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?
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
Supply: QtS=a1+a2Pt+a3Pt-1+et Demand: QtD=b1+b2Pt+b3Yt+ut Equilibrium: QtS=QtD
Simultaneous-Equation Model
We can see the endogeneity of the Pt and Qt variables graphically in the figure.
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.
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
SEM consists of a series of equations with each equation serving to explain one variable which is determined in the model. Consider a threeequation supply-demand model described as follows:
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)
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.
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
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.
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

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.