EndogeneityinEconometrics:

InstrumentalVariableEstimationMingLUEndogeneityinEconometrics:

I1EndogeneityOmittingvariablebiasSimultaneityMeasurementerrorEndogeneityOmittingvariableb2Canweignoretheomittedvariablesbias?Itcanbesatisfactoryiftheestimatesarecoupledwiththedirectionofthebiasesforthekeyparameters.Canweuseproxytoeliminateomittedvariablebias?–Sometimes.CanFEestimationsolveomittingvariableproblem?Firstdifferencingorfixedeffectsestimationeliminatestime-constantvariables.Inaddition,thepaneldatamethodsdonotsolvetheproblemoftime-varyingomittedvariablesCanweignoretheomittedvari3IdeaofIVEstimationExogenousvariable.IndirecteffectsofIV.IdeaofIVEstimationExogenous4ExampleExample5在計量經(jīng)濟學的Endogeneity：有助易變的估計課件6WhatcanserveasIVforedu?Mother’seducation?Numberofsiblings?Thereportofothers?Adummyvariablethatisequalto1ifamanisborninthefirstquarteroftheyear.AngristandKrueger(1991).(Problematic.)InChina,theyearsofprimaryedu?WhatcanserveasIVforedu?M7IVforskippedclass?Thedistancefromhometoschool.IVforskippedclass?Thedista8OtherexamplesofIVIVforinstitution:Language?History?Mauro(1995)使用人口的種族和語言構成作為腐敗的工具變量，HallandJones(1999)用距離赤道的距離和以西歐語言為第一語言的程度作為制度質(zhì)量的工具變量，LaPortaetal.(1997,1998,1999)把法律的起源作為各種法律結(jié)構的工具變量。Acemoglu,Johnson,andRobinson(2001,2002)使用殖民地時代(1500年前后)的死亡率和人口密度作為制度的工具變量IVforschoolchoice:Numberofsteams?OtherexamplesofIVIVforins9IdentificationReferto(15.9)and(15.10)IdentificationReferto(15.9)10The(asymptotic)standarderrorof

SSTisthetotalsumofsquaresofthexiThe(asymptotic)standarderro11Self-selectionAngrist(1990)studiedtheeffectthatbeingaveteranintheVietnamwarhadonlifetimeearnings.DraftlotterynumberisagoodIVcandidateforveteran.SomeadditionalwordsaboutnaturalexperimentandDIDSelf-selectionAngrist(1990)s12PropertiesofIVwithaPoorInstrumentalVariable

PoorIVcancauseseriousbias.PropertiesofIVwithaPoorI13R2MostregressionpackagescomputeanR-squaredafterIVestimation,usingthestandardformula:R2=1-SSR/SST,whereSSRisthesumofsquaredIVresiduals,andSSTisthetotalsumofsquaresofy.R2canbenegativeinthiscase.R2Mostregressionpackagescom14IVESTIMATIONOFTHEMULTIPLEREGRESSION

MODELstructuralequationIVESTIMATIONOFTHEMULTIPLE15EstimationEstimation16EfficientIVEquation(15.26)isanexampleofareducedformequation,whichmeansthatwehavewrittenanendogenousvariableintermsofexogenousvariables.EfficientIVEquation(15.26)i17TWOSTAGELEASTSQUARESTWOSTAGELEASTSQUARES182SLSinwordsThefirststageistoruntheregressionin(15.36),whereweobtainthefittedvaluesy?2.ThesecondstageistheOLSregression(15.38).Becauseweusey?2inplaceofy2,the2SLSestimatescandiffersubstantiallyfromtheOLSestimates.Anotherinterpretation:2SLSinwordsThefirststagei19MultipleEndogenousExplanatoryVariablesORDERCONDITIONFORIDENTIFICATIONOFANEQUATION:Weneedatleastasmanyexcludedexogenousvariablesasthereareincludedendogenousexplanatoryvariablesinthestructuralequation.MultipleEndogenousExplanator20IVSOLUTIONSTOERRORS-IN-VARIABLESPROBLEMSOnepossibilityistoobtainasecondmeasurementonX*1,say,z1,asIV.AnalternativeistouseotherexogenousvariablesasIVsforapotentiallymismeasuredvariable.IVSOLUTIONSTOERRORS-IN-VARI21TESTINGFORENDOGENEITYANDTESTINGOVERIDENTIFYINGRESTRICTIONSThe2SLSestimatorislessefficientthanOLSwhentheexplanatoryvariablesareexogenous;aswehaveseen,the2SLSestimatescanhaveverylargestandarderrors.TESTINGFORENDOGENEITYANDTE22Howtotestendogeneity?1.ComparingtheOLSand2SLSestimatesanddeterminingwhetherthedifferencesarestatisticallysignificant.(Hausman,1978)2.Aregressiontest:Howtotestendogeneity?1.Com23Anotherinterpretationof2SLSIncludingv?2intheOLSregression(15.51)clearsuptheendogeneityofy2.Wecanalsotestforendogeneityofmultipleexplanatoryvariables.Foreachsuspectedendogenousvariable,weobtainthereducedformresiduals.Then,wetestforjointsignificanceoftheseresidualsinthestructuralequation,usinganFtest.Anotherinterpretationof2SLS24TestingOveridentificationRestrictionsIfwehavemorethanoneinstrumentalvariable,wecaneffectivelytestwhethersomeofthemareuncorrelatedwiththestructuralerror.UseoneIVandgetthepredictedresidual,thentestthecorrelationbetweenotherIVsandtheresidual.TestingOveridentificationRes25TESTINGOVERIDENTIFYINGRESTRICTIONS:(i)Estimatethestructuralequationby2SLSandobtainthe2SLSresiduals,u?1.(ii)Regressu?1onallexogenousvariables.ObtaintheR-squared,sayR12.(iii)UnderthenullhypothesisthatallIVsareuncorrelatedwithu1,nR12~aX2(q),whereqisthenumberofinstrumentalvariablesfromoutsidethemodelminusthetotalnumberofendogenousexplanatoryvariables.IfnR12exceeds(say)the5%criticalvalueintheX2(q)distribution,werejectH0andconcludethatatleastsomeoftheIVsarenotexogenous.TESTINGOVERIDENTIFYINGRESTRI26IsitbettertohavemoreIVs?Addinginstrumentstothelistimprovestheasymptoticefficiencyofthe2SLS.Butthisrequiresthatanynewinstrumentsareinfactexogenous.Withthetypicalsamplesizesavailable,addingtoomanyinstruments—thatis,increasingthenumberofoveridentifyingrestrictions—cancauseseverebiasesin2SLS.IsitbettertohavemoreIVs?272OmittedTopics2SLSWITHHETEROSKEDASTICITYAPPLYING2SLSTOTIMESERIESEQUATIONS2OmittedTopics2SLSWITHHETE28APPLYING2SLSTOPOOLEDCROSSSECTIONS

ANDPANELDATAForpooledcrosssectionsdata:addtimedummy.Forpaneldata:Inthefirststage,usethedifferencedIVtogetanestimateoftheendogenousvariable.Question:IfthepanelmodelisaFEone,howtochecktheefficiencyofIViftheIVistimeinvariant?APPLYING2SLSTOPOOLEDCROSS29STATAcommandsTocompareOLSand2SLSivregy(x=iv)x2eststoref2regyxx2hausmanf2Thesequenceisimportant.STATAcommandsTocompareOLSa30STATAcommandsTocompareFEandIV-FExtivregy(x=iv)x2,feeststoref2xtregyxx2,fehausmanf2STATAcommandsTocompareFEan31Theend.Theend.32EndogeneityinEconometrics:

InstrumentalVariableEstimationMingLUEndogeneityinEconometrics:

I33EndogeneityOmittingvariablebiasSimultaneityMeasurementerrorEndogeneityOmittingvariableb34Canweignoretheomittedvariablesbias?Itcanbesatisfactoryiftheestimatesarecoupledwiththedirectionofthebiasesforthekeyparameters.Canweuseproxytoeliminateomittedvariablebias?–Sometimes.CanFEestimationsolveomittingvariableproblem?Firstdifferencingorfixedeffectsestimationeliminatestime-constantvariables.Inaddition,thepaneldatamethodsdonotsolvetheproblemoftime-varyingomittedvariablesCanweignoretheomittedvari35IdeaofIVEstimationExogenousvariable.IndirecteffectsofIV.IdeaofIVEstimationExogenous36ExampleExample37在計量經(jīng)濟學的Endogeneity：有助易變的估計課件38WhatcanserveasIVforedu?Mother’seducation?Numberofsiblings?Thereportofothers?Adummyvariablethatisequalto1ifamanisborninthefirstquarteroftheyear.AngristandKrueger(1991).(Problematic.)InChina,theyearsofprimaryedu?WhatcanserveasIVforedu?M39IVforskippedclass?Thedistancefromhometoschool.IVforskippedclass?Thedista40OtherexamplesofIVIVforinstitution:Language?History?Mauro(1995)使用人口的種族和語言構成作為腐敗的工具變量，HallandJones(1999)用距離赤道的距離和以西歐語言為第一語言的程度作為制度質(zhì)量的工具變量，LaPortaetal.(1997,1998,1999)把法律的起源作為各種法律結(jié)構的工具變量。Acemoglu,Johnson,andRobinson(2001,2002)使用殖民地時代(1500年前后)的死亡率和人口密度作為制度的工具變量IVforschoolchoice:Numberofsteams?OtherexamplesofIVIVforins41IdentificationReferto(15.9)and(15.10)IdentificationReferto(15.9)42The(asymptotic)standarderrorof

SSTisthetotalsumofsquaresofthexiThe(asymptotic)standarderro43Self-selectionAngrist(1990)studiedtheeffectthatbeingaveteranintheVietnamwarhadonlifetimeearnings.DraftlotterynumberisagoodIVcandidateforveteran.SomeadditionalwordsaboutnaturalexperimentandDIDSelf-selectionAngrist(1990)s44PropertiesofIVwithaPoorInstrumentalVariable

PoorIVcancauseseriousbias.PropertiesofIVwithaPoorI45R2MostregressionpackagescomputeanR-squaredafterIVestimation,usingthestandardformula:R2=1-SSR/SST,whereSSRisthesumofsquaredIVresiduals,andSSTisthetotalsumofsquaresofy.R2canbenegativeinthiscase.R2Mostregressionpackagescom46IVESTIMATIONOFTHEMULTIPLEREGRESSION

MODELstructuralequationIVESTIMATIONOFTHEMULTIPLE47EstimationEstimation48EfficientIVEquation(15.26)isanexampleofareducedformequation,whichmeansthatwehavewrittenanendogenousvariableintermsofexogenousvariables.EfficientIVEquation(15.26)i49TWOSTAGELEASTSQUARESTWOSTAGELEASTSQUARES502SLSinwordsThefirststageistoruntheregressionin(15.36),whereweobtainthefittedvaluesy?2.ThesecondstageistheOLSregression(15.38).Becauseweusey?2inplaceofy2,the2SLSestimatescandiffersubstantiallyfromtheOLSestimates.Anotherinterpretation:2SLSinwordsThefirststagei51MultipleEndogenousExplanatoryVariablesORDERCONDITIONFORIDENTIFICATIONOFANEQUATION:Weneedatleastasmanyexcludedexogenousvariablesasthereareincludedendogenousexplanatoryvariablesinthestructuralequation.MultipleEndogenousExplanator52IVSOLUTIONSTOERRORS-IN-VARIABLESPROBLEMSOnepossibilityistoobtainasecondmeasurementonX*1,say,z1,asIV.AnalternativeistouseotherexogenousvariablesasIVsforapotentiallymismeasuredvariable.IVSOLUTIONSTOERRORS-IN-VARI53TESTINGFORENDOGENEITYANDTESTINGOVERIDENTIFYINGRESTRICTIONSThe2SLSestimatorislessefficientthanOLSwhentheexplanatoryvariablesareexogenous;aswehaveseen,the2SLSestimatescanhaveverylargestandarderrors.TESTINGFORENDOGENEITYANDTE54Howtotestendogeneity?1.ComparingtheOLSand2SLSestimatesanddeterminingwhetherthedifferencesarestatisticallysignificant.(Hausman,1978)2.Aregressiontest:Howtotestendogeneity?1.Com55Anotherinterpretationof2SLSIncludingv?2intheOLSregression(15.51)clearsuptheendogeneityofy2.Wecanalsotestforendogeneityofmultipleexplanatoryvariables.Foreachsuspectedendogenousvariable,weobtainthereducedformresiduals.Then,wetestforjointsignificanceoftheseresidualsinthestructuralequation,usinganFtest.Anotherinterpretationof2SLS56TestingOveridentificationRestrictionsIfwehavemorethanoneinstrumentalvariable,wecaneffectivelytestwhethersomeofthemareuncorrelatedwiththestructuralerror.UseoneIVandgetthepredictedresidual,thentestthecorrelationbetweenotherIVsandtheresidual.TestingOveridentificationRes57TESTINGOVERIDENTIFYINGRESTRICTIONS:(i)Estimatethestructuralequationby2SLSandobtainthe2SLSresiduals,u?1.(ii)Regressu?1onallexogenousvariables.ObtaintheR-squared,sayR12.(iii)UnderthenullhypothesisthatallIVsareuncorrelatedwithu1,nR12~aX2(q),whereqisthenumberofinstrumentalvariablesfromoutsidethemodelminusthetotalnumberofendogenousexplanatoryvariables.IfnR12exceed