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1、第第11章联立方程模型章联立方程模型Introduction to simultaneous-equation modelContentlIntroduction to simultaneous-equation systemslThe indentification problemlIndirect least-squares estimation (ILS)lTwo-stage least squares (2SLS)Introduction to simultaneous-equation systemslSo far, we concerned ourselves primarily
2、with single-equation models. In this chapter we turn our attention to models consisting of several equations, in which the behavior of the variables is jointly determined.lConsidering a supply-demand models, in which the price of a products is simultaneously determined by the interaction of producer
3、s and consumers in a market.Simultaneous-Equation ModellSEM consists of a series of equations with each equation serving to explain one variable which is determined in the model. Consider a three-equation supply-demand model described as follows:lSupply: QtS=a1+a2Pt+a3Pt-1+etlDemand: QtD=b1+b2Pt+b3Y
4、t+utlEquilibrium: QtS=QtDSimultaneous-Equation ModellThe supply equation, demand equation, and equilibrium condition determine the market price and the quantity supplied (and demanded) when the market is in equilibrium.lFor this reason, the variables QtD, QtS, and Pt are often called endogenous vari
5、ables; they are determined within the system of equations.lThe model also contains two variables whose values are not determined directly within the system, which is often called predetermined variables. Pt-1 and Yt are both predetermined variables in the model.lThe Pt-1 is determined within the sys
6、tem-by past values of the variables, thus, lagged endogenous variables are predetermined variables. The variable Yt is determined completely outside the model and is called an exogenous variable.Simultaneous-Equation ModellWe can see the endogeneity of the Pt and Qt variables graphically in the figu
7、re.SEM: structural modellConsidering following supply-demand systemSupply: QtS=a1+a2Pt+etDemand: QtD=b1+b2Pt+b3Yt+utEquilibrium: QtS=QtD lThe model is often called a structural model because its form is given by the underlying theory. A structural model contains endogenous variables on the left-hand
8、 side(LHS) and contains endogeneous as well as predetermined variables on the RHS.SEM: reduced modellIf we solve the structural model for each of the endogenous variables as a function solely of the predetermined variables in the model. Then, the transformed model is called reduced form model.Simult
9、aneous-Equation ModellBecause Pt and Qt are endogenous, applying ordinary least squares to the estimation of the supply (or the demand) equation will generate biased and inconsistent estimators.lIn the SEM, where (endogenous) variables in one equation feed back into variables in another equation, th
10、e error terms are correlated with the endogenous variables and least squares is both biased and inconsistent.Simultaneous-Equation ModellSuppose we estimate the supply equation in the SEM model by using OLS. The slope parameter estimate will belRearrange the equation, we find thatSimultaneous-Equati
11、on ModellNow see a simple model of national income determination.lThe reduced form of the modelSimultaneous-Equation ModellUsing OLS, we haveThe Identification ProblemlSuppose we know the reduced form of a system of equations. Is this sufficient to allow us to discern the value of the parameters in
12、the original set of structural equation? lThe problem of determining the structural equations, given knowledge of the reduced form, is called the identification problem.The Identification ProblemlAn equation is unidentified, if there is no way of estimating all the structural parameters from the red
13、uced form.lAn equation is identified, if it is possible to obtain values of the parameters from the reduced form equation system.lAn equation is exactly identified, if a unique parameter value exists and is overidentified if more than one value is obtainable for some parameters.lIn a structural mode
14、l, some equations may be identified while others may not. lIn a single equation, it is possible that some parameters may be identified while others may remain unidentified.The Identification ProblemlConsidering following supply-demand systemSupply: Qt=a1+a2Pt+etDemand: Qt=b1+b2Pt+utlThe reduced form
15、The Identification ProblemlConsidering following supply-demand systemSupply: Qt=a1+a2Pt+etDemand: Qt=b1+b2Pt+b3Yt+utlThe reduced formThe Identification ProblemlConsidering following supply-demand systemSupply: Qt=a1+a2Pt+a3Tt+etDemand: Qt=b1+b2Pt+b3Yt+utlThe reduced formThe Identification ProblemlOr
16、der condition (necessary condition)k, the number of all variables excluded from the equation be greater than or equal to m, the number of endogenous variables in the model system minus 1. that is, km-1.g, the number of predetermined variables excluded from the equation must be greater than or equal
17、to m, the number of included endogenous variables minus 1. that is, gm-1.Indirect least-squares estimation (ILS)lConsider a simple model of national income determinationCt=b0+b1Yt+utYt=Ct+ItlThe reduced formlExample:(ex154.txt)Two-stage least squares (2SLS)lTwo-stage squares (2SLS) provides a very u
18、seful estimation procedure for obtaining the values of structural parameters in overidentified equations.lConsider following model:Income function: Yt=a1+a2Mt+a3It+a4Gt+etMoney supply function: Mt=b1+b2Yt+utWhere, I and G are exogenous.Its easy to see the money supply function is overidentified, bec
19、ause k=2m-1=1 and income function is unidentified.Two-stage least squares (2SLS)lFirst, regress Y on all predetermined variables, in this example, reg Y on I and G.Yt=p1+p2It+p3Gt+wt, wt is error term. Estimate it, we get =p1+p2It+p3Gt.lSecond, money supply function is estimated by replacing the var
20、iable Yt with the first-stage fitted variable . That is, reg Mt on ,Mt=b1+b2( +wt)+ut=b1+b2 +(b2wt+ut)Two-stage least squares (2SLS)lreg m y (i g)livreg m (y=i g) livregress 2sls m (y=I g), small stata 11lreg3 (m y), 2sls exog(i g) endog(m y)Two-stage least squares (2SLS)lAnother examplelSTATA comma
21、ndreg exp aid inc pop (inc pop ps)reg aid exp ps (inc pop ps)Ivreg exp (aid =inc pop ps) inc popIvreg aid (exp= inc pop ps) psreg3 (exp aid inc pop) (aid exp ps), 2slssummarylEndogenous variable, exogenous variable, predetermined variablelThe indentification problemlIndirect least-squares estimation (ILS)lTwo-stage least squares (2SLS)
