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1、ApEc8212:EconometricAnalysisII--Lecture#4InstrumentalVariables(Part1)Instrumentalvariable(IV)methodsareusedtodealwithproblemsofomittedvariablebias,measurementerrorandsimultaneity.Theyareusedveryofteninappliedeconometrics(~75outofthe140empiricalpapersinAER,2003-2007).Thislectureandthenextl
2、ecturepresentadetailedanalysisofIVmethods.I.InstrumentalVariables:1EndogenousVariableOfthe3assumptionsofthesingle-equationlinearmodel,themosttroublesomeisAssumptionOLS.1,thatistheassumptionthatE[xu]=0.Thereasonforthistroubleis:1.Itislikelytobeviolated.2.Ifviolated,thenβˆOLS,theOLSestimate
3、of,willbebiasedandinconsistent(thisisnottruewhenAssumptionsOLS.3isviolated).3.Itisnoteasytotestwhetherithasbeenviolated.ThemostcommonlyusedmethodtogetaroundtheproblemthatE[xu]≠0isinstrumentalvariables.1ToseetheneedforIVmethods,considerthelinearpopulationmodel:y=x′β+u,E[u]=0,SupposethatCo
4、v(u,xj)=0istrueforj=1,2,…K-1butwearenotsurewhetherCov(u,xK)=0.Thatis,assumethatthefirstK-1variablesinthexvectorareexogenous,butxKmaybeendogenous.Oneexampleofthisisomittedvariablebias,wheresomeomittedvariable,q,ispartofuandiscorrelatedwithxK.InLecture3wesawthatifCov(u,xK)≠0thenalltheelemen
5、tsinβarelikelytobeinconsistent.Instrumentalvariable(IV)methodsworkbyintroducinganothervariable(an“instrument”),callitz1,thatsatisfiestwofundamentalconditions:1.Cov(z1,u)=0(“exclusionrestriction”)2.Cov[z1,xK
6、L[xK
7、1,x1,…xK-1]]≠0Thefirstconditionamountstorequiringthatz1beexogenouswithrespect
8、totheerrorterminthe(structural)equationofinterest.Itdoesnot“belong”inthatequationandthuscanbe“excluded”).2Thesecondconditionsaysthatz1musthavesome“explanatorypower”forxKbeyondwhatisobtainedfromalinearprojectionofxKontheothervariablesinx.Anequivalent(andperhapsmoreintuitive)waytostatethisi
9、stogeneratealinearprojectionofxKontheothervariablesinxandonz1,whichyields:xK=δ0+δ1x1+…δK-1xK-1+θ1z1+rK,E[rK
10、x1,…xK-1,z1]=0Theconditionamountstorequiringthatθ1≠0.Peopleoftensaythatthesecondconditionrequiresthat“z1shouldbecorrelatedwithxK”.Butthisisnotenough.Morepreci
