第11章 联立方程模型,Introduction to simultaneous-equation model,Content,Introduction to simultaneous-equation systems The indentification problem Indirect least-squares estimation (ILS) Two-stage least squares (2SLS),Introduction to simultaneous-equation systems,So far, we concerned ourselves primarily 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. Considering a supply-demand models, in which the price of a products is simultaneously determined by the interaction of producers and consumers in a market.,Simultaneous-Equation Model,SEM 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: Supply: QtS=a1+a2Pt+a3Pt-1+et Demand: QtD=b1+b2Pt+b3Yt+ut Equilibrium: QtS=QtD,Simultaneous-Equation Model,The supply equation, demand equation, and equilibrium condition determine the market price and the quantity supplied (and demanded) when the market is in equilibrium. For this reason, the variables QtD, QtS, and Pt are often called endogenous variables; they are determined within the system of equations. The 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. The Pt-1 is determined within the system-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 Model,We can see the endogeneity of the Pt and Qt variables graphically in the figure.,SEM: structural model,Considering following supply-demand system Supply: QtS=a1+a2Pt+et Demand: QtD=b1+b2Pt+b3Yt+ut Equilibrium: QtS=QtD The 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 side(LHS) and contains endogeneous as well as predetermined variables on the RHS.,SEM: reduced model,If 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.,Simultaneous-Equation Model,Because 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. In the SEM, where (endogenous) variables in one equation feed back into variables in another equation, the error terms are correlated with the endogenous variables and least squares is both biased and inconsistent.,Simultaneous-Equation Model,Suppose we estimate the supply equation in the SEM model by using OLS. The slope parameter estimate will be Rearrange the equation, we find that,Simultaneous-Equation Model,Now see a simple model of national income determination. The reduced form of the model,Simultaneous-Equation Model,Using OLS, we have,The Identification Problem,Suppose we know the reduced form of a system of equations. Is this sufficient to allow us to discern the value of the parameters in the original set of structural equation? The problem of determining the structural equations, given knowledge of the reduced form, is called the identification problem.,The Identification Problem,An equation is unidentified, if there is no way of estimating all the structural parameters from the reduced form. An equation is identified, if it is possible to obtain values of the parameters from the reduced form equation system. An equation is exactly identified, if a unique parameter value exists and is overidentified if more than one value is obtainable for some parameters. In a structural model, some equations may be identified while others may not. In a single equation, it is possible that some parameters may be identified while others may remain unidentified.,The Identification Problem,Considering following supply-demand system Supply: Qt=a1+a2Pt+et Demand: Qt=b1+b2Pt+ut The reduced form,The Identification Problem,Considering following supply-demand system Supply: Qt=a1+a2Pt+et Demand: Qt=b1+b2Pt+b3Yt+ut The reduced form,The Identification Problem,Considering following supply-demand system Supply: Qt=a1+a2Pt+a3Tt+et Demand: Qt=b1+b2Pt+b3Yt+ut The reduced form,The Identification Problem,Order 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 to m, the number of included endogenous variables minus 1. that is, gm-1.,Indirect least-squares estimation (ILS),Consider a simple model of nationa