联立方程模型

联立方程模型ContentIntroductiontosimultaneous-equationsystemsTheindentificationproblemIndirectleast-squaresestimation(ILS)Two-stageleastsquares(2SLS)Introductiontosimultaneous-equationsystemsSofar,weconcernedourselvesprimarilywithsingle-equationmodels.Inthischapterweturnourattentiontomodelsconsistingofseveralequations,inwhichthebehaviorofthevariablesisjointlydetermined.Consideringasupply-demandmodels,inwhichthepriceofaproductsissimultaneouslydeterminedbytheinteractionofproducersandconsumersinamarket.Simultaneous-EquationModelSEMconsistsofaseriesofequationswitheachequationservingtoexplainonevariablewhichisdeterminedinthemodel.Considerathree-equationsupply-demandmodeldescribedasfollows:Supply:QtS=a1+a2Pt+a3Pt-1+etDemand:QtD=b1+b2Pt+b3Yt+utEquilibrium:QtS=QtDSimultaneous-EquationModelThesupplyequation,demandequation,andequilibriumconditiondeterminethemarketpriceandthequantitysupplied(anddemanded)whenthemarketisinequilibrium.Forthisreason,thevariablesQtD,QtS,andPtareoftencalledendogenousvariables;theyaredeterminedwithinthesystemofequations.Themodelalsocontainstwovariableswhosevaluesarenotdetermineddirectlywithinthesystem,whichisoftencalledpredeterminedvariables.Pt-1

andYtarebothpredeterminedvariablesinthemodel.ThePt-1isdeterminedwithinthesystem-bypastvaluesofthevariables,thus,laggedendogenousvariablesarepredeterminedvariables.ThevariableYtisdeterminedcompletelyoutsidethemodelandiscalledanexogenousvariable.Simultaneous-EquationModelWecanseetheendogeneityofthePt

andQt

variablesgraphicallyinthefigure.SEM:structuralmodelConsideringfollowingsupply-demandsystemSupply:QtS=a1+a2Pt+etDemand:QtD=b1+b2Pt+b3Yt+utEquilibrium:QtS=QtD

Themodelisoftencalledastructuralmodelbecauseitsformisgivenbytheunderlyingtheory.Astructuralmodelcontainsendogenousvariablesontheleft-handside(LHS)andcontainsendogeneousaswellaspredeterminedvariablesontheRHS.SEM:reducedmodelIfwesolvethestructuralmodelforeachoftheendogenousvariablesasafunctionsolelyofthepredeterminedvariablesinthemodel.Then,thetransformedmodeliscalledreducedformmodel.Simultaneous-EquationModelBecausePtandQtareendogenous,applyingordinaryleastsquarestotheestimationofthesupply(orthedemand)equationwillgeneratebiasedandinconsistentestimators.IntheSEM,where(endogenous)variablesinoneequationfeedbackintovariablesinanotherequation,theerrortermsarecorrelatedwiththeendogenousvariablesandleastsquaresisbothbiasedandinconsistent.Simultaneous-EquationModelSupposeweestimatethesupplyequationintheSEMmodelbyusingOLS.TheslopeparameterestimatewillbeRearrangetheequation,wefindthatSimultaneous-EquationModelNowseeasimplemodelofnationalincomedetermination.ThereducedformofthemodelSimultaneous-EquationModelUsingOLS,wehaveTheIdentificationProblemSupposeweknowthereducedformofasystemofequations.Isthissufficienttoallowustodiscernthevalueoftheparametersintheoriginalsetofstructuralequation?Theproblemofdeterminingthestructuralequations,givenknowledgeofthereducedform,iscalledtheidentificationproblem.TheIdentificationProblemAnequationisunidentified,ifthereisnowayofestimatingallthestructuralparametersfromthereducedform.Anequationisidentified,ifitispossibletoobtainvaluesoftheparametersfromthereducedformequationsystem.Anequationisexactlyidentified,ifauniqueparametervalueexistsandisoveridentifiedifmorethanonevalueisobtainableforsomeparameters.Inastructuralmodel,someequationsmaybeidentifiedwhileothersmaynot.Inasingleequation,itispossiblethatsomeparametersmaybeidentifiedwhileothersmayremainunidentified.TheIdentificationProblemConsideringfollowingsupply-demandsystemSupply:Qt=a1+a2Pt+etDemand:Qt=b1+b2Pt+utThereducedformTheIdentificationProblemConsideringfollowingsupply-demandsystemSupply:Qt=a1+a2Pt+etDemand:Qt=b1+b2Pt+b3Yt+utThereducedformTheIdentificationProblemConsideringfollowingsupply-demandsystemSupply:Qt=a1+a2Pt+a3Tt+etDemand:Qt=b1+b2Pt+b3Yt+utThereducedformTheIdentificationProblemOrdercondition(necessarycondition)k,thenumberofallvariablesexcludedfromtheequationbegreaterthanorequaltom,thenumberofendogenousvariablesinthemodelsystemminus1.thatis,km-1.g,thenumberofpredeterminedvariablesexcludedfromtheequationmustbegreaterthanorequaltom,thenumberofincludedendogenousvariablesminus1.thatis,gm-1.Indirectleast-squaresestimation(ILS)ConsiderasimplemodelofnationalincomedeterminationCt=b0+b1Yt+utYt=Ct+ItThereducedformExample:(ex154.txt)Two-stageleastsquares(2SLS)Two-stagesquares(2SLS)providesaveryusefulestimationprocedureforobtainingthevaluesofstructuralparametersinoveridentifiedequations.Considerfollowingmodel:Incomefunction:Yt=a1+a2Mt+a3It+a4Gt+etMoneysupplyfunction:Mt=b1+b2Yt+utWhere,IandGareexogenous.It’seasytoseethemoneysupplyfunctionisoveridentified,becausek=2>m-1=1andincomefunctionisunidentified.Two-stage