoincludesathirdstepestimatorforAPEs.Procedure3.1Moreformally,atwo-stepsieveM-estimationaimstoconsistentlyestimatethetrueparameters(ho;go)inAssumptions(3.2)and(3.3),alongwithathirdstepaimstoestimatethetrueAPEsfory2andy3inEquations(2.12a)and(2.12b),goesasfollows.(a)RuntheOLSofy2onpk1(n)1(z)asin(2.7)atachoiceofk1(n)byaresearcher,toobtaintheresidualbv2:(b)Plugbv2intopl(n)2(bv2),runabiprobitusinghx1;pl(n)2(bv2)iasregressorsinthestequationandhz;pl(n)2(bv2)iasregressorsinthesecondequation,toobtainthecoeftsonregressorshx1;pl(n)2(bv2)i:(c)Foreachobservation,constructthecounterfactualAPEsasinequations(2.14a)and(2.14b)byholdingx1butaverageacrossbvi2.Thefollowingsessiongivestheconsistency,asymptoticvariance,andconsistentestimatesofthevariance.2.4AsymptoticProperties2.4.1ConsistencyThefollowingtwotheoremsprovideconsistencyresultsforthetwo-stepsieveM-estimatesoftheparameters(ho;go)andathird-stepmethodofmomentsestimationofAPEsry2(ho;go)andry3(ho;go).Theorem4.1(a)UnderAssumptions3.1,3.2and3.4,letbhnbethesieveleastsquaresestimatorintheststep,byequation(2.8),wehavedhbhn;ho=op(1).63(b)Giventheststepbhnestimatorisconsistent,underAssumptions3.1,3.3and3.5,letbgnbethesievemaximumlikelihoodestimatorinthesecondstep,byequation(2.3.3),wehavedg(bgn;go)=op(1).TheassumptionsforconsistencyoftheestimatorbhnareintendedtolinkthetheoreticalconditionsinTheorem3.1ofChen(2007)tothisparticularmodelin(2.6).Sincetheassumptionsandestimationequationslistedabovehavealreadycriterionfunctionswhichthecontinuityanduniformconvergencecondition,wedon'tneedtorestatetheirgeneralconditions.Theconsistencyofthesecond-stepestimatorbgntakesintoaccountthefactthataconsistentestimatorbhnhasbeenpluggedintothelikelihoodfunction.Byasimilarargument,sincethelikelihoodfunctionistheCDFofabivariatedistribution,whichisafunctioncontinuousinbothofitsarguments,giventheconsistentestimatorbhn,theaboveassumptionsarealsosuftoguaranteeconsistency.Theorem4.2AssumethatthetrueAPEsofinterestarenonlinearfunctionals,denotedbyry2(ho;go)andry3(ho;go)asinequations(2.12a)and(2.12b).Underconsistencyofthetwo-stepsieveM-estimatorsbhn;bgn,amethodofmomentsestimatorofAPEsbry2(;)andbry3(;)withthetwo-stepM-estimatorsbhn;bgnpluggedinastheirarguments,asbyequations(2.14a)and(2.14b),areconsistent.Formally,bry2bhn;bgnp!ry2(ho;go)andbry3bhn;bgnp!ry3(ho;go):AsshowninTheorem4.2,consistencyoftheestimatesofAPEsnotonlyreliesonconsistencyoftheestimatorforthefunctionals,whichisundertheframeworkofmethodofmoments,butalsoonconsistencyofthetwoplugged-inestimatorsbhn;bgn.Atheoreticalframeworkthatnestsmodel(2.6)isprovidedinChen,LintonandVanKeilegom(2003).SincetherandomfunctionalinvolvedinconstructingAPEsisastandardnormalCDF,theiruniformcontinuityconditionis642.4.2AsymptoticnormalityTheasymptoticnormalityforestimatesofAPEsdrawsonthesieveinferenceliteraturefornon-linearfunctionals.Inparticular,Theorem3.1inHahn,LiaoandRidder(2015)providesageneralasymptoticnormalityresultforaknownfunctionalthattakesthesievetwo-stepM-estimatesasitsarguments.Inourcase,thedifferenceliesinthatthenonlinearfunctionalneedstobeestimatedbymethodofmoments,insteadofbeingknowntoaresearcher.Sincemethodofmomentsentitlesconsistencybyusualargumentwhenanestimatesofeddimensionalparametersareplugged-in,wehavebry2bhn;bgnp!ry2bhn;bgnandbry3bhn;bgnp!ry3bhn;bgn.Then,byapplyingastochasticequi-continuityunderstandardregularityconditions,onecanshowthat,suph;g2Nh;nNg;npnbry2(h;g)ry2(h;g)=op(1)(2.15)andhencetheasymptoticdistributionofbry2bhn;bgnisequivalenttothatofry2bhn;bgn.Bythesamereasoning,bry3bhn;bgnisequivalenttothatofry2(h;g).Inthefollowing,Iintroducetheforvy2;hn;vy2;Gn;vy2;gnandvy3;hn;vy3;Gn;vy3;gn,whicharecomponentsoftheasymp-toticvariancevy2;n2sdandvy3;n2sdforAPEsfory2andy3.Assumethatbhn;bgn,theestimatesfromequations(2.8)and(2.3.3),belongtotheshrinkingneighborhoodNn=(h;g):h2Nh;nandg2Ng;nwpa1,whereNh;n=fh2Hn:khhokHd1ngNg;n=fg2Gn:kggokGd2ngd1n=d1n(log(log(n)))d1ndenotestheconvergencerateofthesieveMestimateunderthemetrickkH.Thesameisford2n.Forallh2Nh;n,denotethecriterionfunctionasj(wi1;ho)[yi2ho(zi)]2.Then,thedifferenceinthecriterionfunctionj(wi1;h)j(wi1;ho)canbeapproximatedlinearlybyDj(wi1;ho)[hho],where65Dj(wi1;ho)[vh]¶j(wi1;ho+tvh)¶tt=0foranyvh2Nh;nfhog:(2.16)Foranyvh1;vh22Nh;n,aninner-productonNh;nasDvh1;vh2Ej¶EhDjwi1;ho+tvh2hvh1ii¶tt=0:LetV1betheHilbertspacegeneratedbyHfhogundertheinnerproducth;ij,andkvk2j=hv;vij.Duetoasymptoticequivalencetheorem,wewillfocusonry3bhn;bgntoderivetheasymp-totics.Assumetherearelinearfunctionals¶ry2(ho;go)¶h[]:V1!Rsuchthat¶ry2(ho;go)¶h[v]¶ry2(ho+tv;go)¶tt=0forallv2V1:Letho;ndenotetheprojectionofhoonHnunderthenormkkj.LetV1;ndenotetheHilbertspacegeneratedbyNh;nfho;ng.ThendimV1;n=k1(n)<¥.ByRieszrepresentationtheo-rem,therearesieveRieszrepresentersvy2;hn2V1;nsuchthat¶ry2(ho;go)¶h[v]˝vy2;hn;v˛jforallv2V1;n:Foranyg2Ng;n,denotethesecond-stepcriterionfunctionasy(wi2;g;ho)logF2(di1[xi1b+m(yi2ho(zi))];di2[zig+q(yi2ho(zi))];di1di2r):Thenthedifferenceinthecriterionfunctiony(wi2;g;ho)y(wi2;go;ho)canbeapproxi-matedlinearlybyDy(wi2;go;ho)[ggo],whereDy(wi2;go;ho)vg¶ywi2;go+tvg;ho¶tt=0foranyvg2Ng;nfgog:Foranyvg1;vg22Ng;n,weaninner-productonNg;nasDvg1;vg2Ey¶EhDywi2;go+tvg2;hohvg1ii¶tt=0:66LetV2betheHilbertspacegeneratedbyGfgogundertheinnerproducth;iy,andkvk2y=hv;viy.Weassumethatthereisalinearfunctional¶ry2(ho;go)¶g[]:V2!Rsuchthat¶ry2(ho;go)¶g[v]¶ry2(ho;go+tv)¶tt=0:Letgo;ndenotetheprojectionofgoonGnunderthenormkky:LetV2;ndenotetheHilbertspacegeneratedbyNg;nfgo;ng:ThendimV2;n=k2(n)<¥.ByRieszrepresentationtheo-rem,therearesieveRieszrepresentersvy2;gn2V2;nsuchthat¶ry2(ho;go)¶g[v]Dvy2;gn;vEyforallv2V2;n:(2.17)LetV=V1V2.Foranyv=vh;vg2V,wedenoteao(ho;go)¶ry2(ao)¶a[v]¶ry2(ao)¶h[vh]+¶ry2(ao)¶gvg¶ry2ho+tvh;go+tvg¶tt=0Toevaluatetheeffectoftheestimationontheasymptoticvarianceofthesecond-stepsieveMestimator,weafunctionalG(ao)[;]onVasG(ao)[v1;v2]=¶2E[y(wi2;go+t2v2;ho+t1v1)]¶t1¶t2t1=0;t2=0forany(v1;v2)2V:WeassumethatG(ao)[;]isabilinearfunctionalonV.GiventheRieszrepresentersin(2.17),vy2;Gn2V1;nG(ao)hvh;vy2;gniDvh;vy2;GnEforanyvh2V1;n:UsingthesieveRieszrepresentervy2;hn;vy2;gnandvy2;Gn,wetheasymptoticvarianceforAPEofy2asvy2;n2sdVar"n12nåi=1Dj(wi1;ho)vy2;hn+vy2;Gn+Dy(wi2;go;ho)hvy2;gni#:(2.18)Bythesameprocedure,usingthesieveRieszrepresentervy2;hn;vy2;gnandvy2;Gn,wetheasymptoticvarianceforAPEofy3asvy3;n2sdVar"n12nåi=1Dj(wi1;ho)vy3;hn+vy3;Gn+Dy(wi2;go;ho)hvy3;gni#:(2.19)67AfterverifyingtheassumptionsinTheorem3.1ofHahn,LiaoandRidder(2015)asinAp-pendixB.1,thefollowingTheorem4.3statesthatathirdstepestimatorforAPEshasarootnasymptoticnormality.Theorem4.3MethodofmomentsestimatorofAPEsbry2(;)andbry3(;)withthetwo-stepM-estimatorsbhn;bgnpluggedinastheirargumentshavethefollowingasymptoticdistribution,pnhbry2bhn;bgnry2(ho;go)ivy2;nsdd!N(0;1);pnhbry3bhn;bgnry3(ho;go)ivy3;nsdd!N(0;1):wherevy2;nsdandvy3;nsdarein(2.18)and(2.19).2.4.3ConsistentvarianceestimationInthefollowing,Iintroducethenotationofbvy2;nn;sdasanaturalsampleanalogofvy2;nsdforbry2bhn;bgn.Thenotationofbvy3;nn;sdfollowsthesameprocedureandthusisomitted.DenoteDj(wi1;h)hvh1i¶jwi1;h+tvh1¶tandrj(wi1;h)hvh1;vh2i¶Djwi1;h+tvh1hvh2i¶tt=0foranyvh1;vh22V1;n:Similarly,Dy(wi2;g;h)hvg1iandry(wi2;g;h)hvg1;vg2iforvg1;vg22V2;n:theempiricalRieszrepresenterbvy2;hnby¶ry2(ban)¶h[vh]˝vh;bvy2;hn˛n;jforanyvh2V1;n68whereDvh1;vh2En;j1nåni=1rjwi1;bhnhvh1;vh2i.Similarly,webvy2;gnas¶ry2(ban)¶gvgDvg;bvy2;gnEn;yforanyvg2V2;nwhereDvg1;vg2En;y1nåni=1rywi2;bhn;bgnhvg1;vg2i;andbvy2;Gnandbvy3;GnasGnbhn;bgnhbvy2;gn;vhiDvh;bvy2;GnEn;jforanyvh2V1;nwhereGnbhn;bgnhbvy2;gn;vhi1nnåi=1¶Dywi2;bgn;bhn+tvhhbvy2;gni¶tt=0Asasampleanalogof(2.18)and(2.19),consistentvariancearebvy2;n2n;sd=1nnåi=1Djwi1;bhnhbvy2;hn+bvy2;Gni+Dywi2;bgn;bhnhbvy2;gn2;bvy3;n2n;sd=1nnåi=1Djwi1;bhnhbvy3;hn+bvy3;Gni+Dywi2;bgn;bhnhbvy3;gn2:Theorem4.4Supposethatthedataarei.i.d.andtheconditionsinTheorem4.3areThenunderAssumptionB.4,B.5andB.6inAppendixB.2,wehavebvy2;nn;sdvy2;nsd1=op(1);bvy3;nn;sdvy3;nsd1=op(1):Therefore,pnhbry2bhn;bgnry2(ho;go)ibvy2;nsdd!N(0;1);pnhbry3bhn;bgnry3(ho;go)ibvy3;nsdd!N(0;1):SeeAppendixB.2forproofofthetheorem.692.5PracticalInferenceAshintedbyProcedure3.1,aresearchercanproceedbyaddingacertainnumberofbasisfunc-tionstotheparametrictwo-stepbiprobit,ignoringthefactthattrueunknownfunctionsaredimensional,butassumingtheunknownfunctionsisinfactdimensionalwithbasisfunc-tionsatthelengththeresearcherhappenstochoose.Thisway,inferenceforparametersofinterestcaneasilybedonebyastandarddeltamethodasforexampleinWooldridge(2010).ThisideaofnumericalequivalencebetweenthesemiparametricvarianceandpracticalinferencehasbeendevelopedinAckerberg,ChenandHahn(2012)forasemiparametricandsecond-stepparametricestimator.Foraprocedurewherebothstepsarenonparametric,Hahn,LiaoandRidder(2015)showstheequivalenceholdsgenerally.Applyingthisnumericalequivalenceresult,thefollowingprovidesaconsistentestimatoroftheasymptoticvariance.Iassumethat,foragivendataset,aresearcherbelievestheunknownparameters(ho;go)areinfactdimensionalwithdimensionsedatk1=k1(n)andl=l(n).Formally,ho()=pk11()0dho;(2.22a)go=(q;mo();qo())=q;pl2()0dmo;pl2()0dqo:(2.22b)wherepk11()isbasisfunctionforHnin(2.7)andpl2()isbasisfunctionforMnandQn.Hence,foraparametricproblem,dhoisestimatedbyaparametricleastsquarebdh=argmaxdh2Dh1nnåi=1hyi2pk11(zi)0dhi2;wheretheparameterspaceDhisacompactsetinRk1:Havingpluggedintheestimatebdh,letdgoq;dmo;dqobeparametersestimated70inthesecond-step,bdg=(bq;bdm;bdq)=argmax(b;r;g;dm;dq)2QDmDq1nnåi=1logF2di1xi1b+pl2hyi2pk11(zi)0bdhi0dm;di2zig+pl2hyi2pk11(zi)0bdhi0dq;di1di2r;whereQ,DmandDqarecompactsetsinRdq,RlandRl,Byachangeofnotation,theAPEsofinterestatdimensionry2ho;n;go;nandry3ho;n;go;ncanbeintermsofthecoefonthebasisfunctionpy2dho;dgoandpy3dho;dgo,ry2ho;n;go;n=ry2pk11()0dho;q;pl2()0dmo;pl2()0dqopy2dho;dgo=b2Ewi1hF(1)i;pi;andry3ho;n;go;n=ry3pk11()0dho;q;pl2()0dmo;pl2()0dqopy3dho;dgo=Ewi1Fi1;pFi0;p:whereF(1)i;p¶Fp(t)¶tt=x1b+pl2yi2pk11(zi)0dho0dmo=fx1b+pl2hyi2pk11(zi)0dhoi0dmo;Fi1;pFx1(3)b(3)+b3+pl2hyi2pk11(zi)0dhoi0dmo;Fi0;pFx1(3)b(3)+pl2hyi2pk11(zi)0dhoi0dmo:71Now,absorbtherandomnessofthestepdatawi1intosubscripts,andletthepartialeffectforanindividualidenotedbyri;y2dho;dgob2F(1)i;p;ri;y3dho;dgoFi1;pFi0;pBythesamechangeofnotation,thesampleanalogsofthefunctionalpy2dho;dgoandpy3dho;dgoareasfollows,bpy2bdh;bdgbb2"n1nåi=1bF(1)i;p#;bpy3bdh;bdgn1nåi=1hbFi1;pbFi0;pi;wherebF(1)i;p¶Fp(t)¶tt=x1bb+pl2yi2pk11(zi)0bdh0bdm=fx1bb+pl2hyi2pk11(zi)0bdhi0bdm;bFi1;pFx1(3)bb(3)+bb3+pl2hyi2pk11(zi)0bdhi0bdm;bFi0;pFx1(3)bb(3)+pl2hyi2pk11(zi)0bdhi0bdm:Sincewecannotobservethetrueparametersdho;dgo,weplugintheirestimatesintotheindividualpartialeffectsinstead,ri;y2bdh;bdgbb2bF(1)i;p;ri;y3bdh;bdgbFi1;pbFi0;p.Understandardregularityconditionsforparametricestimation,itiseasytoshowthatfollowingpropositionholdsforinferenceoftheAPEsintheparametricproblem.NoteAPEsisasampleanalogofnonlinearfunctionthatcontainstheparameterfrombothstepsasarguments.72Proposition5.1Whenaresearcherbelievesthattheparametersdho;dgoareatdi-mensionasinequations(2.22a)and(2.22b),estimatesofAPEsofy2andy3hasaasymptoticdistributionasfollows,pnhbpy2bdh;bdgpy2dho;dgoid!N0;Vy2;pnhbpy3bdh;bdgpy3dho;dgoid!N0;Vy3;whereVy2andVy3aretheasymptoticvarianceasVy2Varhri;y2dho;dgopy2dho;dgoRy2o;hH1o;hSi;hRy2o;gH1o;gSi;g+Fo;ghH1o;hSi;h;Vy3Varhri;y3dho;dgopy3dho;dgoRy3o;hH1o;hSi;hRy3o;gH1o;gSi;g+Fo;ghH1o;hSi;h:Proof.SeeAppendixB.3fortheoftherestofthetermsandderivation.Sincenowtheparametersliveinthedimensionalspace,bytheusualargument,thecon-sistentestimatorofthevarianceVy2andVy3aresampleanalogs,bVy21nnåi=1hri;y2bdh;bdgbpy2bdh;bdgbRy2hbH1hbSi;hbRy2gbH1gbSi;g+bFghbH1hbSi;h2;bVy31nnåi=1hri;y3bdh;bdgbpy3bdh;bdgbRy3hbH1hbSi;hbRy3gbH1gbSi;g+FghbH1hbSi;h2;where‹denotesevaluatedasbdh;bdgandreplaceexpectationE[]withsampleaverage1nnåi=1.Sincetheparametersareinspace,theconsistencyfollowseasily.2.6SimulationStudy2.6.1DesignsThissectionprovidesasmallscaleMonteCarlosimulationtoassessthesamplebehavioroftheproposedestimatorcomparedwithotheralternativemethods.ThenumberofMonteCarloreplicationis1000.Ineachreplication,sievetwo-stepM-estimatorsareperformedwithbasisfunctionofdimensionK=1,K=2andK=3forunknownfunctionsinbothsteps.Thesenumbers73arepickedtomimicasimple,reasonableprocedurethatareoftenseeninempiricalstudies.Nooptimalityisinvolvedintheselectionforthelengthofsieve.Theoretically,solongasbothofthesievespacesgrowwiththesamplesize,thedimensionofsievespacesareallowedtobedifferentineachstepwithoutchangingtheasymptoticbehavior.HereIthemtobethesameforsimplicity.Bothcommonlyusedparametricestimatorsandsemiparametricestimatorsarecomparedtotheproposedsieveestimator.ThedegeneratedcaseofK=1istheparametrictwo-stepbiprobit.APEsfromthosenonlinearmodelsarecomputed.Thecoeffromaparametriclinearprob-abilitymodelestimatedwithtwo-stageleastsqauresarealsoreportedtobecomparedwithAPEs.AlternativesemiparametricestimatorslikethespecialregressormethodbyLewbel(2000)doesnotpermitestimatingaveragepartialeffectsasitdoesnotrecoverthestructuralerror.Inthosecases,ratiosofthecoefforbinaryexplanatoryvariableoverthecoefofcontinuousexplanatoryvariablearecomparedacrosssemiparametricmethods.Althoughheretheratiosarenotthesameasrelativeeffects,theyaremoremeaningfulquantitiesthancoefentsinthebinaryresponsemodel,whicharesubjecttoscalingandnormalization.Specialregressorestimatorsus-ingkerneldensityestimationandK-NearestNeighborestimationsarebothreported.Atrimmingparameterof1%ischosentoeliminateoutliers.Whilenotrimmingleadstolargebias,atoolargetrimmingleadstoabigvariance.Onlyresultsforsamplesizeof1000isreportedinthetables.LimitedexperimentswithothersamplesizesindicatethatthebiasesforsemiparametricmethoddiminishasNincreases,thoughratherslowly.Theestimatorsarecomparedunderthreedesigns.Thesethreedesignsdiffersinitslevelofmis-oftheerrorterms.Design1servesasamiddlegroundwheretheerrortermsfollowaconditionalnormalityasinAssumption2.2andonlythetwo-stepsieveestimatorsareconsistent.Design2istheextremecaseoffullymisswhereerrortermsfollowajointChi-squareddistributionsothatAssumption2.2isviolatedandnoneoftheestimatorsareconsistent.Design3istheextremecaseofcorrectwhereerrortermsfollowatrivariatenormaldistributionasinAssumption2.1andallestimatorsareconsistent.Design174Thedatageneratingprocessisasfollows:y1=1[z1+y2+y3+mo(v2)+v10];y2=ho(z2;z3)+v2;y3=1[0:1z2+z3+qo(v2)+v30];mo(v2)=11+exp(v2);ho(z2;z3)=log(jz2+0:1z3j=2:5);qo(v2)=v2;v1=v3+e1;e1˘Normal(0;1);v2˘0:8Normal(1;:6)+0:2Normal(4;2);v3˘Normal(0;1);z1˘Normal(0;9);e2;z3˘i.i.d.Normal(0;1);z2=1[e2>0]:Usingthesamesetofnotationasinmodel(2.6),lety1bethebinaryoutcome,y2thecontinuousEEVandy3thedummyEEV.Thecontrolfunctionsmo(),qo()andho()areapproximatedbysievespacesusinglinearspansofpowerseriesasbasisfunctions.Letv2beNormal(1;:6)withprobability0.8andNormal(4;2)withprobability0.2.Themixtureoftwonormaldistributionstisdesignedtoyieldadistributionthatisbothskewedandbimodal,butstillhasmeanzeroandvarianceone.(z1;z2;z3)arethewholesetofexogenousvariables.z1istheincludedinstrumentinthestructuralequationandisdesignedtomeettheconditionsofthespecialregressorinthespecialregressormethod:z1isindependentofallvariables,hasalargesupport(representedbybeingdrawnfromadistributionwithvarianceofnine),andisexcludedfromthereducedformsforEEVsy2andy3.Toillustratetheimportanceoflargesupportofz1inreducingthebiasofthespecialregressormethod,aDGPwithasmallersupportisalsopresentedinDesign1where75z1˘Normal(0;1):z2isbinaryandtheprimaryinstrumentforthecontinuousEEVy2.z3iscontinuousandtheprimaryinstrumentforthebinaryEEVy3.Letho()denotetheconditionalmeanfunctionfory2givenz2andz3andisdesignedtobealogtransformation,asoftenseeninempiricalapplications.Letmo()betheconditionalmeanofu1givenv2andisdesignedtobeasquashingfunctionthatshrinksitsoutputtotheunitinterval.ThissquashingfunctionistakenfromthepartiallinearmodelinthesimulationstudyofAiandChen(2003).Letqo()betheusuallevelformthatdoesnotputarestrictionontherangeofitsoutput.Inaddition,theformulaforsimulatedAPEsfory2andy3ataed(z1;y2;y3)isry2=1p2nåi=1f1p2(z1+y2+y3+mo(vi2));ry3=nåi=1F1p2(z1+y2+1+mo(vi2))F1p2(z1+y2+mo(vi2)):Theratioweareinterestedinforsemiparametricmethodsisb3b2=1.Design2Thedatageneratingprocessisasfollows:y1=1[z1+y2+y3+u10];y2=z2+0:1z3+v2;y3=1[0:1z2+z3+u30];u1=v2+v3+e1˘c233;u3=v2+v3˘c222;v2˘c211;z1˘Normal(0;9);e2;z3˘i.i.d.Normal(0;1);z2=1[e2>0]:76Design2sharesthesamestructuralequationasinDesign1,butthejointdistributionoftheerrortermsandthereducedformsfortheEEVsaredifferent.TheerrortermsfollowajointChi-squareddistributionofc211.Namely,errortermu1isconstructedasthesumofv2;v3andanadditiveer-rore1,eachwithademeanedChi-squareddistribution.SincealinearcombinationofChi-squareddistributionstillbelongstotheChi-squaredfamily,u1isademeanedChi-squaredwithdegreeoffreedomthree.Similarly,u3isademeanedChi-squaredwithdegreeoffreedomtwo.Thereducedformfory2isnowalinearcombinationoftheexogenousvariablesz2andz3,insteadofanonlinearfunction.Asc211hasalargervariancethanNormal(0,1),theerrortermshavealargervarianceinDesign2.Soz1˘Normal(0;9)becomesasmallsupportthatisnotsuftocoverrestofthevariationinthestructuralequation.Instead,z1˘Normal(0;16)isdesignedtobethecaseofalargesupport.LetFc23()denotetheCDFofc23withdegreeoffreedomthreeandfc23()denotethecorrespondingpdf.Sincealltheerrortermsbelongtothesamedistributionalfamily,APEsfory2andy3ataed(z1;y2;y3)canbeexpressedanalytically,ry2=fc23[3(z1+y2+y3)];ry3=Fc23[3(z1+y2)]Fc23[3(z1+y2+1)]:Theratioofcoefofy3overy2isstillb3b2=1.Design3Thedatageneratingprocessisasfollows:y1=1[z1+y2+y3+u10];y2=z2+0:1z3+v2;y3=1[0:1z2+z3+u30]0BBBB@u1v2u31CCCCA˘Normal2666640BBBB@0001CCCCA;0BBBB@10:50:6250:510:50:6250:511CCCCA37777577z1˘Normal(0;9);e2;z3˘i.i.d.Normal(0;1);z2=1[e2>0]:Design3istheoppositeofdesign2whereallerrortermsaretri-variatenormallydistributed,soaparametricbiprobitisdesignedtodeliverthebestresult.However,Iexperimentwithvariouslengthofsievestodemonstratetheconsequenceofaddingtoomanysievetermsinpractice.Likeindesign2,theAPEsfory2andy3ataed(z1;y2;y3)doesnotneedtobesimulatedandcanalsobeexpressedanalytically,ry2=12nåi=1f12(z1+y2+y3);ry3=nåi=1F12(z1+y2+1)F12(z1+y2):2.6.2ResultsFigure2.1,Figure2.2andFigure2.3presenttheempiricaldistributionsofAPEsfory2andy3andtheratioofb3b2forsixestimatorsinDesign1,Design2andDesign3,respectively.CFBipro-bitK=3isthecontrolfunctionestimatorwithunknownfunctionsapproximatedbysievespacesofdimensionthree.CFBiprobitK=2isthecontrolfunctionestimatorwithunknownfunctionsapproximatedbysievespacesofdimensiontwo.CFBiprobitisthecaseofK=1andsimplyapara-metrictwo-stepcontrolfunctionestimator.2SLSisthelinearprobabilitymodelestimatedwithusualIVapproach.SRKernelisthespecialregressormethodusingkernalmethodandSRKNNisthespecialregressormethodusingK-NearestNeighbormethod.CFBiprobitK=3,CFBiprobitK=2,SRKernelandSRKNNarethesemiparametricestimators.CFBiprobitand2SLSaretheparametricestimators.TheredverticalbardepictsthetrueAPEineachcase.Figure2.1describestheperformanceoftheestimatorsinDesign1whenz1hasalargesupport.Forallthreequantitiesofinterest(APEfory2,APEfory3andb3b2),CFBiprobitK=3,CFBiprobit78K=2andCFBiprobitareallhighlypreciseandslightlybiased.Thebiasesdecreasewiththeincreaseofthedimensionofsievespace,althoughtoasmallermagnitude.Whenz1isdesignedtohavealargesupport,specialregressormethodsaresupposedtohaveasmallerbiascomparedtothecasewherez1hasasmallsupport.Thisisillustratedbytheempiricaldistributionoftheb3b2onthelowerleftcorner.Intermsoftherelativeeffects,SRKNNhasasmallerbiaswhileSRKernelhasahigherprecision(peak).2SLShasthelargestbiastotheright.Intermsofpartialeffects,AIFisusedforspecialregressormethodsforcomparisonwithAPEsfromCFBiprobitK=3,CFBiprobitK=2andCFBiprobit.However,bydesign,AverageIndexFunctions(AIFs)fromspecialregressormethodarebasedonadifferentresponsefunctionfromtheAPEs.SeeLinandWooldridge(2015b)forfurtherdiscussionsondifferencesbetweenAIFsandAPEs.SofortheAPEofy2,AIFfromspecialregressorhasalargebiastotheright.SRKNNhasaslightlylargerbiasthanthatofSRKernel.Thebiasofcoeffrom2SLSisbetweenthespecialregressormethodsandthesievemethods.FortheAPEofy3,AIFsfromSRKernelandSRKNNalmostcenteraroundthetrueAPE,butwithlessprecision.2SLSisbiasedtotheright.Figure2.2describestheperformanceoftheestimatorsinDesign1exceptnowz1hasasmallsupport.Thistime,specialregressormethodsarefarbiasedinallcases,evenforrelativeeffectsasb3b2.Whencomparingpartialeffects,astheAIFisinvolved,itislessclearwhetherthehugebiasisduetothenatureofthespecialregressormethodorthewaythepartialeffectisconstructed.Forthesievemethods,withasmallerspreadofz1,ahigherweightisplacedontheerrortermwhenestimatingtheparameters.Itismoreclearnowhowtheincreaseofdimensionimprovesapproximationineachcase.TheimprovementfromaddingonedimensionislargewhenKgoesfromtwotoone,butbecomesverysmallwhenKgoesbeyondthree,indicatingK=2wouldbeaoptimalsievelengthforthegivensamplesize.2SLShasmoderatebiasforpartialeffectsfory2andy3andahugebiasfortherelativeeffects.Figure2.3describesthecasewhenthesupportofz1isrelativelysmallanderrortermsarefullymisasjointChi-squared,violatingthetrivariatenormalityorconditionalnormalityassumption.TheCFBiprobitK=3,CFBiprobitK=2andCFBiprobitstillperformsthebest79withsmallbiasandhighprecision.However,increasingofdimensioninthesievespacedoesnotimprovetheapproximationduetotheFurther,althoughspecialregressormethodsdoesnotseemtohavealargebiasintermsofthepartialeffectsfory2andy3,theyhavehugevariancesintherelativeeffects.2SLShasmoderatebiasandarelativelyhighprecisioninallcases.Thebehaviorsoftheestimatorsdoesnotchangemuchwhenthesupportofz1islargerasinFigure2.4duetotheFigure2.5describestheidealcasewheretheerrortermsarejointlynormaldistributedwithalargesupportofz1.SpecialregressormethodsarecenteredaroundthetruevaluefortherelativeeffectbuthavemoderatebiasforthepartialeffectsduetothewayAIFisCFBiprobitK=3,CFBiprobitK=2andCFBiprobitareoverlappedwitheachbecausethereisnoimprovementinaddingmoredimensions.2SLShaveasmallerbiascomparedtothepreviousdesigns.Table2.1(Design1)reportsthebiasandRMSEforthethreequantities(APEfory2,APEfory3andb3b2)withbothlargeandsmallsupportofz1.Whenthesupportissmall,notonlythespecialregressorsbutalsoalltheotherestimatorshavealargerbias.TheRMSEisalsobiggerforalltheestimatorswhenthesupportissmall,exceptfortherelativeeffectestimatedbyCFBiprobit.Theimprovementfromaddingmoredimensionsinthesievespaceislessobviousinthecasewheresupportislargethanthecasewheresupportissmall.Theimprovementfromsieveapproximationismoreobviousforthepartialeffectsofy3thanfory2.Table2.2(Design2)reportsthebiasandRMSEforthethreequantitieswithbothlargeandsmallsupportofz1.Althoughalloftheestimatorsaremithereisstillaimprovementinbiasforthepartialeffectofy2.Thereisnoobservedimprovementsforthepartialeffectofy3andtherelativeratio.Also,thevarianceforSRKernelandSRKNNarehugefortherelativeeffects.Thislargevariancedecreaseswiththeincreaseofthesupportofz1.Table2.3(Design3)reportsthebiasandRMSEforthethreequantitiesjustwithalargesupportsothatalltheestimatorsareconsistent.Althoughalltheestimatorsareconsistent,asbefore,therearestillimprovementsinincreasingthedimensionofthesievespaceforthepartialeffectofy2,y3andtheratio.80Tosummarize,thesimulationresultinDesign1showsthattheCFBiprobitwithunknownfunctionsapproximatedbysievemethodshassuperiorityoverotherexistingestimationmethods,whetherparametricornonparametric,inthattheestimatesofthetrueAPEsconvergesquicklyasthenumberofbasisfunctionsincreasesŒŒsufclosetothetrueAPEbysimplyaddingtwoorthreebasisfunctions.Design2demonstratestherobustnessofCFBiprobitwithunknownfunc-tionsapproximatedbysievemethodsunderAddingmoresievebasisfunctionsthoughnotidealdoesnotimpairefy.Design3illustratesthatundertrivariatenormalitybyaddingmoresievetermswouldincreasethemeanofAPEs,whichmightcorrectforsomebias.NotethatasdocumentedinAiandChen(2003),powerseriesmighthaveerratictailbehaviors,splinesievewillalsobeusedinfutureworktoexaminetheperformance.SincethesecondstepbiprobitisaMLEestimator,inpractice,softwarepackagesusespenalizationsmethodsinMLEtoovercomenumericalinstability,whichcreatesbias.Sothesimulationresultsmighthavesufferedfromthisembeddedproblem.2.7ConclusionandFutureWorkThispaperproposesatwo-stepsieveM-estimationviaacontrolfunctionapproachtoaccountforendogeneityinatriangularsystemforasingleindexbinaryresponsemodel.TheendogeneitycomesfromonedummyEEVandpotentiallymanycontinuousEEVs.Thissemiparametrictwo-stepprocedureservesasanextensionandimprovementoftheparametrictwo-stepbiprobitinLinandWooldridge(2015a),byaddingmorexibilityandrobustness.Inparticular,thetwo-stepsieveM-estimationrelaxesthetrivariatejointnormalityassumptionintheparametriccasetoaconditionalbivariatenormalityassumption.ThelatenterrorsforthebinaryoutcomeanddummyEEVaredecomposedinapartiallinearfashion.Eachofthelatenterrorscontainsanunknownfunctionasitsconditionalmean,plusanindependentremainderterm.Theremaindertermsofthelatenterrorsareassumedtobebivariatenormallydistributed.Thestagesieveleastsquaresregressionerrorenterstheunknownfunctionasarguments.Thesecond81stepsimultaneouslyestimatesthelinearindexesandtheunknowncontrolfunctions,byusingasievejointMLEofthebinaryoutcomeanddummyEEV.MethodofsievesmakesfunctionalformsmorexibleforreducedformsofthecontinuousEEVandimposelessrestrictionondistributionofthejointdistributionoferrorterms.Itistheshaperestrictionfeatureofthesievesthatenablesustodecomposetheerrortermsinthispartiallinearfashion,andthusthistwo-stepsieveprocedureisessentiallyturnedintoaparametricprocedureforagivensamplesize.Aresearchercouldsimplyaddacertainnumberofsievebasisfunctionstotransformvariablesbyhandinbothsteps.Averagepartialeffects(APEs)basedontheaveragestructuralfunction(ASF)isproposedasameasureofcausaleffects.Athird-stepestimationoftheAPEsemploysamethodofmomentsestimator,withtheprevioustwo-stepestimatorspluggedin.AsymptoticpropertiesfortheAPEsandtwo-stepestimators,suchasconsistencyandasymptoticnormality,areprovidedbymappingthismodeltoastreamoftheoreticalsieveliterature.Moreover,IshowapracticalinferencefortheAPEs,usingastandarddeltamethodforaparametricmultiplestepestimationinWooldridge(2010).Thenumericalequivalenceoftheparametricinferenceandthesemiparametricinferece,asestablishedinHahn,LiaoandRidder(2015),allowsresearcherstoturntothepracticalinferenceforsimplicity.IntheMonteCarloexperiments,IillustratehowthesieveapproximationperformswithavariationofsievelengthsunderdifferentdegreesofAsatopicofon-goingresearch,thistwo-stepsieveM-estimatorforthebinaryresponsemodelwithvariousnatureofendogeneityembracesnumerouspossibilitiesoffutureextensions.First,inordertofurtherrelaxthelevelofdistributionalassumptions,itiscomputationllyeasytoapplythesemi-nonparametricmaximumlikelihoodestimatorbyGallantandNychka(1987)towardsap-proximatingthemultivariatedensityofthelikelihoodfunctioninthesecondstep.Inthatcase,asymptoticsforatwo-stepestimatorneedtobecarefullyderived.Anotheravenueforrelaxingthejointdistributionistousesieveapproximationsforthecopulafunctionwhichisnowrequiredtobeparametricinliterature.Second,toaccommodateforotherpracticalcomplicationsineconomicdata,interactionsbetweentheerrortermsandexplanatoryvariablescanbetoarriveattheendoge-nousswitchingorrandomcoefmodel.Itisalsointerestingtoaddweakinstruments(Staiger82andStock,1997)intothisframework.Third,itisimportanttodiscussintheorytheconditionsforpointinthismodel.Partialcanbeemployedinthescenarioswherepointarenotenabled.Fourth,forcausalinference,onecouldderiveAPEsoverthewholesampleandstudyitsasymptoticpropertyusingUstatistics,insteadofnowatagivenobser-vation.Distributionaltreatmenteffects,whichrecovertreatmenteffectsfortheentiredistributionofoutcomes,arealsoapromisingdevisetoprovidemorecausalstatisticstoempiricalresearchers.83APPENDIX84APPENDIXFORCHAPTER2B.1AssumptionsandProofofResultsinSection2.4.2Proof.ofTheorem4.3.ThefollowingborrowsfromHahn,LiaoandRidder(2015)AppendixB.AssumptionB.1(a)liminfnvy2;nsd>0;liminfnvy3;nsd>0(b)Letao(ho;go)bethetrueunknownfunctions,thefunctionalry2(;)andry3(;)supa2Nnry2(a)ry2(ao)¶ry2(ao)¶h[hho]¶ry2(ao)¶g[ggo]vy2;nsd=on12;supa2Nnry3(a)ry3(ao)¶ry3(ao)¶h[hho]¶ry3(ao)¶g[ggo]vy3;nsd=on12;(c)Thereexistsgn2GnsuchthatkgngokG=Od2;nandforanyvh2V1andvg2V2,kvhkjcjkvhkHandvgycykvhkGwherecjandcyaresomegenericpositiveconstants;(d)1vy2;nsdmaxˆ¶ry2(ao)¶hho;nho;¶ry2(ao)¶ggo;ngo˙=on12;1vy3;nsdmaxˆ¶ry3(ao)¶hho;nho;¶ry3(ao)¶ggo;ngo˙=on12AssumptionB.1(a)ensuresthathesievevarianceisasymptoticnonzero.AssumptionB.1(b)impliesthatthereisalinearapproximationforry2(ao)andry3(ao)uniformlyovera2Nnwithapproximationerrorovy3;nsdn12.AssumptionB.1(c)impliesthatkkjandkkymaybeweakerthanthepseudo-metricskkHandkkGrespectively.ByAssumptionB.1(c)andthe85ofgo;m,wehavegogo;njkgognkjkgognkH=Od2;nwhichindicatesthatgo;n2Ng;n.Similarly,wehaveho;n2Nh;n,whichtogetherwiththeformerresultimpliesthatho;n;go;n2Nn.AssumptionB.1(d)alsorequiresthatthesieveapproximationerrorconvergestozeroataratefasterthanvy2;nsdn12andvy3;nsdn12,whichisanunder-smoothingconditiontoderivethezeromeanasymptoticnormalityofthesieveplug-inestimatorr(ban).uhn;ugn;uGn=kvnk1sdvhn;vgn;vGnandg=genugnforanyg2Ng;n,whereen=on1=2issomepositivesequence.Letmn[]betheempricalprocesssuchthatmnyZ2;g;h1nåni=1yZ2;i;g;hEyZ2;g;h.AssumptionB.2(a)Thefollowingstochasticequicontinuityconditionhold:supa2Nnmnny(Z2;g;h)y(Z2;g;h)Dy(Z2;g;h)henugnio=Ope2nandsupa2NnmnnDy(Z2;g;h)hugniDy(Z2;go;ho)hugnio=Op(en)(b)letKy(g;h)E[y(Z2;g;h)y(Z2;go;ho)],thenKy(g;h)Ky(g;h)=enG(ao)hhho;ugni+kggok2ykggok2y2+Oe2n(B.1)uniformalyover(h;g)2Nn:Thestochasticequicontinuityconditionsareregularassumptionsinthesievemethodliterature,e.g.,Shen(1997),ChenandShen(1998)andChen,LiaoandSun(2012).AssumptionB.2(b)impliesthattheKullback-LeiblertypeofdistancehasalocalquadraticapproximationuniformlyovertheshrinkingneighborhoodNn.Whenthereisnoestimatebhn,i.e.h=hoin(B.1),AssumptionB.2(b)willbereducedtosupa2NnKy(g;h)Ky(g;h)kggok2ykggok2y2=Oe2n86whichistheconditionusedinChen,LiaoandSun(2012)toderivetheasymptoticnormalityofone-stepsieveplug-inestimater(bgn).Asaresult,wecanviewtheextratermin(B.1)astheestimationeffectthattheestimatebhnintroducestotheasymptoticdistributionofthesecond-stepsieveMestimatorbgn.AssumptionB.3(a)Thest-stepsieveMestimatorbhnDbhnho;uhn+uGnEjmnnDj(Z1;ho)huhn+uGnio=Op(en)(b)thefollowingcentrallimittheorem(CLT)holds:n12nåi=1nDjZ1;i;hohuhn+uGni+DyZ1;i;go;hohugniod!N(0;1)whereNormal(0;1)denotesastandardnormalrandomvariable;(c)e2;n=O(en),end12;n=o(1)andugny=O(1).AssumptionB.3(a)isahighlevelcondition,whichisestablishedinChen,LiaoandSun(2012)underasetofsufconditions.AssumptionB.3(b)isimpliedbythetrianglearrayCLTs.AssumptionB.3(b)impliesthattheoptimizationerrore2;ninthesecond-stepsieveMestimationisofthesameorlargerorderasen.Asd12;nistheconvergencerateofthesecond-stepsieveMestimatorbgn,underthestationarydataassumption,itisreasonabletoassumethatd12;ncon-vergestozeroattheratenotfasterthanroot-n,whichexplainstheassumptionend12;n=o(1).ByAssumptionB.3(c),end12;n=o(1)andthetriangleinequality,wehavekbgngokykbgngok+enugny=Opdg;n(B.2)whichimpliesthatbgn2Ng;nwpa1.87Bytheofbgn,wehaveOpe22;n1nnåi=1yZ2;i;bgn;bhn1nnåi=1Z2;i;bgn;bhn=mnnyZ2;bgn;bhnyZ2;bgn;bhn+DyZ2;bgn;bhnhenugnio+mnnDy(Z2;go;ho)henugniDyZ2;bgn;bhnhenugniomnnDy(Z2;go;ho)henugnio+hKybgn;bhnKybgn;bhn(B.3)ByAssumptionB.2(a),wehavemnnyZ2;bgn;bhnyZ2;bgn;bhn+DyZ2;bgn;bhnhenugnio=Ope2n(B.4)andmnnDy(Z2;go;ho)hugniDyZ2;bgn;bhnhugnio=Op(en)(B.5)NotethatG(ao)[;]isabilinearfunctional.Using(B.2),AssumptionB.2(b)andB.3(c),wededucethatKybgn;bhnKybgn;bhn=enG(ao)hbhnho;ugni+kbgngok2ykbgngok2y2+ope2n=DenuGn;bhnhoEj+e2nugn2y2+Denugn;bgngoEy+ope2n=DenuGn;bhnhoEj+Denugn;bgngoEy+Ope2n(B.6)Frome2;n=O(en),(B.3),(B.4),(B.5)and(B.6),wegetOpe2nenmnnDy(Z2;go;ho)hugnioenDuGn;bhnhoEjenDugn;bgngoEy:Dividingbyen,weobtainDugn;bgngoEyDuGn;bhnhoEjmnnDy(Z2;go;ho)hugnio=Op(en)(B.7)Bygo;nistheprojectionofgoonV2;nunderthesemi-normkky.HencethereisDgo;ngo;ugnEy=0andDbgngo;ugnEy=Dbgngo;n;ugnEy(B.8)88From(B.7),(B.8)anden=o(n12),wegetDbgngo;n;ugnEyDuGn;bhnhoEjmnnDy(Z2;go;ho)hugnio=opn12:(B.9)ByAssumptionB.1(a)-(c)andtheRieszrepresentationtheorem,pnry2bhn;bgnry2ho;n;go;nvy2;nsd=pn¶ry2(ao)¶hhbhnho;ni+¶ry2(ao)¶hbgngo;nvy2;nsd+pnry2bhn;bgnry2(ho;go)¶ry2(ao)¶hhbhnhoi¶ry2(ao)¶g[bgngo]vy2;nsdpnry2ho;n;go;nry2(ho;go)¶ry2(ao)¶hho;nho¶ry2(ao)¶ggo;ngovy2;nsd=pnDbhnho;n;uhnEj+Dbgngo;n;ugnEy+op(1)(B.10)Byho;nistheprojectionofhoonV1;nunderthesemi-normkkj.HencethereisDho;nho;uhnEj=0andDbhnho;n;uhnEj=Dbhnho;uhnEj(B.11)Fromtheresultsin(B.10)and(B.11),wegetpnry2bhn;bgnry2ho;n;go;nvy2;nsd=pnDbhnho;uhnEj+Dbgngo;n;ugnEy+op(1)which,togetherwith(B.9)andAssumptionB.3(a),impliesthatpnry2bhn;bgnry2ho;n;go;nvy2;nsd=pnDbhnho;uhn+uGnEj+mnnDy(Z2;go;ho)hugnio+op(1)=n12nåi=1nDjZ1;i;hohuhn+uGni+DyZ2;i;go;hohugnio+op(1)(B.12)89Furthermore,fromAssumptionB.1,wegetry2ho;n;go;nry2(ho;go)vy2;nsdry2ho;n;go;nry2(ho;go)¶ry2(ao)¶hho;nho¶ry2(ao)¶ggo;ngovy2;nsd+1vy2;nsd¶ry2(ao)¶hho;nho+¶ry2(ao)¶hgo;ngo=on12(B.13)whichcombinedwith(B.12)impliespnhbry2bhn;bgnry2(ho;go)ivy2;nsdd!N(0;1):Samethinggoesforpnhbry3bhn;bgnry3(ho;go)ivy3;nsdd!N(0;1):B.2AssumptionsandProofofResultsinSection2.4.3Proof.ofTheorem4.4ThefollowingborrowsfromHahn,LiaoandRidder(2015)AppendixC.AssumptionB.4LetW2;nng2V2;n:kgky1o,thenthereis(a)Dvg1;vg2Ey=Enry(Z2;ao)hvg1;vg2ioforanyvg1;vg22V2;(b)supa2Nn;vg1;vg22W2;nEnry(Z2;a)hvg1;vg2iry(Z2;ao)hvg1;vg2io=o(1)(c)supa2Nn;vg1;vg22W2;nmnnry(Z2;a)hvg1;vg2io=op(1)90(d)supa2Nn;vg2W2;n¶r(a)¶gvg¶r(ao)¶gvg=o(1)UnderAssumptionB.4,theempiricalRieszrepresenterbvgnbvgnvgnyvgny=op(1):(B.14)AssumptionB.5LetW1;nnh2V1;n:khkj1o,andB2;nˆv2V2;n:vvgnyvgn1ydvg;n˙,wheredvg;n=o(1)issomepositivesequence.Then(a)Dvh1;vh2Ej=Enrj(Z1;h)hvh1;vh2ioforanyvh1;vh22V1;(b)suph2Nh;n;vh1;vh22W1;nEnrj(Z1;h)hvh1;vh2irj(Z1;h)hvh1;vh2io=o(1)(c)suph2Nh;n;vh1;vh22W1;nmnnrj(Z1;h)hvh1;vh2io=op(1)(d)supa2Nn;vh2W1;n¶r(a)¶h[vh]¶r(ao)¶h[vh]=o(1)(e)supa2Nn;vg2B2;n;vh2W1;nGn(a)vh;vgG(ao)vh;vgn=op(1)UnderAssumptionB.5,theempiricalRieszrepresentersbvhnandbvGnbvhnvhnjvhnj=op(1)andbvGnvGnjvGnj=op(1)(B.15)AssumptionB.6Letkk2denotetheL2(dFZ)-norm,then:(a)thefunctionalDj(Z1;h)[vh]suph2Nh;n;vh2W1;nDj(Z1;h)[vh]Dj(Z1;ho)[vh]2=o(1)andsuph2Nh;n;vh2W1;nmnnD2j(Z1;h)[vh]o=op(1)91(b)thefunctionalDy(Z2;ao)vgsupa2Nn;vg2W2;nDy(Z2;a)vgDy(Z2;ao)vg2=o(1)andsupa2Nn;vg2W2;nmnnD2y(Z2;a)vgo=op(1)(c)thefollowingULLNholdssupa2Nn;vh2W1;n;vg2W2;nmnDj(Z1;h)[vh]Dy(Z2;a)vg=op(1)(d)supvh2W1;nDj(Z1;ho)[vh]2=O(1)andsupvg2W2;nDy(Z2;ao)vg2=O(1);more-overvhnj+vGnj+vgnyDj(Z1;ho)hvhn+vGni+Dy(Z2;ao)hvgni2=O(1)Bythetriangleinequality,Holderinequality,AssumptionB.6(i)and(iv)wehavesuph2Nh;n;vh2W1;n1nnåi=1D2jZ1;i;h[vh]EhD2jZ1;i;ho[vh]isuph2Nh;n;vh2W1;nmnD2jZ1;i;ho[vh]+suph2Nh;n;vh2W1;nEhD2jZ1;i;h[vh]iD2jZ1;i;h[vh]op(1)+suph2Nh;n;vh2W1;nDjZ1;i;h[vh]DjZ1;i;ho[vh]22+2suph2Nh;n;vh2W1;nDjZ1;i;h[vh]DjZ1;i;ho[vh]2DjZ1;i;ho[vh]2=op(1)(B.16)Similarly,applyingtriangleinequaltiyandHolderinequalitytoAssumptionB.6(ii)and(iv),supa2Nn;vg2W2;n1nnåi=1D2yZ2;i;avgEhD2yZ2;i;aovgi=op(1):92Bythetriangleinequality,wehaveEDj(Z1;h)[vh]Dj(Z2;a)vgDj(Z1;ho)[vh]Dy(Z2;ao)vgEDj(Z1;h)[vh]Dj(Z1;ho)[vh]Dy(Z2;a)vgDy(Z2;ao)vg+EDj(Z1;h)[vh]Dj(Z1;ho)[vh]Dy(Z2;ao)vg+EDj(Z1;ho)[vh]Dy(Z2;a)vgDy(Z2;ao)vgforanya2Nn,vh2W1;nandvg2W2;n.UsingHolderinequality,AssumptionB.6(i)and(ii),wehaveforanya2Nn,vh2W1;nandvg2W2;nsupEDj(Z1;h)[vh]Dj(Z1;ho)[vh]Dj(Z2;a)vgDy(Z2;ao)vgsupDj(Z1;h)[vh]Dj(Z1;ho)[vh]2Dj(Z2;a)vgDy(Z2;ao)vg2=o(1)SimilarlybyAssumptionB.6(i),(ii)and(iv),wehavesuph2Nh;n;vh2W1;n;vg2W2;nDj(Z1;h)[vh]Dj(Z1;ho)[vh]2Dy(Z2;ao)vg2=o(1)(B.17)andsupa2Nn;vh2W1;n;vg2W2;nDy(Z2;a)[vh]Dy(Z2;ao)[vh]2Dj(Z1;ho)[vh]2=o(1)Thuswehavesupa2Nn;vh2W1;n;vg2W2;nEDj(Z1;h)[vh]Dj(Z2;a)vgDj(Z1;ho)[vh]Dy(Z2;ao)vg=op(1)(B.18)Letcdenotesomegenericpositiveconstant.Asthedataisi.i.d,bywehavekvnk2sd=Var"n12nåi=1DjZ1;i;hohvhn+vGni+DyZ2;i;aohvgn#=EDjZ1;i;hohvhn+vGni+DyZ2;i;aohvgni2=DjZ1;i;hohvhn+vGni+DyZ2;i;aohvgni22(B.19)93Bythetriangleinequality,kbvnvnksdkvnksdDj(Z1;ho)hbvhnvhni2+Dj(Z1;ho)hbvGnvGni2kvnksd+Dy(Z2;ao)hbvgnvgni2kvnksd(B.20)Using(B.19),AssumpionB.6(iv)andtheresultin(B.14),wehaveDy(Z2;ao)hbvgnvgni2kvnksd=vgnykvnksdbvgnvgnyvgnyDy(Z2;ao)264bvgnvgnbvgnvgny3752andbecausebvgnvgn=bvgnvgny2Wgn,wehaveDy(Z2;ao)hbvgnvgni2kvnksd=vgnykvnksdbvgnvgnyvgnysupvg2WgnDy(Z2;ao)vg2=op(1)(B.21)Similarly,wehaveDj(Z1;ho)hbvhnvhni2+Dj(Z1;ho)hbvGnvGni2kvnksd=op(1)whichtogetherwith(B.20)and(B.21)impliesthatkbvnvnksdkvnksd=op(1):(B.22)Using(B.22)andthetriangleinequality,wegetop(1)=kbvnvnksdkvnksdkbvnksdkvnksd1=kbvnksdkvnkn;sdkvnkn;sdkbvnksd1=kbvnksdkvnkn;sd kvnkn;sdkbvnksd1!+ kbvnksdkvnkn;sd1!(B.23)94Wenextshowthatkbvnksdkvnkn;sd1=op(1).Forthispurpose,wenotethatbvhn+bvGnjkbvnksdvhn+vGnj+bvhnvhnj+bvGnvGnjkbvnksdvhnj+vGnj+vhnjbvhnvhnjvhnj+vGnjbvGnvGnjvGnjkbvnksd=vhnjkbvnksd0B@1+bvhnvhnjvhnj1CA+vGnjkbvnksd0B@1+bvGnvGnjvGnj1CA=kvnksdkbvnksd264vhnjkbvnksd1+op(1)+vGnjkbvnksd1+op(1)375=kvnksdkbvnksdOp(1)=1kbvnksd=kvnksd1Op(1)+Op(1)=Op(1)(B.24)wherethetwoinequalitiesarebythetriangleinequality,thesecondequalityisby(B.15),thethirdequalityisby(B.19)andAssumptionB.6(iv),andthelastequalityisbytheinequalityin(B.23).Similarly,wecanshowthatvgnykbvnk1sd=Op(1):(B.25)Bythetriangleinequality,wegetkbvnk2n;sdkbvnk2sdkbvnk2sd=1nnåi=1hDjZ1;i;bhnhbvhn+bvGni+DyZ2;i;banhvgnii2kbvnk2sdEZhDjZ1;i;hohbvhn+bvGni+DyZ2;i;aohvgniikbvnk2sdI1;n+I2;n+2I3;n(B.26)whereEZ[]denotestheexpectationtakingwithrespecttothedistributionofZ(EZ1[]andEZ2[]95aresimilarlyI1;n=1nnåi=1D2jZ1;i;bhnhbvhn+bvGniEZ1hD2jZ1;i;hohbvhn+bvGniikbvnk2sd;I2;n=1nnåi=1D2yZ2;i;banhbvgniEZ1hD2yZ2;i;aohbvgniikbvnk2sdI3;n=1nnåi=1DjZ1;i;bhnhbvhn+bvGniDyZ2;i;banhbvgnikbvnk2sdEZhDjZ1;i;bhnhbvhn+bvGniDyZ2;i;banhbvgniikbvnk2sdBy(B.16)and(B.24),wehaveI1;n=bvhn+bvGn2jkbvnk2sd1nnåi=1D2jZ1;i;bhnhbvhn+bvGniEZ1hD2jZ1;i;hohbvhn+bvGniibvhn+bvGn2jOp(1)suph2Nh;n;vh2W1;n1nnåi=1D2jZ1;i;h[vh]EZ1hD2jZ1;i;ho[vh]i=op(1)(B.27)Similarly,by(B.17)and(B.25),wehaveI2;n=bvgn2ykbvnk2sd1nnåi=1D2yZ2;i;banhbvgniEz2hD2jZ2;i;aohbvgniibvgn2y=op(1)(B.28)96ForthelasttermI3;n,notethatforanya2Nn,vh2W1;nandvg2W2;nI3;nmnnDjZ1;bhnhbvhn+bvGniDy(Z2;ban)hbvgniokbvnk2sd+EZhDjZ1;bhnhbvhn+bvGniDy(Z2;ban)hbvgnikbvnk2sdDj(Z1;ho)hbvhn+bvGniDy(Z2;ao)hbvgniikbvnk2sdbvhn+bvGnjbvgnykbvnk2sdsupmnDj(Z1;h)[vh]Dy(Z2;ao)vg+supEDj(Z1;h)[vh]Dy(Z2;a)vgDj(Z1;ho)[vh]Dy(Z2;ao)vg=bvhn+bvGnjbvgnykbvnk2sdop(1)=op(1)(B.29)wheretheinequalityisbythetriangleinequality,theequalityisbyAssumptionB.6(iii)and(B.18),thelastequalityisby(C18)and(C19).Fromtheresultsin(B.26),(B.27),(B.28)and(B.29),wededucethatkbvnk2n;sdkbvnk2sdkbvnk2sd=op(1)(B.30)Itisclearthat(B.23)and(C.24)implythatkbvnkn;sd=kbvnk2sd1=op(1),whichtheproof.B.3ProofforResultsinSection2.5Proof.ofProposition5.1.Thefollowingaimstoderivetheasymptoticsforpnhbpy2bdh;bdgpy2dho;dgoiandpnhbpy3bdh;bdgpy3dho;dgoi.Noticethatwehavepy2dho;dgo=Ewi1hry2wi1;dho;dgoi;py3dho;dgo=Ewi1hry3wi1;dho;dgoi:97andbpy2bdh;bdg=n1nåi=1ri;y2bdh;bdg;bpy3bdh;bdg=n1nåi=1ri;y3bdh;bdg:Nowwecouldfocusonthepropertyofri;y2bdh;bdginsteadtoderivetheasymptoticsforbpy2bdh;bdg.Bythemeanvalueexpansion,n12nåi=1ri;y2bdh;bdg=n12nåi=1ri;y2dho;dgo+n1nåi=1¨Rhpnbdhdho+n1nåi=1¨Rgpnbdgdgo;where¨Ry2i;h5dhri;y2¨dh;¨dg;¨Ry2i;g5dgri;y3¨dh;¨dg:Weknow¨dh;¨dgis"trapped"between‹dh;‹dganddho;dgo.Aswehave‹dh;‹dgp!dho;dgo,itiseasytosee¨dh;¨dgconvergesinprobabilitytodho;dgo,too.Afterverifyingsomeregularityconditions,wehaven1nåi=1¨Ry2i;hp!Ry2o;hEh5dhri;y2dho;dgoi;n1nåi=1¨Ry2i;gp!Ry2o;gEh5dgri;y2dho;dgoi:Thestepbdhisaleastsquaresestimator,bystandardarguments,whichhasanfunctiontakingtheform,pnbdhdho=n12nåi=1H1o;hSi;h+op(1)whereHo;hEHi;hEhpk1(n)1(zi)pk1(n)1(zi)0iandSi;hpk1(n)1(zi)yi2pk1(n)1(zi)0dho:Itsasymptoticdistributionforpnbdhdhoispnbdhdhod!N(0;Vh):98whereVhH1o;hVarSi;hH1o;hhastheheteroskedasticrobustsandwichform.Thesecondstep‹dgisamaximumlikelihoodestimatorofthebivariateprobit,withthestepbdhpluggedin.Absorbtherandomnessofallthedatawi(wi1;wi2)intothesubscript,useastandarddeltamethodforafunctioncontainstwo-stepsieveM-estimation(Wooldridge,2010,Chapter12,forexample),thefunctionispnbdgdgo=n12nåi=1H1o;gSi;g+Fo;ghH1o;hSi;h+op(1):F(j)i;2isthepartialderivativeofbivariateCDFFi;2di1xi1b+m0idmo;di2zig+m0idqo;di1di2rwithrespecttoitsj-thargument,wherej=1;2;3;andmiistheshorthandformipl(n)2hyi2pk1(n)1(zi)0dhoi:F(j;k)i;2isthecrossderivativewithrespecttoitsj-thandk-tharguments,j;k=1;2;3,Si;gÑdgFi;2Fi;2=1Fi;20BBBBBBBBBBB@¶Fi;2¶b¶Fi;2¶g¶Fi;2¶r¶Fi;2¶dmo¶Fi;2¶dqo1CCCCCCCCCCCA=1Fi;20BBBBBBBBBB@x0i1di1F(1)i;2z0idi2F(2)i;2di1di2F(3)i;2midi1F(1)i;2midi2F(2)i;21CCCCCCCCCCA;Ho;gEHi;gEÑdgSi;g;ÑdgSi;g=1Fi;20BBBBBBBBBB@x0i1xi1di1F(11)i;2x0i1zidi1di2F(12)i;2x0i1di2di1F(13)i;2x0i1mi0di1F(11)i;2x0i1mi0di1di2F(12)i;2z0ixi1di1di2F(21)i;2z0izidi2F(22)i;2z0idi2di1F(23)i;2z0imi0di1di2F(21)i;2z0im0idi2F(22)i;2xi1di1di2F(31)i;2zidi1di2F(32)i;2di1di2F(33)i;2m0idi1di2F(31)i;2m0idi1di2F(32)i;2mixi1di1F(11)i;2mizidi1di2F(12)i;2midi1di2F(13)i;2mim0idi1F(11)i;2mim0idi1di2F(12)i;2mixi1di2F(21)i;2mizdi2F(22)i;2midi1di2F(23)i;2mim0idi2di1F(21)i;2mim0idi2F(22)i;21CCCCCCCCCCA;Fo;ghEFi;ghEÑdhSi;g;99pl(n)(1)2()=(p(1)2;1()0;:::;p(1)2;l(n)()0)0istheorderderivativeforeachelementofthebasisfunction,shorthandm(1)ipl(n)(1)2hyi2pk1(n)1(zi)0dhoi;ÑdhSi;g=1Fi;20BBBBBBBBBB@x0i1d0mom(1)ipk1(n)1(zi)0di1F(11)i;2x0i1d0qom(1)ipk1(n)1(zi)0di1di2F(12)i;2z0id0mom(1)ipk1(n)1(zi)0di1di2F(21)i;2z0id0qom(1)ipk1(n)1(zi)0di2F(22)i;2d0mom(1)ipk1(n)1(zi)0di1di2F(31)i;2d0mom(1)ipk1(n)1(zi)0di1di2F(32)i;2m(1)ipk1(n)1(zi)0di1F(1)i;2mid0mom(1)ipk1(n)1(zi)0di1F(11)i;2mid0qom(1)ipk1(n)1(zi)0di1di2F(12)i;2m(1)ipk1(n)1(zi)0di2F(2)i;2mid0mom(1)ipk1(n)1(zi)0di1di2F(21)i;2mid0qom(1)ipk1(n)1(zi)0di2F(22)i;21CCCCCCCCCCA:Theasymptoticdistributionforpn‹dgdgothenispn‹dgdgod!N(0;Vg);whereVgH1o;gDoH1o;gandDoVarSi;g+Fo;ghH1o;hSi;h:Thefunctionforpnhbpy2bdh;bdgpy2dho;dgoithenhasthreecomponentspnhbpy2bdh;bdgpy2dho;dgoi=n12nåi=1hri;y2bdh;bdgpy2dho;dgoi+op(1)=n12nåi=1hri;y2dho;dgopy2dho;dgoRy2o;hH1o;hSi;hRy2o;gH1o;gSi;g+Fo;ghH1o;hSi;h+op(1):Therefore,pnhbpy2bdh;bdgpy2dho;dgoid!N(0;Vy2);whereVy2=Varhri;y2dho;dgopy2dho;dgoRy2o;hH1o;hSi;hRy2o;gH1o;gSi;g+Fo;ghH1o;hSi;h:Similarly,followexactlythesameprocedure,butwithaslightchangeofnotation,wehavepnhbpy3bdh;bdgpy3dho;dgoid!N(0;Vy3);where100Vy3=Varhri;y3dho;dgopy3dho;dgoRy3o;hH1o;hSi;hRy3o;gH1o;gSi;g+Fo;ghH1o;hSi;h:101B.4FiguresandTablesforSection2.6Figure2.1EmpiricalDistributionForDesign1withz1˘Normal(0;9)102Figure2.2EmpiricalDistributionForDesign1withz1˘Normal(0;1)103Figure2.3EmpiricalDistributionForDesign2withz1˘Normal(0;9)104Figure2.4EmpiricalDistributionForDesign2withz1˘Normal(0;16)105Figure2.5EmpiricalDistributionForDesign3withz1˘Normal(0;9)106Table2.1SimulationResultsforDesign1(1)(2)(3)(4)(5)(6)CFBiprobitK=3CFBiprobitK=2CFBiprobit2SLSSRKernelSRKNNz1˘Normal(0;9),APEy2=.0806Bias-.0001-.0002-.0002-.0057-.0143-.0174RMSE.0053.0053.0054.0090.0160.0195z1˘Normal(0;9),APEy3=.0851Bias-.0014.0005-.0049.0255-.0009-.0066RMSE.0334.0331.0355.0649.0638.0791z1˘Normal(0;9),b3=b2=1Bias-.0098.0122-.0494.4993.3227.3258RMSE.4021.3985.4245.97431.0981.429z1˘Normal(0;1),APEy2=.0932Bias.0003.0010.0012-.0188-.0328-.0350RMSE.0065.0066.0064.0200.0339.0365z1˘Normal(0;1),APEy2=.1046Bias-.0071-.0114-.0208.0640.0203.0123RMSE.0353.0371.0426.0895.0749.0908z1˘Normal(0;1),b3=b2=1Bias-.0603-.1031-.19041.2961.2011.163RMSE.3371.3548.40361.5821.9132.352aSequentialaveragingofthecontrolfunctiontermv2andxisappliedtocomputeestimatesofAPEs.bThebiasisasthedifferencebetweenthetrueAPEs(thetrueratioone)andtheestimates.RMSEistherootmeansquarederror.cEstimator(1)istheCFapproachwithunknownfunctionsapproximatedbysievespacesofdimensionthree.Estimator(2)istheCFapproachwithunknownfunctionsapproximatedbysievespacesofdimensiontwo.Estimator(3)isthestandardCFapproachwithnounknownfunctions.Estimator(4)isthelinearprobabilitymodelestimatedbyusualtwo-stageleastsquares.Estimator(5)isthesemiparametricspecialregressormethodwithdensityestimatedbykernelmethods.Estimator(6)isthesemiparametricspecialregressormethodwithdensityestimatedbyK-Nearest-Neighbormethod.107Table2.2SimulationResultsforDesign2(1)(2)(3)(4)(5)(6)CFBiprobitK=3CFBiprobitK=2CFBiprobit2SLSSRKernelSRKNNz1˘Normal(0;9),APEy2=.0806Bias-.0004-.0023-.0038-.0056-.0143-.0165RMSE.0213.0213.0224.0244.0160.0345z1˘Normal(0;9),APEy3=.0851Bias.0129.0134.0117.0070-.0009-.0096RMSE.0430.0427.0429.0613.0638.0787z1˘Normal(0;9),b3=b2=1Bias.2539.2148.2623.3152.81921.085RMSE.8013.6549.81991.03621.97514.040z1˘Normal(0;16),APEy2=.0815Bias-.0001-.0013-.0025-.0045-.0074-.0152RMSE.0188.0187.0193.0216.0258.0315z1˘Normal(0;16),APEy2=.0828Bias.0163.0157.0138.0067.0021-.0058RMSE.0417.0407.0408.0604.0642.0720z1˘Normal(0;16),b3=b2=1Bias.3069.2947.3121.3314.2678.3837RMSE.7561.7430.80331.18162.9246.094aSequentialaveragingofthecontrolfunctiontermv2andxisappliedtocomputeestimatesofAPEs.bThebiasisasthedifferencebetweenthetrueAPEs(thetrueratioone)andtheestimates.RMSEistherootmeansquarederror.cEstimator(1)istheCFapproachwithunknownfunctionsapproximatedbysievespacesofdimensionthree.Estimator(2)istheCFapproachwithunknownfunctionsapproximatedbysievespacesofdimensiontwo.Estimator(3)isthestandardCFapproachwithnounknownfunctions.Estimator(4)isthelinearprobabilitymodelestimatedbyusualtwo-stageleastsquares.Estimator(5)isthesemiparametricspecialregressormethodwithdensityestimatedbykernelmethods.Estimator(6)isthesemiparametricspecialregressormethodwithdensityestimatedbyK-Nearest-Neighbormethod.108Table2.3SimulationResultsforDesign3(1)(2)(3)(4)(5)(6)CFBiprobitK=3CFBiprobitK=2CFBiprobit2SLSSRKernelSRKNNAPEy2=.1108Bias-.0001-.0006-.0011-.0065-.0156-.0220RMSE.0144.0144.0146.0212.0271.0337APEy3=.1133Bias.0013.0013.0015.0006-.0080-.0187RMSE.0275.0273.0272.0372.0420.0494b3=b2=1Bias.0356.0400.0484.1585.2187.3097RMSE.3077.3096.3141.5311.75401.850aSequentialaveragingofthecontrolfunctiontermv2andxisappliedtocomputeestimatesofAPEs.bThebiasisasthedifferencebetweenthetrueAPEs(thetrueratioone)andtheestimates.RMSEistherootmeansquarederror.cEstimator(1)istheCFapproachwithunknownfunctionsapproximatedbysievespacesofdimensionthree.Estimator(2)istheCFapproachwithunknownfunctionsapproximatedbysievespacesofdimensiontwo.Estimator(3)isthestandardCFapproachwithnounknownfunctions.Estimator(4)isthelinearprobabilitymodelestimatedbyusualtwo-stageleastsquares.Estimator(5)isthesemiparametricspecialregressormethodwithdensityestimatedbykernelmethods.Estimator(6)isthesemiparametricspecialregressormethodwithdensityestimatedbyK-Nearest-Neighbormethod.109BIBLIOGRAPHY110BIBLIOGRAPHYAi,C.,Chen,X.,2003.EfEstimationofModelswithConditionalMomentRestrictionsContainingUnknownFunctions.Econometrica71,1795Œ1843.Ai,C.,Chen,X.,2007.Estimationofpossiblysemiparametricconditionalmomentrestricti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ables(EEVs),onemustusecareininterestingpartialeffects.Ataminimum,theshouldreducetowhatiswidelyacceptedasthequantityofinterestinalinearmodel.Partialeffectsbasedonthepropensityscorearerarelyofinterest,asthatcorrespondstocomputingE(YjX)inmodelswhereXcontainsendogenousvariables.IfE(YjX)weretheobjectofinterestthendecadesofpublishedresearchonaccountingforEEVsineconometricmodelswouldbeirrelevant.Partialeffectsbasedontheaveragestructuralfunction(ASF),asbyBlundellandPow-ell(2003,2004),havethedesirablefeaturethattheycorrespondtotheparametersofinterestinlinearmodelswithEEVs.Inaddition,incommonnonlinearmodelstheyareintuitivelyappealing114andcanbeobtainedviacounterfactualreasoning.TotheASF,letYbetheresponsevariable,Xasetofexplanatoryvariables(possiblycon-tainingendogenousaswellasexogenousexplanatoryvariables),andUavectorofunobservables.AssumethatY=g(X;U)forsomefunctiong(;).TheASFisasASF(x)EU[g(x;U)];(3.1)wherexisanonrandomplaceholder.Inotherwords,theASFaveragesouttheunobservables,U,acrossthepopulationandisafunctionofpotentialvaluesoftheobservedexplanatoryvariables,x.IttakesnostandonwhetherUandXaredependent.OnecanthenpartialeffectsbasedonASF(x)usingpartialderivativesordiscretedifferences.SeeWooldridge(2010,Section15.6and15.7.4)forfurtherdiscussion,includinghowstandardmethodsofcomputingAPEscanbeobtainedbytheASF.IfwehaveasinglebinarycovariateX,anditsvaluesrepresentcontrolandtreatmentstates,thenASF(1)ASF(0)=EU[g(1;U)]EU[g(0;U)]istheaveragetreatmenteffect,asinRosenbaumandRubin(1983).Recently,Lewbel,DongandYang(2012)(LDY)haveproposedadifferentapproachtoesti-matingpartialeffectsforaclassofsemiparametricindexbinaryresponsemodels.Moreprecisely,LDYanewfunctioncalledtheaverageindexfunction(AIF).LDYarguethattheAIFisafimiddlegroundflbetweenthepropensityscoreandtheASF.IntheLDYsetupthemodelhasanindexstructure,sothatthebinaryresponse,D,dependsonXthroughalinearindexXb,wherebisacolumnvectorofparameters.BecauseXbisascalar,estimationoftheAIFiscomparedwithfullynonparametricestimationoftheaveragestructuralfunction.AsdiscussedbyLDY,theAIFcanbegenerallyunderanindexstructureinthepresenceofafispe-cialregressor.flBycontrast,oftheASFrequiresadditionalassumptionswhensomeendogenouselementsofXarediscrete(althoughitdoesnotassumeaspecialregressor).AsnotedbyLDY,theoftheAIFdoesnotrelyonthespecialregressorsetup;thepresenceofthespecialregressorsimplyallowsonetoidentifytheindexcoefb.HereweareinterestedininterpretingtheAIF.Therefore,inmostofwhatfollowswedonotseparatelyshow115aspecialregressor.ThenwecanwriteanindexmodelasD=1[Xb+U0];(3.2)where1[]denotestheindicatorfunctionand(X;U)isarandomvector.Giventheindexstructurein(3:2),threefunctionsthatcanbeusedtopartialeffectsarethepropensityscore,AIF,andASF:PS(x)=FUjX(xbjx);(3.3a)AIF(x)=FUjXb(xbjxb);(3.3b)ASF(x)=FU(xb);(3.3c)whereF()denotesacumulativedistributionfunction.NotethatASF(x)dependsonlyontheunconditionalCDFofU.InthinkingaboutAIF(x),notethatitcanbeobtainedbytherandomvariableW=XbandthenobtainingtheCDFofUgivenW.Then,onereplacestheplaceholder,w,withxb.Thepropensityscore,PS(x),dependsontheCDFofUgivenX.Generally,PS,AIF,andASFalldiffer,althoughitfollowsimmediatelythattheyareallthesamewhenUandXareindependent.BlundellandPowell(2003)suggestestimatingpartialeffectsfromASF(x)whereasLDYsuggestusingAIF(x).ItiswidelyagreedthatPS(x)isusuallynotofinterestwhenXhaselementscorrelatedwithU:thepartialeffectsobtainedfromPS(x)generallyhavenothingtodowiththecausaleffects.Bycontrast,asarguedbyBlundellandPowell(2003,2004),ASF(x)isofconsiderableinterestwhetherornotXincludesEEVs;seealsoWooldridge(2010,Section2.2).LDYarguethatAIFisausefulfimiddlegroundflbetweenthepropensityscoreandASF.ThepurposeofthisnoteistoshowthatinstandardcasestheAIFsuffersfromessentiallythesameshortcomingsasthepropensityscorebecauseitisaffectedbycorrelationbetweentheunobserv-ablesandtheobservedEEVs.Infact,insomesimplecaseswithendogenousvariablestheAIFandpropensityscorearethesame.116InSection3.2weusealinearmodeltoillustratetherelationshipsamongthethreepartialeffects.Section3.3coversbinaryresponsemodelsandshowsthat,understandardassumptions,theAIFandpropensityscoreareidentical.Section3.4containsabriefconclusion.3.2ALinearModelWestartwithalinearmodeltoshowhowtheAIFdoesnotidentifyquantitiesofinterestinthemostcommonsetting.ConsiderthemodelY=Xb+U;(3.4)whereY;X;andUareallscalarsforsimplicity.Nothingsubstantivechangesifweincludeaninterceptorallowmultipleexplanatoryvariables.Assumethat(X;U)hasazeromeanbivariatenormaldistribution:XU˘Normal26400;0B@s2XrsXsUrsXsUs2U1CA375;(3.5)wherer=Corr(X;U)isthecorrelationcoefTheaveragestructurefunctioninthismodelissimplyASF(x)=xb;(3.6)becauseE(U)=0.ItisimportanttounderstandthattheASFhasnothingtodowiththeconditionaldistributionofUgivenX.Whether(3:4)representsasingleequationinasimultaneoussystemorUcontainsanomittedvariablecorrelatedwithX,alleconomistswouldagreethatbisthequantityofinterest.Forthissimplelinearmodel,thepropensityscoreisanalogoustoE(YjX=x).TheanalogoftheAIFistoobtain117E(YjXb)=Xb+E(UjXb);(3.7)andthentopluginx.Assumingthatb6=0toruleoutdegeneracies,byjointnormality,E(UjXb)=t(Xb);(3.8)wherethescalartisast=Cov(Xb;U)Var(Xb)=bCov(X;U)b2s2X=rsUbsX:(3.9)Pluggingthisexpressionfortinto(3:8)givesE(UjXb)=rsUsXX;(3.10)andsoAIF(x)=xb+rsUsX=E(YjX=x):(3.11)Inthissimplelinearcase,theAIFisidenticaltotheconditionalexpectation,E(YjX=x).Asinalmostanycasewithendogenousexplanatoryvariables,theconditionalmeanisnotofinterest.Wehaveshownexplicitlyin(3:11)thatthethepartialeffectintermsoftheAIFdoesnotdelivertheeffectofinterest,b.IfweaddexogenousvariablestothemodelthenthelinkbetweentheconditionalexpectationandAIFisbroken,butthelatterisstillofdoubtfulinterest.LetVbeanormallydistributedrandomvariableindependentof(X;U).NowassumeY=Xb+Vg+U;(3.12)118whichmeanstheaveragestructuralfunctionisASF(x;v)=xb+vg,andsothepartialeffectofxontheASFisb.Underthesameassumptionsasbefore,itiseasilyshownthatE(YjXb+Vg)=X b+rsUsXs2X+g2s2V=b2!+V g+rsUsXbs2X=g+gs2V=b!;(3.13a)E(YjX;V)=X b+rsUsXs2X!+Vg:(3.13b)Wheng6=0(asinthespecialregressorsetup,whereg=1),theinconsistencyintheAPEofXbyAIF(x)islessthanthatbasedonE(YjX;V).PerhapsthisisthesenseinwhichAIFconstitutesafimiddlegroundflbetweentheconditionalexpectationandtheASF.Nevertheless,theAIFstilldoesnotidentifytheparameterofinterest,b.And,unliketheconditionalexpectation,theAIFgetsthecoefonexogenousvariableVwrong,too.3.3ABinaryResponseModelNowweturntoabinaryresponsemodel,thecontextinwhichLDYtheAIF.Aswiththelinearcase,thepresenceofaspecialregressorhasnothingtodowiththeoftheAIF.Therefore,wedonotexplicitlyincludeaspecialregressor.SpecifythebinaryresponseDasD=1[Xb+U0]:(3.14)IfUandXareindependent,itiseasytoseethatthePS,AIF,andASFareallthesame,andequaltoASF(x)=P(Uxb)=FU(xb):(3.15)AssumingthatFU()isdifferentiablewithdensityfU(),foracontinuousX,thepartialeffectis119¶ASF(x)¶x=bfU(xb);(3.16)whichhasthesamesignasb.Importantly,equations(3:15)and(3:16)remainvalidforanydependencebetweenUandX.Therefore,itisinterestingtoseewhathappenstoAIF(andPS)whenweallowvariousformsofdependencebetweenUandX.ThefollowingconsiderstwocasesthatUandXarenotindependent.WithoutspecifyingtheunconditionaldensityofU,supposethatUjX˘Normal[0;exp(2Xb)];(3.17)sothatUhasazeromeanconditionalonXbutisheteroskedastic,withitsvariancedependingonXb.Then,withF()thestandardnormalCDF,P(UXbjXb)=PUexp(Xb)Xbexp(Xb)Xb=F[exp(Xb)Xb];(3.18)becauseU=exp(Xb)isindependentofXwithastandardnormaldistribution.Withb6=0wegetthesameexpressionifweconditiononXratherthanXb.Therefore,AIF(x)=PS(x)=F[exp(xb)xb]:(3.19)Lettingf()bethestandardnormalPDF,thepartialeffectcalculatedfromtheAIFis¶AIF(x)¶x=bexp(xb)(1xb)f[exp(xb)xb];(3.20)whichhasthesamesignasbonlywhenxb<1.ComparedwiththeASF,thepartialeffectbasedontheAIFismorecomplicated,anditdoesnotalwayshavethesamesignasb.ThepartialeffectbasedontheASFissimplyascaledversionofb,soitspropertiesaremoreappealingandmoreinthespiritofwhatweexpectwithalinearindexmodel.IfXiscorrelatedwithUthen,withthestandardbinaryresponsemodelnormalizations2U=1,theASFin(3:15)isstillvalid.TheAPEbasedontheASFis120APE(ASF)X=bEX[f(Xb)];(3.21)whichhasthesamesignasbbecausef(z)>0forallz2R.FortheAIF,wefollowthederivationforthelinearmodel.ByjointnormalitywecanwritethestructuralerrorUasU=t(Xb)+R;(3.22a)RjX˘Normal(0;s2R);(3.22b)t=rsUbsX;(3.22c)s2R=1t2b2s2X=1r2:(3.22d)NotethatRisindependentofX,andthusofXb.Therefore,D=1[(1+t)Xb+R0];(3.23)andE(DjXb)=P(R(1+t)XbjXb)=PRsR(1+t)XbsRXb=F(1+t)Xb)sR:(3.24)ItfollowsthattheaveragepartialeffectbasedontheAIFisAPE(AIF)X=(1+t)bsREXˆf(1+t)Xb)sR˙;(3.25)whichhasthesamesignasbonlyif1+t>0.Infact,itiseasytochoosevaluesofthepopulationparameterssothatt=1,inwhichcaseAPE(AIF)X=0foranysignormagnitudeofb.Alterna-tively,APE(AIF)XcouldbemuchlargerthanAPE(AIF)X.Eitherway,itishardtoseehowAPE(AIF)Xisofanyusebecauseithaslittletodowithstructuralorcausaleffects.1213.4ConclusionWehavecomparedthreeofaveragepartialeffectsinbinaryresponsemodelsthatcon-tainendogenousexplanatoryvariables.Asiswellknown,thepropensityscoreisnotusefulforsummarizingpartialeffectswhensomeexplanatoryvariablesarecorrelatedwiththeunobserv-ables.APEsbasedontheaveragestructuralfunctionofBlundellandPowell(2003)haveproventobeusefulinbothparametricandnonparametricapproaches.Recently,Lewbel,DongandYang(2012)introducedtheaverageindexfunctionforaclassofsemiparametricbinaryresponsemod-els,andestimationoftheAIFhasbeenimplementedinStata(Baum,2012).Unfortunately,wehaveshownthatAIFhasessentiallythesameshortcomingsasthepropensityscore.WhiletheASFisagenerallyusefulconcept,itisnotalwaysnonparametricallyorsemiparamet-ricallyinmodelswithdiscreteEEVs.Ongoingresearch,forexampleChesher(2010),succeedsinboundingaveragepartialeffectsincertainmodelswithdiscreteEEVs.Chesherfo-cusesonthestructuralfunction,anditisdiftoseehowtheAIFwillhavearoleinsearchingforusefulbounds.122APPENDIX123BIBLIOGRAPHY129BIBLIOGRAPHYBaum,C.F.,2012.sspecialreg:Statamoduletoestimatebinarychoicemodelwithdiscreteendogenousregressorviaspecialregressormethod.http://ideas.repec.org/c/boc/bocode/s457546.htmlBlundell,R.W.,Powell,J.L.2003.Endogeneityinnonparametricandsemiparametricregressionmodels.InAdvancesinEconomicsandEconometrics:TheoryandApplications.CambridgeUniversityPress:Cambridge,UK,312-357.Blundell,R.W.,Powell,J.L.2004.Endogeneityinsemiparametricbinaryresponsemodels.TheReviewofEconomicStudies71,655-679.Chesher,A.,2010.Instrumentalvariablemodelsfordiscreteoutcomes.Econometrica78,575-601.Lewbel,A.,1998.Semiparametriclatentvariablemodelestimationwithendogenousormismea-suredregressors.Econometrica66,105-121.Lewbel,A.,Dong,Y.,Yang,T.,2012.Comparingfeaturesofconvenientestimatorsforbinarychoicemodelswithendogenousregressors.CanadianJournalofEconomics45,809-829.Rosenbaum,P.R.,Rubin,D.B.,1983.Thecentralroleofthepropensityscoreinobservationalstudiesforcausaleffects.Biometrika70,41-55.Wooldridge,J.M.,2010.Econometricanalysisofcrosssectionandpaneldata,secondedition.MITPress:Cambridge,MA.130