KeywordsValidinstrumentsWeakidentiWeshowthatwheninstrumentsarenearlyexogenousthetwostageleastsquaresstatisticunpredictablyoverrejectsorunderrejectsthenullhypothesisthattheendogenousregressorisins ID: 266058
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AreNearlyExogenousInstrumentsreliable?DanielBerkowitz,MehmetCaner,YingFangDepartmentofEconomics,UniversityofPittsburgh,Pittsburgh,PA15260,USADepartmentofEconomics,NorthCarolinaStateUniversity,POBox8110,4168NelsonHall,Raleigh,NC,27695,USAWangYananInstituteforStudiesinEconomics,WISE,XiamenUniversity,ChinaABSTRACTARTICLEINFOArticlehistory: Keywords:ValidinstrumentsWeakidentiWeshowthatwheninstrumentsarenearlyexogenous,thetwostageleastsquares-statisticunpredictablyover-rejectsorunder-rejectsthenullhypothesisthattheendogenousregressorisinsignicantandRubintestover-rejectsthenull.Weprovethatinthelimitthesetestsarenolongernuisanceparameterfree.©2008ElsevierB.V.Allrightsreserved.1.IntroductionInstrumentalvariablemethodsareusedtoidentifycausalrelation-ships.Researcherspickrelevantinstrumentsthatshouldberelatedto EconomicsLetters101(2008)20 Wejustconsideronekindofinstitutionand,hence,oneendogenousvariableforexpositionalsimplicity.Ourmethodalsoworksformultipleendogenousvariables.SeeAcemogluandJohnson(2006)forananalysisofhowinstrumentalvariablescanbeusedtoidentifyhowtwoendogenousinstitutions,propertyrights(measuredbyasurveyofriskofexpropriation)andefciencyofcontracts(measuredbyanindexoflegalformalism),canaffectlongrungrowth. ContentslistsavailableatScienceDirectEconomicsLettersjournalhomepage:www.elsevier.com/locate/econbase ,areunknowns,and,fornotationalconventional,wedenote}.Othercovariates,forexample,population,latitudeandeducation,canbeaddedtothesysteminEqs.(1)and(2)withoutlossofgenerality.Inordertodeterminewhetherornotinstitutionsmatter,weestimatetheunknownparameteranduseteststatisticstocheck=0.Todothisproperly,weneedvalidinstrumentsthatarebothrelevantandexogenous.Aspreviouslydiscussed,relevantinstrumentsarepickedonthebasisofatheoretical,institutionaland/orhistoricalargument,andarevalidatedexpostbyestimatingthereducedform.Thesecondcriterionforvalidityisthatinstrumentsareexogenous,whichimpliestheyareorthogonaltotheerrorterminthestructuralequation:Itisgenerallydifcult,aswehavepreviouslyargued,tondinstru-mentsthatsatisfythisstrongcondition.Inparticular,whiletheseinstru-mentsinuencelongrungrowthinthestructuralequationprimarilythroughinstitutions,theymayalsobeweaklycorrelatedwithunobservedfactorsthatcanalsoinuencelongtermgrowth.Wemodelthispotentialsmallcorrelationasnearlyexogenouswhichisalocaltozerosetup:NearlyExogenouswhereisank1vectorofconstantsthatiscontainedincompactset.IfwechooseCovtocapturenearexogeneity,thentheteststatisticsalwaysdivergeinthelimit.Thus,thisassumptiondoesnotprovideanyguidancefornitesamplebehaviorwhenthereissomemildcorrelationbetweentheinstrumentanderror.Inwhatfollows,smallsamplesimulationmethodsareusedtoshowthatevenaslightrelaxationoftheexogeneityassumptioninEq.(3)makesthestandardteststatisticsunreliable.SupposeweemploytheTSLS-testtodeterminewhetherornotinstitutionsmatter.DenotingtheHandHasthenullandthealternativeandTSLSastheTSLSestimatorof,weusethe-statistictotestwherethestatisticisgivenbyWegeneratei.i.d.datafortheoneinstrument,thestructuralerrortermandreducedform,(),fromajointnormaldistribution1Cov1Cov0CovwhereCovmeasuresthecorrelationbetweentheinstrumenttheerrorterm,andCovmeasurestheendogeneityofinstitutions,whichissetto0.25inallsimulations.Whenthei.i.d.data()aregenerated,wecanderivetheobservationsofINSTandLRGrbyusingEqs.(1)and(2)andspeciedtruevaluesof.Basedontheinformationof(LRGrINST),wecomputethe-statisticandthentestwhetherthenullof=0canberejectedatthe5%levelbyusingthecriticalvalue1.95.Wereplicatethesimulationby1000timestoderivethedistributionofthe-statisticandcalculatetheactualrejectionprobabilitywhichisreportedinTable1Table1reportsratesofrighthandsideandlefthandfalserejectionwhentheinstrumentismoreweaklycorrelatedwiththeerrorterm:Cov=0.06or0.06andillustratesthatastheabsolutevalueofthecorrelationdecreases,thesizeproblemsofthetwo-sided-testaremitigated.Whenthecorrelationispositivethereisa9.4%falserejectionrateontherighthandside,aconservative0.4%ratefromthelefthandsideandanoverall9.8%falserejectionrate.When,thecorrelationisnegative,theratesoffalserejectionontherighthandandlefthandsidesare0.6%and7.2%,respectively,andtheoverallfalserejectionrateis7.9%.SupposewetestthenullagainstthealternativeusingAndersonRubin(AndersonandRubin,1949)test:ðÞ¼ðÞðHere,AR(=0)istheteststatisticforthenull,istheprojectionmatrixandTable1illustratesthatthesmallsampleproblemsassociatedwiththeAndersonRubintest(forherein,denotedtheARtest)arealsodiminishedwhentheinstrumentislessendogenous.Whenthecor-relationdecreasesto0.06,theARtestfalselyrejects9.4%ofthetime.Sinceitisnotpossibletocalculatetheabsolutevalueofthecorrelationbetweentheinstrumentsandstructuralerror,itisnotpossibletoadjustforthissmallsampledistortionandtheARtestisalsounreliable.3.LargesampledistributionsThissectionaddstothebadnews:weshowthattheshiftsintest-statisticdistributionsobservedinthesmallsamplesimulationsalsoholdinlimit.Fortherestofthepaper,wegeneralizethesimultaneousequationsystemEqs.(1)and(2)tomodelamoregeneralsystemwith1endogenousexplanatoryvariables,andinstruments:wherearerespectivelynx1vectorandnxmmatrixofendoge-nousexplanatoryvariables,isannxkmatrixofinstruments,isannx1vectorofstructuralerrors,isannxmmatrixofreducedformerrors,andtheerrorshavezeromeansandnitevariance,andarecorrelatedwitheachother.Asnotedbefore,otherexogenouscovariatescanbeaddedtothesystem.Inthenexttheorem,weshowthatnearexogeneityshiftstheas-ymptoticdistributionofthe-statistictoanormaldistributionwithnon-zeromean.Theorem1.SupposethattheinstrumentisnearlyexogenousaccordingtoEq.,andthestandardAssumption2intheholds.Then,whereisthesquarerootof,andisthesecondmomentmatrixofinstruments. BytheFrischWaughLovellTheorem,wecanalwaysprojectoutthesecovariatesandobtainthesysteminEqs.(1)and(2)(seeDavidsonandMcKinnnon,1993,p.19). Table1TeststatisticsSamplesize=100,and1000simulationsTruthisthatinstitutionsdonotmatterTestNominal5%criticalvaluesrejectionrateActualrejectionrate(RHS)Actualrejectionrate(LHS)-statistic±1.950.069.8%9.4%0.4%-statistic±1.950.067.9%0.6%7.2%ARtest3.85±0.069.4%n.a.n.a.-statistic±1.950.1019.4%19.2%0.2%-statistic±1.950.1014.3%0.3%14.0%ARtest3.85±0.1017.7%n.a.n.a. Wecangeneralizethisteststatistictoallowformultipleendogenousexplanatoryvariablesandatleastasmanyinstruments.D.Berkowitzetal./EconomicsLetters101(2008)20 Proof.SeetheAppendix.AccordingtoTheorem1,themeanofthedistributiondependsupontheparameter,which,byEq.(4),isrelatedtothesmallcorrelationbetweenstructuralerrorandinstruments.When=0andtheinstrumentsareexogenous,the-statisticconvergestothestandardnormaldistribution.When0(given0),thedistribu-tionshiftstotheright.When0(given0),thedistributionshiftstotheleft.Sincewecannotconsistentlyestimateletaloneknowitssign,wecannotusethislargesampletheoremtoimproveinference.ThenexttheoremcharacterizestheimpactofnearexogeneityonthedistributionoftheARtest,whichisnowmoregenerallydefromEq.(7)forinstrumentsandendogenousexplanatoryvariables:ðÞ¼ðÞðWeusethisstatistictotestHagainstHisthetruevalue.Theorem2.SupposethattheinstrumentisnearlyexogenousaccordingtoEq.,andthestandardAssumption2intheholds.Ifthenullhypothesisis,thenðÞðwhere)isanon-centralchi-squaredistributionwithkdegreesoffreedomandthenon-centralityparameter/2,whereProof.SeetheAppendix.AccordingtoTheorem2,themeanofthenon-centralityparameterisquadraticinparameter.Therefore,when=0theARtestconvergestothecenteredchi-squaredistribution,andwhenthedistributionshiftstotheright.Again,sincewedonotknow,wecannotusethesetheoremstoobtainappropriatecriticalvalues.Theconvergenceisuniform.4.ConclusionThisarticleanalyzesthelimittheorywhentherearebothweaklyedaswellasnearlyexogenousinstruments.WeshowthatRubintestisnolongerasymptoticallypivotal.Infutureresearchweconsiderhowtoremedythisproblembyusingadelete-djackknifebootstrapprocedure.AppendixAInthebeginningofthisAppendix,werstdescribethenearexogeneityassumptionandsomemomentconditionsthatarerequiredtoobtainthetheoremsinthepaper.Assumptions1and2aresufcientforLemma1,Theorem1andTheorem2.Assumption1.Nearexogeneity½¼,whereisaxed×1vector.Assumption2.Thefollowinglimitsholdjointlywhenthesamplesizeconvergestoinnity:,wherearerespectivelya1×1scalar,an×1vectorandanmatrix.whereisapositivedenite,nitematrix.,andwhereTheseconvergencesinAssumption2arenotprimitiveassump-tionsbutholdunderweakprimitiveconditions.Parts(a)and(b)followfromtheweaklawoflargenumbers,andPart(c)followsfromtriangulararrayscentrallimittheorem.InsteadofameanzeronormaldistributioninStaigerandStock(1997),thein(c)isanormaldistributionwithnon-zeromean,whichisadrifttermcomingfromthenearexogeneityassumption.Foranyindependentsequence,iffZiui]2+ b bforsome0forall=1,2,3,,thenLiapunov'stheoremleadstothelimitingresultsin(c);seeJamesDavidson(1994)Lemma1.IfAssumptionsholdforthemodeldenedbyEqs.and(2,thentheTSLSestimatorTSLSisconsistentandwhereðÞ¼ProofofLemma1.WeknowthatSowehave YVZN VZN1 ZVYN1 YVZN VZN1 ByAssumption2andEq.(2),wecanobtainthat YVZN VZN1 Now,weconsider ½þ CombiningAssumptions(1)and(2),weobtain Thentheresultinthelemmafollowsdirectly.Q.E.D.Lemma1summarizesthelimitingresultsoftheTSLSestimatorundernearexogeneity.Thereasonwhywecanobtainaconsistentestimatorundernearexogeneityisbecausethecorrelationbetweeninstrumentsandstructuralerrorsshrinkstowardzeroasymptoti-cally.When=0,wecanobtaintheregularresultsoftheTSLSestimatorundertheorthogonalitycondition.Insteadofanormaldistributionwithazeromean,nearexogeneitycanshiftthedistributionawayfrommeanzero.Thenon-zeromeandependsonanunknownlocaltozeroparameterwhichisimpossibletobeestimatedconsistently(DonaldW.K.Andrews,2000ProofofTheorem1.TheresultinthetheoremdirectlyfollowsfromLemma1.Q.E.D.ProofofTheorem2.TheAndersonRubintestisgivenbyðÞ¼ rstobservethatðÞ¼.Parts(b)and(c)inAssumption2implies:D.Berkowitzetal./EconomicsLetters101(2008)20 Next,notethat 1NKyYb0MZyYb0 1NKuVMZu¼ 1NKuVu Bypart(a)inAssumption2,thersttermconvergesinprobability,andthelasttermtendstozero.Wehave 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