计量经济学工具变量IV(2SLS).ppt

Week14InstrumentVariableRegressionModels SimultaneousEquationUsing2SLS Chapter16 IVEstimationinMultipleRegressionmodels 15 1 3 计量经济学 研究生 ANewApproachtotheOmittedVariableProblem Wehavetalkedabouttheproblemofomittedvariablebias inCh 3 andhaveshownthatitwillleadtoinconsistency forIfwehaveasuitableproxy wecanminimizethebias tosomedegree seeChapter9 Furthermore iftheomittedvariableistimeinvariant thenwecanuseapaneldatamodelwithoutmuchhesitation Withoutasuitableproxy nopaneldata oriftheomittedvariabledoeschangewithtimeweneedanewapproach InstrumentalVariablesRegression Threeimportantthreatstointernalvalidityare omittedvariablebiasfromavariablethatiscorrelatedwithXbutisunobserved socannotbeincludedintheregression 遗留变量偏差 simultaneouscausalitybias XcausesY YcausesX 联立因果 errors in variablesbias Xismeasuredwitherror 变量误差 Instrumentalvariablesregressioncaneliminatebiasfromthesethreesources Terminology endogeneityandexogeneity Anendogenousvariableisonethatiscorrelatedwithu Anexogenousvariableisonethatisuncorrelatedwithu Historicalnote Endogenous literallymeans determinedwithinthesystem thatis avariablethatisjointlydeterminedwithy Inotherwords itisavariablesubjecttosimultaneouscausality However thisdefinitionisnarrowandIVregressioncanbeusedtoaddressOVbiasanderrors in variablebias notjusttosimultaneouscausalitybias WhatisSimultaneousCausality SupposewehavetwoendogenousvariablesY1 Y2andtwoexogenousvariablesX1 X2suchthatY1i 0 1X1i 2Y2i u1i 1 Y2i 0 1Y1i 2X2i u2i 2 LetsseewhyY2 orY1 isendogenousSupposeu1i 0andu2i 0 thenwehaveY1i E Y1i from 1 Butin 2 if 2 0 thiswillcauseachangeinY2i soY2iiscorrelatedwithu1ithrough 2 ThesameistrueforY1iandu2iin 2 through 1 SimultaneousBias Canweestimatethesetwoequationsconsistently y1 a1y2 b1z1 u1y2 a2y1 b2z2 u2Forconsistency weneedcov y2 u1 0 andcov y1 u2 0However alargeu2meansalargery2 whichimpliesalargery1 ifa1 0 socov y1 u2 0Thesameistrueforcov y2 u1 duetothecirculareffectofu1 TheIVEstimatorwithaSingleRegressorandaSingleInstrument yi 0 1xi uiLoosely IVregressionbreaksxintotwoparts apartthatmightbecorrelatedwithu andapartthatisnot Byisolatingthepartthatisnotcorrelatedwithu itispossibletoestimate 1 Thisisdoneusinganinstrumentalvariable zi whichisuncorrelatedwithui Theinstrumentalvariabledetectsmovementsinxithatareuncorrelatedwithui andusethesetoestimate 1 Twoconditionsforavalidinstrument yi 0 1xi uiForaninstrumentalvariable an instrument ztobevalid itmustsatisfytwoconditions Instrumentrelevance cov zi xi 0Instrumentexogeneity cov zi ui 0Inotherwords IVvariablezimustbeanexogenousvariablethatiscorrelatedwithxOr zi seffectonyisonlythroughxWhichconditioncanwetest A 1B 2C BothD NeitherE Don tknowWecantestthe1stbuthavetoassumethe2nd Example LaborEconomics Supposelog wage 0 1educ u u 2abil vWhenabilisunobserved howcanweestimate 1consistentlyifcov educ abil 0 Ifwehaveaproxyforabil suchasIQandsubstituteitintoourmodel thenwearefineOtherwise weneedsomethingthatiscorrelatedwitheducbutnotwithabilParent seducation ornumberofsiblingsmightbeaninstrumentforeduc Supposewehave yi 0 1xi uicov x ui 0Ourestimateof 1willbeinconsistentEitherwefindtheomittedvariableinuiandadditintoourmodeltoovercometheinconsistencyOrwefindaninstrumentzifortheincludedvariableSupposefornowthatyouhavesuchazi we lldiscusshowtofindinstrumentalvariableslater Howcanyouusezitoestimate 1 Wewillexplainthisintwoways InstrumentVariableRegression TheIVEstimator onexandonez Explanation 1 TwoStageLeastSquares TSLS Asitsounds TSLShastwostages tworegressions 1 Firstisolatesthepartofxthatisuncorrelatedwithu regressxonzusingOLSxi 0 1zi vi 1 Becauseziisuncorrelatedwithui 0 1ziisuncorrelatedwithui Wedon tknow 0or 1butwehaveestimatedthem so Computethepredictedvaluesofxi xi wherexi 0 1zi i 1 n 2 Replacexibyxiintheregressionofinterest regressyonxiusingOLS yi 0 1xi ui 2 Becausexiisuncorrelatedwithuiinlargesamples sothefirstleastsquaresassumptionholdsThus 1canbeestimatedbyOLSusingregression 2 Thisargumentreliesonlargesamples so 0and 1arewellestimatedusingregression 1 Thistheresultingestimatoriscalledthe TwoStageLeastSquares TSLS estimator TheIVEstimator onexandonez ctd Explanation 2 only alittlealgebrayi 0 1xi uiButxi 0 1zi viThus cov yi zi cov 0 1xi ui zi cov 0 zi cov 1xi zi cov ui zi 0 cov 1xi zi 0 1cov xi zi wherecov ui zi 0 instrumentexogeneity thus 1 inlargesamplesTheinstrumentrelevancecondition cov x z 0 ensuresthatyoudon tdividebyzero SupplyandDemandExample Startwithanequationyou dliketoestimate sayasupplyfunctioninamarket qs a1p b1z u1 wherepisthepriceandzisasupplyshifter Callthisastructuralequation it sderivedfromeconomictheoryandhasacausalinterpretationwherepdirectlyaffectsqs Example cont Problemthatcan tjustregressobservedquantityonprice sinceobservedquantityaredeterminedbytheequilibriumofsupplyanddemandConsiderasecondstructuralequation inthiscasethedemandfunctionqd a2p u2SoquantityaredeterminedbyaSEM Example cont Bothqandpareendogenousbecausetheyarebothdeterminedbytheequilibriumofsupplyanddemandzisexogenous andit stheavailabilityofthisexogenoussupplyshifterthatallowsustoidentifythestructuraldemandequationWithnoobserveddemandshifters supplyisnotidentifiedandcannotbeestimated IdentificationofDemandEquation p q D S z z1 S z z2 S z z3 UsingIVtoEstimateDemand Givenqs a1p b1z u1 qd a2p u2So wecanestimatethestructuraldemandequation usingzasaninstrumentforpFirststageequationisp p0 p1z v2Secondstageequationisq a2p u2Thus 2SLSprovidesaconsistentestimatorofa2 theslopeofthedemandcurveWecannotestimatea1 theslopeofthesupplycurve TheGeneralSEM Supposeourstructuralequationsare y1 a1y2 b1z1 u1y2 a2y1 b2z2 u2Thus y2 a2 a1y2 b1z1 u1 b2z2 u2So 1 a2a1 y2 a2b1z1 b2z2 a2u1 u2 whichcanberewritten ifa2a1 1 asy2 p1z1 p2z2 v2v2 a2u1 u2 1 a2a1 Thisisthesocalled reduced formHowever inthereducedform wedon tknowwhatisthevalueofa1ora2 Example 1 Supplyanddemandforbutter IVregressionwasoriginallydevelopedtoestimatedemandelasticitiesforagriculturalgoods forexamplebutter log Qbutter 0 1log Pbutter ui 1 priceelasticityofbutter percentchangeinquantityfora1 changeinprice recalllog logspecificationdiscussion Data observationsonpriceandquantityofbutterfordifferentyearsTheOLSregressionoflog Qbutter onlog Pbutter suffersfromsimultaneouscausalitybias why SimultaneouscausalitybiasintheOLSregressionoflog Qbutter onlog Pbutter arisesbecausepriceandquantityaredeterminedbytheinteractionofdemandandsupply Asidenote Whatistherelationshipbetween sayMarxianconceptoflabortheoryofvalueandtheMicroeconomicstheoryofpriceformation Whatisthelong termsupplycurveanditsdetermination AQuickNoteonMarxianEconomics AtQ1 theproductionislessthensociallynecessary andiscausingashortageThecompetitionwilldrivethepriceaboveitvalue untilmoreproducersentersthemarketormoreproductisbeingproducedThisleadstoanincreaseinthelevelofoutput allthewaytoQ AtQ2 theproductionismorethensociallynecessary andiscausingasurplus Thecompetitionwilldrivethepricebelowitvalue untilsomeproducersleavesthemarketorlessproductisbeingproducedThisleadstoadropinthelevelofoutput allthewaytoQ SListhelong termsupplycurvethatisconsistentwiththeMarxianconceptofsociallynecessarylabortimeIsittruethatmainstreameconomichasnotheorytoexplainwhyitisatSLratherthensomeotherlevel BacktooursupplyanddemandforbutterThisinteractionofdemandandsupplyproduces Wouldaregressionusingthesedataproducethedemandcurve A DemandB SupplyC Neither Whatwouldyougetifonlysupplyshifted TSLSestimatesthedemandcurvebyisolatingshiftsinpriceandquantitythatarisefromshiftsinsupply Zisavariablethatshiftssupplybutnotdemand TSLSinthesupply demandexample log Qbutter 0 1log Pbutter uiLetZ rainfallindairy producingregions IsZavalidinstrument Letscheck2conditions 1 Exogenous corr raini ui 0 A YesB NoC InsufficientinformationPlausibly whetheritrainsindairy producingregionsshouldn taffectdemand 2 Relevant corr raini log Pbutter 0 A YesB NoC InsufficientinformationPlausibly insufficientrainfallmeanslessgrazingmeanslessbutter log Qbutter 0 1log Pbutter uiZ raini rainfallindairy producingregions Stage1 regresslog Pbutter onrain getlog Pbutter log Pbutter isolateschangesinlogpricethatarisefromsupply partofsupply atleast Stage2 regresslog Qbutter onlog Pbutter Theregressioncounterpartofusingshiftsinthesupplycurvetotraceoutthedemandcurve TSLSinthesupply demandexample ctd TSLS 2stagelestsquares inEViews EverythingthesameasinOLSexcept In EstimationMethods select TSLS Two stagelestsquares TSNLSandARMA Providealistofinstrumentvariables besuretoincludeallexogenousvariablesaswell Onlythevariablesontherighthandsidenotinthelistofinstrumentsareconsideredendogenous InOptions select Heteroskedasiticityconsistentcoefficientcovariance Example15 5using2SLS DependentVariable LOG WAGE Method Two StageLeastSquaresSample 1753IFINLFIncludedobservations 428Instrumentlist EXPEREXPERSQFATHEDUCMOTHEDUCVariableCoefficientStd Errort StatisticProb EDUC0 0613970 0314371 9530240 0515EXPER0 0441700 0134323 2883290 0011EXPERSQ 0 0008990 000402 2 2379930 0257C0 0481000 4003280 1201520 9044R squared0 135708Meandependentvar1 190173AdjustedR squared0 129593S D dependentvar0 723198S E ofregression0 674712Sumsquaredresid193 0200F statistic8 140709Durbin Watsonstat1 945659Prob F statistic 0 000028 Note RedareinstrumentsBlueareexogenousGreenisendogenous Example DemandforCigarettes Howmuchwillahypotheticalcigarettetaxreducecigaretteconsumption Toanswerthis weneedtheelasticityofdemandforcigarettes thatis 1 intheregression log Qcigarettes 0 1log Pcigarettes uiWilltheOLSestimatorplausiblybeunbiased Whyorwhynot Example Cigarettedemand ctd log Qcigarettes 0 1log Pcigarettes uiPaneldata Annualcigaretteconsumptionandaveragepricespaid includingtax 48continentalUSstates 1985 1995Proposedinstrumentalvariable Zi generalsalestaxperpackinthestate GSTaxiIsthisavalidinstrument 1 Relevant corr GSTaxi log Pcigarettes 0 2 Exogenous corr GSTaxi ui 0 Example Cigarettedemand twoinstruments DependentVariable LOG PACKPC Method Two StageLeastSquaresSample 1528IFYEAR 1995Includedobservations 48WhiteHeteroskedasticity ConsistentStandardErrors CovarianceInstrumentlist LOG INCOME POP TAX TAXS CPITAXS CPIVariableCoefficientStd Errort StatisticProb LOG INCOME POP 0 2804050 2538901 1044360 2753LOG AVGPRS CPI 1 2774240 249610 5 1176800 0000C9 7768100 96176310 165510 0000R squared0 429422Meandependentvar4 538837AdjustedR squared0 404063S D dependentvar0 243346S E ofregression0 187856Sumsquaredresid1 588044F statistic13 28079Durbin Watsonstat1 946351Prob F statistic 0 000029 Identification ThegeneralIVregressionmodel ctd Y1 0 1Y2 kYk 1 k 1Z1 k rZr uWeneedtointroducesomenewconceptsandtoextendsomeoldconceptstothegeneralIVregressionmodel Terminology identificationandoveridentificationTSLSwithincludedexogenousvariablesoneendogenousregressormultipleendogenousregressorsAssumptionsthatunderliethenormalsamplingdistributionofTSLSInstrumentvalidity relevanceandexogeneity GeneralIVregressionassumptions Identification ctd Thecoefficients 1 karesaidtobe exactlyidentifiedifm k Therearejustenoughinstrumentstoestimate 1 k overidentifiedifm k Therearemorethanenoughinstrumentstoestimate 1 k Ifso youcantestwhethertheinstrumentsarevalid atestofthe overidentifyingrestrictions we llreturntothislaterunderidentifiedifm k Therearetoofewenoughinstrumentstoestimate 1 k Ifso youneedtogetmoreinstruments Identification Ingeneral aparameterissaidtobeidentifiedifdifferentvaluesoftheparameterwouldproducedifferentdistributionsofthedata InIVregression whetherthecoefficientsareidentifieddependsontherelationbetweenthenumberofinstruments m andthenumberofendogenousregressors k Intuitively iftherearefewerinstrumentsthanendogenousregressors wecan testimate 1 kForexample supposek 1butm 0 noinstruments IdentificationofGeneralSEM Onceagain ourstructuralequationsare y1 a1y2 b1z1 u1y2 a2y1 b2z2 u2Letz1bealltheexogenousvariablesinthefirstequation andz2bealltheexogenousvariablesinthesecondequationIt sokayfortheretobeoverlapinz1andz2Howareweabletoidentifywhichequationiswhich Weneedtostatetherankcondition IdentificationofGeneralSEM Givenourtwoequations y1 a1y2 b1z1 u1y2 a2y1 b2z2 u2Toidentifyequation1 theremustbesomevariables atleast1 inz2thatarenotinz1Toidentifyequation2 theremustbesomevariables atleast1 inz1thatarenotinz2WerefertothisastherankconditionWeareabletoidentifythetwoequationsiftherankconditionissatisfied Example LaborMarket Supposethestructuralequationsforthelabormarketare hours a1log wage b10 b11educ b12age b13kidslt6 b14nwifeinc b15exper b16exper2 u1log wage a2hours b20 b21educ b22age b23kidslt6 b24nwifeinc b25exper b26exper2 u2Canweidentifywhichisthesupply demandequationforlabor No Thatisthereasonfortherankcondition Example LaborMarket Supposethestructuralequationsforthelabormarketinsteadareasfollows hours a1log wage b10 b11educ b12age b13kidslt6 b14nwifeinc u1log wage a2hours b20 b21educ b22exper b23exper2 u2Whichisthesupply demandequationforlabor 1 issupplyand2 isdemandequationsforlabor forage kidslt6andnwifeincaffectssupplybutnotdemandforlabor whileexperienceaffectsdemandbutnotsupplyoflabor OrderCondition Notethattheexogenousvariableexcludedfromthefirstequationmusthaveanon zerocoefficientinthesecondequationfortherankconditiontoholdOrderconditionstatesthattheremustbeatleastasmanyexogenousvariablesexcludedinthefirstequationasthereareendogenousvariablesinthefirstequation seepage529 Notethattheorderconditionclearlyholdsiftherankconditiondoes therewillbeanexogenousvariablefortheendogenousoneRankcondition Ordercondition Ordervs RankConditions Orderconditiononlycountsthenumberofvariables whiletheRankconditionisconcernedwiththesignificanceofthecoefficientsoftheexcludedvariablefromthefirstequationonthesecondequationWecanchecktheorderconditionbycountingthenumberofexogenousvariablesexcludedineachequation whiletherankconditionrequirestorFtest EstimationoftheGeneralSEM EstimationofSEMisstraightforwardTheinstrumentsfor2SLSaretheexogenousvariablesfrombothequationsCanextendtheideatosystemswithmorethan2equationsForagivenidentifiedequation theinstrumentsarealloftheexogenousvariablesinthewholesystem Example LaborMarketLetsreplicateexample16 5onpage564usingMrozdatasetHomeworkforChapter16 16 316 7C16 2 Mroz InferenceUsingTSLS InferenceusingTSLS Inlargesamples thesamplingdistributionoftheTSLSestimatorisnormalInference hypothesistests confidenceintervals proceedsintheusualway e g 1 96SETheideabehindthelarge samplenormaldistributionoftheTSLSestimatoristhat likealltheotherestimatorswehaveconsidered itinvolvesanaverageofmeanzeroi i d randomvariables towhichwecanapplytheCLT InferencewithIVEstimation ThehomoskedasticityassumptioninthiscaseisE u2 z s2 Var u AsintheOLScase giventheasymptoticvariance wecanestimatethestandarderrorThehigheristhecorrelationbetweenxandz thesmalleristhevarianceof 1IfXisit sowninstrument thenwearebacktoouroldOLSBLUEresults IVversusOLSestimation StandarderrorinIVcasediffersfromOLSonlyintheR2fromregressingxonzSinceR2 1 IVstandarderrorsarelargerHowever IVisconsistent whileOLSisinconsistent whenCov x u 0Thestrongerthecorrelationbetweenzandx thesmallertheIVstandarderrors InferenceusingTSLS ctd isapprox distributedN 1 Statisticalinferenceproceedsintheusualway Thejustificationis asusual basedonlargesamplesThisallassumesthattheinstrumentsarevalid we lldiscusswhathappensiftheyaren tvalidshortly Importantnoteonstandarderrors TheOLSstandarderrorsfromthesecondstageregressionaren tright theydon ttakeintoaccounttheestimationinthefirststage isestimated Instead useasinglespecializedprocedurethatcomputestheTSLSestimatorandthecorrectSEs asusual useheteroskedasticity robustSEs TheEffectofPoorInstruments WhatifourassumptionthatCov z u 0isfalse TheIVestimatorwillbeinconsistent tooCancompareasymptoticbiasinOLSandIVGiven WepreferIVifCorr z u Corr z x Corr x u Note R2fromIVcanbemisleading Givenourlinearmodel y 0 1x uiWhenxanduarecorrelated thenv y 12v x v u Sowecan tbreaktheSSTintoSSEandSSR ForreasonsthatIdon tfullyunderstandyet EViewusesSSRfromIVthatmightbelargerthenSSTofytocomputetheR2 ThusR2mightbelessthan0 anddonothavethesamemeaningasinthecaseforOLS Seeexample15 3usingbwghtdata TheGeneralIVRegressionModel SofarwehaveconsideredIVregressionwithasingleendogenousregressor X andasingleinstrument Z Weextendthisto usingWooldridge snotation multipleendogenousregressors Y2 Yk 1 multipleincludedexogenousvariables Z1 Zr TheseneedtobeincludedfortheusualO V reasonmultipleinstrumentalvariables Zr 1 Zr m More relevant instrumentscanproduceasmallervarianceofTSLS theR2ofthefirststageincreases soyouhavemorevariationin ThegeneralIVregressionmodel notationandjargon Ourstructureequationis Y1 0 1Y2 kYk 1 k 1Z1 k rZr uiY1isthedependentvariableY2 Yk 1aretheendogenousregressors potentiallycorrelatedwithu Z1 Zraretheincludedexogenousvariablesorincludedexogenousregressors uncorrelatedwithui 0 1 k raretheunknownregressioncoefficientsZr 1 Zr maretheminstrumentalvariables theexcludedexogenousvariables TwoStageLeastSquares 2SLS 15 3 Considerouroriginalstructuralmodel y1 0 1y2 2z1 u1 Withthereducedformequationy2 0 1z1 2z2 3z3 v2whereatleastonep2orp3 0ThisreducedformequationregressestheendogenousvariableonallexogenousonesHerewe reassumingthatbothz2andz3arevalidinstruments theydonotappearinthestructuralmodelandareuncorrelatedwiththestructuralerrorterm u1 i e Cov zi u1 0 i 2 3 BestInstrument Wecoulduseeitherz2orz3asaninstrumentThebestinstrumentisalinearcombinationofalloftheexogenousvariables y2 0 1z1 2z2 3z3Wecanestimatey2 byregressingy2onz1 z2andz3 cancallthisthefirststageIfthensubstitute 2fory2inthestructuralmodel getsamecoefficientasIVCouldweusez1aloneasaninstrumentfory2 A YesB NoC InsufficientinformationNo foritwillleadtoperfectmulticolinearityLetsseethisinexample15 5usingMrozdata Multicolinearitywith2SLS Multicolinearitywith2SLS Forouroriginalstructuralmodelreducedformequation y1 0 1y2 2z1 u1 y2 0 1z1 2z2 3z3 v2In2SLS wehaveVar b1 2 SST2 1 R whereSST2isthesumsquaredtotalof 2 andRisfromregressionof 2onz1 22 Multicolinearitywith2SLS Forouroriginalstructuralmodelreducedformequation y1 0 1y2 2z1 u1 y2 0 1z1 2z2 3z3 v2SinceVar b1 2 SST2 1 R IsSST2R A YesB NoC InsufficientinformationMulticolinearityproblemhastodowith2things 1 SST2Corr y2 z1 R R Multicolinearitywith2SLS Again ouroriginalstructural reducedformequations y1 0 1y2 2z1 u1 y2 0 1z1 2z2 3z3 v2WehaveVar b1 2 SST2 1 R Letsseewhy 1 SST2Corr y2 z1 R R TwoStageLeastSquares 2SLS Forouroriginalstructuralmodelreducedformequation y1 0 1y2 2z1 u1 y2 0 1z1 2z2 3z3 v2Theassumptionofatleastonepi 0 i 2 3 istheidentificationassumption Whathappensifp2 p3 0 Canwestillestimate 1 A YesB NoC InsufficientinformationPerfectmulticolinearityoccurswhenwesubstitute 2fory2inthestructuralmodelSimilarly ifF testofp2 p3 0fails z2 z3arepoorIVfory2 Example Demandforcigarettes log Qcigarettes 0 1log Pcigarettes 2log Income uZ1 generalsalestaxZ2 cigarette specifictaxEndogenousvariable log Pcigarettes Includedexogenousvariable log Income Instruments excludedendogenousvariables generalsalestax cigarette specifictaxIsthedemandelasticity 1overidentified exactlyidentified orunderidentified A OverB UnderC ExactD NotSureItisoveridentified fortherearemoreinstrumentsthenendogenousvariables Example Cigarettedemand oneinstrument DependentVariable LOG PACKPC Method Two StageLeastSquaresSample 1528IFYEAR 1995Includedobservations 48WhiteHeteroskedasticity ConsistentStandardErrors CovarianceInstrumentlist LOG INCOME POP TAXS TAX CPIVariableCoefficientStd Errort StatisticProb LOG INCOME POP 0 2145150 3117470 6881070 4949LOG AVGPRS CPI 1 1433750 372303 3 0710900 0036C9 3402751 2100337 7190260 0000R squared0 418934Meandependentvar4 538837AdjustedR squared0 393109S D dependentvar0 243346S E ofregression0 189575Sumsquaredresid1 617235F statistic6 533672Durbin Watsonstat1 960299Prob F statistic 0 003227 OneorTwoInstruments DependentVariable LOG PACKPC Method Two StageLeastSquaresSample 1528IFYEAR 1995Includedobservations 48WhiteHeteroskedasticity ConsistentStandardErrors CovarianceOneinstrumentcase Twoinstrumentscase Instrumentlist TAX TAXS CPI TAX TAXS CPITAXS CPIVariableCoeff Std E Prob Coeff Std E Prob LOG INCOME POP 0 214520 31170 49490 28040 25390 2753LOG AVGPRS CPI 1 1