instrument-variables

ShortGuidestoMicroeconometricsFall2010UnversitatPompeuFabraKurtSchmidheinyInstrumentalVariables1IntroductionThishandoutextendsthehandouton”TheMultipleLinearRegressionModel”andreferstoitsdefinitionsandassumptionsinsection2.Itdis-cussestheviolationoftheexogeneityassumption(OLS3),itsconsequencesandthepotentialsolutionthroughtheuseofinstrumentalvariables.Inmanyapplicationsofthelinearmodel,wesuspectthatsomeregres-sorsareendogenous,i.e.oneormoreregressorsarecorrelatedwiththeerrorterm,cov(xik,ui)6=0.Inthissituation,OLScannotconsistentlyestimatethecausaleffectoftheregressoronthedependentvariable.Sometimes,weareabletofindexogenousvariableszi`whicharecor-relatedwiththeendogenousregressorinxikbutnotcorrelatedwiththeerrorterm,i.e.cov(zi`,ui)=0.Suchvariableszi`arecalledinstru-ments.Ifthereareenoughgoodsuchinstrumentalvariables,weareabletoestimatethecausaleffectoftheregressoronthedependentvariable.2CanonicalExamples2.1MeasurementErrors(ErrorsinVariables)Considerthetrueregressionmodelyi=β0+x∗iβ1+u∗iwhichconformsthestandardOLSassumptions.Supposethatthevariablex∗isonlyobservedwithanerrorxi=x∗i+viVersion:28-10-2010,16:08InstrumentalVariables2wheretheerrorvisuncorrelatedwithx∗.Theestimatedregressionmodelusesxasaproxyforx∗yi=β0+xiβ1+uiwhereui=u∗i−viβ1.Theregressorxisthereforecorrelatedwiththeerrortermuasbothdependonv.Assumingindependencebetweenvandu∗,thecovarianceintheaboveexampleiscov(x,u)=−β1σ2v.Inthisspecialcaseofabivariateregression,theOLSestimatoris“biasedtowardszero”as|plimbβ1|=|β1|11+V(vi)V(xi)|β1|.2.2SimultaneityandReversedCausalityConsiderthefollowingsystemofequationsyi1=z0i1β1+yi2γ1+ui1yi2=z0i2β2+yi1γ2+ui2whereweassumethatbothz1andz2areuncorrelatedwithbothu1andu2.Thissystemiscalledastructuralsimultaneousequationsystemsincey1andy2aresimultenouslydetermined.Theregressory2dependsony1throughthesecondequation.Asy1isdirectlydependentonu1,theregressory2isalsocorrelatedwithu1andhenceendogenousinthefirstequation.Assumingthatu1andu1areuncorrelated,cov(yi2,u1)=γ2/(1−γ1γ2)σ2u1.Theaboveequationsystemisalsodescribedasreversedcausalitybecausethedependentvariabley1hasafeedbackeffectontheregressory2.Intheaboveexamplez2andz1arestraightforwardinstrumentsforIVestimationofthefirstandsecondequation,respectively.11InsteadofestimatingthesinglestructuralequationsdirectlybyIVitispossibletoformulateandestimateaso-calledreducedformoftheaboveequationsystem.TheRHSofthereducedformequationsconsistsofexogenousvariablesonly.Ifthesystemisidentified,theparametersinthestructuralformcanbededucedfromtheestimatedparametersinthereducedform.3ShortGuidestoMicroeconometrics2.3OmittedVariablesConsiderthefollowingregressionmodelyi=x0i1β1+xi2β2+viwhichconformswithstandardOLSassumptions.Supposethatthevari-ablex2isnotobserved.Theestimatedregressionmodelisthereforeyi=x0i1β1+uiwhereui=xi2β2+vi.Regressorsxkinx1arethereforecorrelatedwiththeerrortermuiftheyarecorrelatedwiththeomittedvariablex2.Incasexi1andxi2arescalars,cov(xik,ui)=β2cov(xik,xi2).3TheEconometricModelConsiderthemultiplelinearregressionmodelforobservationsi=1,...,Nyi=x0iβ+uiwhereyiisthedependentvariable,x0iisarowvectorofK+1explanatoryvariablesincludingaconstant,βisa(K+1)-dimensionalcolumnvectorofparameters,anduiistheerrorterm.Eachobservationisalsodescribedbyarowvectorz0iofL+1exogenousvariablesincludingaconstant.Thevariablesxikwhichappearinxibutnotinziarecalledendogenousregressors,theonesthatareincludedinziarecalledexogenousregressors.Additionalvariablesinziwhicharenotincludedinxiarecalledtheinstrumentsorexcludedinstruments.InstrumentalVariables4Thedatagenerationprocess(dgp)isfullydescribedbythefollowingsetofassumptions:IV1:Linearityyi=x0iβ+uiandE(ui)=0IV2:Independence{xi,zi,yi}Ni=1i.i.d.(independentandidenticallydistributed)IV2meansthatregressors,instrumentsanddependentvariablesareinde-pendentacrossobservations.Inpracticeguaranteedbyrandomsampling.IV3:ExogeneityofInstrumentscov(zi,ui)=0(uncorrelated)IV3meansthattheexogenousvariables(exogenousregressorsandinstru-ments)areuncorrelatedwiththeerrorterm.IV4:IdentifiabilityZ0XandE(zix0i)=QZXbothhaverankK+1≤L+1NZisfullrankandE(ziz0i)=QZZispositivedefiniteandfiniteIV4isalsocalledinstrumentrelevanceandrequiresthatthereareatleastasmanyinstrumentsasendogenousregressors,L≥K,andthatthecorrelationbetweenInstrumentsandendogenousregressorsisnotzero,thattheinstrumentsarenotperfectlycollinear,thatallinstruments(buttheconstant)havenon-zerovarianceandnottoomanyextremevalues.IV5:ErrorStructurea)V(ui|xi)=σ2∞(homoscedasticity)b)V(ui|xi)=σ2i=g(xi)∞(conditionalheteroscedasticity)5ShortGuidestoMicroeconometrics4EstimationwithOLSTheOLSestimatorofβisbiasedsinceE(u|X)6=0andinconsistentsinceplim1NX0u6=0.5EstimationwithIV(2SLS)TheinstrumentalvariablesestimatorforβisbβIV=(X0PZX)−1X0PZy=bX0X−1bX0ywherePZ=Z(Z0Z)−1Z0andbX=PZX=Z(Z0Z)−1Z0X.Ifthenumberofinstrumentsislargerthanthenumberofendogenousregressors,LK,theIVestimatoriscalledover-identified.Ifthenumberofinstrumentsequalsthenumberofendogenousregressors,L=K,theIVestimatoriscalledjust-identifiedandreducestobβIV=(Z0X)−1Z0y.TheIVestimatorcanalways