ln5 Instrumental Variables 2.pdf

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1、ApEc8212EconometricAnalysis---Lecture#5InstrumentalVariables(Part2)I.ProblemswithWeakInstrumentsinIVEstimatesThejustificationforalltheIVformulasinLecture4isasymptotic;samplesizeshavetobe“large”forIVtowork.Insmallsamples,IVestimatestendtowardOLSestimate

2、sasthenumberofinstrumentsgoesup.(Theyareidenticalifthenumberofinstruments=thenumberofobservations.)“Weak”IVs,thatisIVswithlowpredictivepowerforendogenousvariables,haveseveralproblems.Thislecturereviewsthisissue.Arecentreviewofmost(butnotall)ofthemateri

3、alinthislectureisCameronandTrivedi(2004,sect.4.9).Bound,JaegerandBaker(1995)ConsiderasimpleIVmodelwithonlyonexvariable,whichyoususpectisendogenous.Forsimplicity,assumethatthemeansofy,xandzequalzero,sothatwecanignoretheconstantterms:y=βx+x=z΄+1Noteth

4、atzcanhaveseveralvariables.CalculateˆOLSandˆIVusingthestandardformulas.AsymptoticProblemswhenCorr(,z)0Bound,etal.show(andyoushouldbeabletoshow)that:x,plim(ˆOLS)=+2xxˆ,plim(ˆIV)=+,wherexˆ=z΄π2xˆIneachexpression,thesecondtermshowsthepotenti

5、alinconsistency.Ifanyofthevariablesinzarecorrelatedwith,thenxˆwillbecorrelatedwith,sothatthenumeratorinthesecondterminplim(ˆIV)willnotbezero.Moreimportantly,thesizeofthisbiaswilldependonthedenominatorofthatsecondterm.Thebetterarethez’satpredictingx,

6、thelargerwillbethedenominatorinthatterm(totalvarianceofxisfixed,andthebetterthefirst-stageregressionfitsthemorethevarianceofxwillbeduetovarianceofxˆandthelesswillbeduetothevarianceofν).Sotheweakerone’sinstrumentsare,thegreatertheinconsistencyifAssumpti

7、on2SLS.1(E[z]=0)does2nothold.Thisresultcanbe(partially)quantifiedbydefiningtheinconsistencyinˆOLSandˆIVasfollows:Incons(ˆOLS)=-plim(ˆOLS)Incons(ˆIV)=-plim(ˆIV)DefinetherelativeinconsistencyofˆIV(relativetoinconsistencyinˆOLS)asIncons(ˆIV)/I

8、ncons(ˆOLS).Usingtheexpressionsforthe“plim’s”ofˆOLSandˆIVgivenabove,therelativeinconsistencyis:xˆ,/x,RelativeinconsistencyofˆIV=2Rx,z2whereRx,zistheR-squaredfromregressingxonz.Infact,itwilloftenbethecasethatthereareotherxvariabl