第6讲 内生性问题

内生性问题15-118-2假定3不成立：导致OLS不一致的四种情况1.OmittedVariablesBias（遗漏变量）2.MeasurementError（测量误差）3.SimultaneousCausality（内生性）4.UsingLaggedValuesoftheDependentVariableasExplanators,inthepresenceofserialcorrelation（自相关且包含被解释变量滞后项）•AnyoftheseconditionswillmakeOLSinconsistent(andbiased).遗漏变量•真实模型•估计量b0,b1,b2•估计模型•估计量a0,a115-301122iiiiyXX011iiiyX遗漏变量的后果•1.假如x1和x2是相关的，即r12≠0，那么a1和a2都是有偏的和不一致的，即E(a1)≠β1,E(a0)≠β0，在大样本的情况下，这种偏差也不消失。15-401122iiiiyXX011iiiyX11221()Eabb21为X2对X1的回归系数遗漏变量的后果•2.即使x1和x2是不相关的，a0还是有偏，虽然a1无偏了。•3.误差项方差不正确估计。•按照残差平方和除以n-k-1估计误差项，两个模型不同残差平方和和n-k-1都不同15-501122iiiiyXX011iiiyX遗漏变量的后果•4.计算的a1的方差是真实估计量b1方差的有偏估计量15-601122iiiiyXX011iiiyX21211var()()iaXX221222111211var()()(1)()iibVIFXXrXX11var()var()ab遗漏变量的后果•5.通常的估计参数的置信区间和假设检验变得不可靠。•6.基于不正确模型的预测和预测区间也不可靠。•注意：假如模型是基于相应的经济理论而构建，不要擅自删除该理论所要求的变量。15-701122iiiiyXX011iiiyX18-8MeasurementError测量误差•解释变量X的测量误差会导致假定三不成立(alsocalled“errorsinvariables”).•模型设定的X与实际观察的X不同，那么系数估计会错误18-9测量误差•测量错误产生的原因:–数据瑕疵(如：记不清)–使用了不完美的代理变量(如：用应税收入代替总收入)–没有正确理解真正的解释变量（如：年收入变化还是永久收入变化）18-10测量误差WemodelYi01XiibutinsteadofobservingXiweobserveMiXiviwhereviissomeerrorE(vi)0,Cov(Xi,vi)0WeregressYi01Mii18-11测量误差WemodelYi01XiiWeregressYi01MiiSubstituting,ourregressionequationisequivalentto01(Xivi)i01Xi1vii18-12测量误差010101111ˆ)WemodelWeregresswhichis,inessence,hastoperformtwojobs.Itestimatesthecoefficienton(butitALSOestimatesthecoefficientiiiiiiiiiiYXYMXvX.onSincedoesnotappearinourmodel,itstruecoefficientis0.iivv18-13测量误差01111ˆ().Inessence,weregresshastoperformtwojobs.ItestimatesthecoefficientonbutitALSOestimatesthecoefficientonSincedoesnotappearinourmodel,iiiiiiiYXvXvv110ˆ0.itstruecoefficientis.willbeaweightedaverageofand18-14测量误差OurmodelisYi01XiiInsteadofXi,weobserveMiXiiYi01(Mivi)i01Mi1vii01Miiwherei1vii18-15测量误差010111111(,)(,)(,)(,)()OurmodelisInsteadof,weobserveWeregresswhereiiiiiiiiiiiiiiiiiiiiiiiiYXXMXvYMvCovMCovXvvCovvvCovvvVarv18-16测量误差•Measurementerrorimpliesacorrelationbetweenourobservedexplanatorandtheerrorterm•WhatbiasdoesthiscorrelationcreateinOLS?MiXivii-1vii18-17带有测量误差的DGPYi01XiiE(i)0Var(i)2Cov(i,j)0forijE(Xii)0,1n(xi2)X2MiXiviE(vi)0Var(vi)v2Cov(vi,vj)0forijCov(vi,Xi)018-18测量误差0110111111?)cov(M,)ˆvar()cov(M,v)cov(M,)var()iiiiiiiYMYMMM是的一致估计量吗（Regress+-v18-19MeasurementError(cont.)12122()()()iiiXXvVarXVarXVarv18-20MeasurementError(cont.)211122212222211ˆlim()1ˆ0XXvvXvvXvp1NoticethatThebiaswillalwayspullPARTWAYfromto.WecallthisbiasATTENUATIONBIAS.18-21MeasurementError(cont.)•MismeasuringXleadstoATTENUATIONBIAS（零向偏误）.Theestimatedcoefficientisbiasedtowards0.•ThemagnitudeofthebiasdependsontherelativevariancesofXandv.•Asmallamountofrandommeasurementnoisewillnotbiastheestimateverymuch.18-22MeasurementError(cont.)•MismeasuringXleadstoATTENUATIONBIAS.Theestimatedcoefficientisbiasedtowards0.•Note:mismeasuringYdoesNOTleadtomeasurementerrorbias(thoughitdoesincreasethevarianceoftheerrorterm,thusincreasingstandarderrors).18-23CheckingUnderstanding(cont.)•Supposeyouareadvisingpolicymakersontheeffectofaone-timetaxrebateonconsumption.Usingacrosssectionof50,000households,youregressreportedcurrentconsumptionagainstreportedcurrentincome.•YourresultssuggestthatthetaxrebateinquestionwillNOThavealargeenoughimpactonconsumptiontojustifythepolicy.(continuedonnextslide)18-24CheckingUnderstanding(cont.)•Supposeyourresultsfoundthatthemarginalpropensitytoconsumewastoosmalltojustifythetaxrebate.•AproponentofthemeasurearguesthatyoushouldberegressingconsumptionagainstFriedman’s“permanentincome,”notcurrentincome.Theresultingmeasurementerrorrendersyourresultsirrelevant.•Assessthisargument.18-25CheckingUnderstanding(cont.)•Answer:Ifyourregressionsuffersfrommeasurementerror,thenthetrueeffectofthetaxrebateonconsumptionislargerthanwhatyoufound.•Thecoefficientwithattenuationbiasistoosmalltojustifythepolicy.Thelarger,truecoefficientmightormightnotbelargeenoughtosupportthetaxrebate.18-26SimultaneousCausality（内生性问题）•AnothercommonsourceofcorrelationbetweenXiandiisSIMULTANEOUSCAUSALITY•ThiscomplicationisalsocalledJOINTLYDETERMINEDVARIABLESorENDOGENEITY18-27SimultaneousCausality(cont.)•BothXandYarejointlydetermined•TheprocessthatgeneratesYalsogeneratesXatthesametime•BecauseXandYaredeterminedsimultaneously,XcanadjustinresponsetoshockstoY()•ThusXwillbecorrelatedwith18-28SimultaneousCausality(cont.)•Theclassicexampleofsimultaneouscausalityineconomicsissupplyanddemand.•Bothpricesandquantitiesadjustuntilsupplyanddemandareinequilibrium.•AshocktodemandorsupplycausesBOTHpricesandquantitiestomove.18-29SimultaneousCausality(cont.)•Thus,anyattempttoestimatetherelationshipbetweenpricesandquantities(say,toestimateademandelasticity)suffersfromSIMULTANEITYBIAS.•Econometricianshaveafrequentinterestinestimatingelasticitie