Economics20 Prof Anderson 1 InstrumentalVariables 2SLS y b0 b1x1 b2x2 bkxk ux1 p0 p1z p2x2 pkxk v Economics20 Prof Anderson 2 WhyUseInstrumentalVariables InstrumentalVariables IV estimationisusedwhenyourmodelhasendogenousx sThatis wheneverCov x u 0Thus IVcanbeusedtoaddresstheproblemofomittedvariablebiasAdditionally IVcanbeusedtosolvetheclassicerrors in variablesproblem Economics20 Prof Anderson 3 WhatIsanInstrumentalVariable Inorderforavariable z toserveasavalidinstrumentforx thefollowingmustbetrueTheinstrumentmustbeexogenousThatis Cov z u 0TheinstrumentmustbecorrelatedwiththeendogenousvariablexThatis Cov z x 0 Economics20 Prof Anderson 4 MoreonValidInstruments WehavetousecommonsenseandeconomictheorytodecideifitmakessensetoassumeCov z u 0WecantestifCov z x 0JusttestingH0 p1 0inx p0 p1z vSometimesrefertothisregressionasthefirst stageregression Economics20 Prof Anderson 5 IVEstimationintheSimpleRegressionCase Fory b0 b1x u andgivenourassumptionsCov z y b1Cov z x Cov z u sob1 Cov z y Cov z x ThentheIVestimatorforb1is Economics20 Prof Anderson 6 InferencewithIVEstimation ThehomoskedasticityassumptioninthiscaseisE u2 z s2 Var u AsintheOLScase giventheasymptoticvariance wecanestimatethestandarderror Economics20 Prof Anderson 7 IVversusOLSestimation StandarderrorinIVcasediffersfromOLSonlyintheR2fromregressingxonzSinceR2 1 IVstandarderrorsarelargerHowever IVisconsistent whileOLSisinconsistent whenCov x u 0Thestrongerthecorrelationbetweenzandx thesmallertheIVstandarderrors Economics20 Prof Anderson 8 TheEffectofPoorInstruments WhatifourassumptionthatCov z u 0isfalse TheIVestimatorwillbeinconsistent tooCancompareasymptoticbiasinOLSandIVPreferIVifCorr z u Corr z x Corr x u Economics20 Prof Anderson 9 IVEstimationintheMultipleRegressionCase IVestimationcanbeextendedtothemultipleregressioncaseCallthemodelweareinterestedinestimatingthestructuralmodelOurproblemisthatoneormoreofthevariablesareendogenousWeneedaninstrumentforeachendogenousvariable Economics20 Prof Anderson 10 MultipleRegressionIV cont Writethestructuralmodelasy1 b0 b1y2 b2z1 u1 wherey2isendogenousandz1isexogenousLetz2betheinstrument soCov z2 u1 0andy2 p0 p1z1 p2z2 v2 wherep2 0Thisreducedformequationregressestheendogenousvariableonallexogenousones Economics20 Prof Anderson 11 TwoStageLeastSquares 2SLS It spossibletohavemultipleinstrumentsConsiderouroriginalstructuralmodel andlety2 p0 p1z1 p2z2 p3z3 v2Herewe reassumingthatbothz2andz3arevalidinstruments theydonotappearinthestructuralmodelandareuncorrelatedwiththestructuralerrorterm u1 Economics20 Prof Anderson 12 BestInstrument Coulduseeitherz2orz3asaninstrumentThebestinstrumentisalinearcombinationofalloftheexogenousvariables y2 p0 p1z1 p2z2 p3z3Wecanestimatey2 byregressingy2onz1 z2andz3 cancallthisthefirststageIfthensubstitute 2fory2inthestructuralmodel getsamecoefficientasIV Economics20 Prof Anderson 13 Moreon2SLS Whilethecoefficientsarethesame thestandarderrorsfromdoing2SLSbyhandareincorrect soletStatadoitforyouMethodextendstomultipleendogenousvariables needtobesurethatwehaveatleastasmanyexcludedexogenousvariables instruments asthereareendogenousvariablesinthestructuralequation Economics20 Prof Anderson 14 AddressingErrors in VariableswithIVEstimation Remembertheclassicalerrors in variablesproblemwhereweobservex1insteadofx1 Wherex1 x1 e1 ande1isuncorrelatedwithx1 andx2Ifthereisaz suchthatCorr z u 0andCorr z x1 0 thenIVwillremovetheattenuationbias Economics20 Prof Anderson 15 TestingforEndogeneity SinceOLSispreferredtoIVifwedonothaveanendogeneityproblem thenwe dliketobeabletotestforendogeneityIfwedonothaveendogeneity bothOLSandIVareconsistentIdeaofHausmantestistoseeiftheestimatesfromOLSandIVaredifferent Economics20 Prof Anderson 16 TestingforEndogeneity cont Whileit sagoodideatoseeifIVandOLShavedifferentimplications it seasiertousearegressiontestforendogeneityIfy2isendogenous thenv2 fromthereducedformequation andu1fromthestructuralmodelwillbecorrelatedThetestisbasedonthisobservation Economics20 Prof Anderson 17 TestingforEndogeneity cont SavetheresidualsfromthefirststageIncludetheresidualinthestructuralequation whichofcoursehasy2init Ifthecoefficientontheresidualisstatisticallydifferentfromzero rejectthenullofexogeneityIfmultipleendogenousvariables jointlytesttheresidualsfromeachfirststage Economics20 Prof Anderson 18 TestingOveridentifyingRestrictions Ifthereisjustoneinstrumentforourendogenousvariable wecan ttestwhethertheinstrumentisuncorrelatedwiththeerrorWesaythemodelisjustidentifiedIfwehavemultipleinstruments itispossibletotesttheoveridentifyingrestrictions toseeifsomeoftheinstrumentsarecorrelatedwiththeerror Economics20 Prof Anderson 19 TheOverIDTest EstimatethestructuralmodelusingIVandobtaintheresidualsRegresstheresidualsonalltheexogenousvariablesandobtaintheR2toformnR2Underthenullthatallinstrumentsareuncorrelatedwiththeerror LM cq2whereqisthenumberofextrainstruments Economics20 Prof Anderson 20 TestingforHeteroskedasticity Whenusing2SLS weneedaslightadjustmenttotheBreusch PagantestGettheresidualsfromt