第11章 联立方程模型

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
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
Now see a simple model of national income determination.
The reduced form of the model
Simultaneous-Equation Model
Using OLS, we have
a1 b2
, v1t
ut
a2
et b2
Qt
21
+ v2t ,
21
a2b1 a1b2 a2 b2
, v2t
a 2ut a2
b2et b2
The Identification Problem
Considering following supplydemand system
– Supply: Qt=a1+a2Pt+et – Demand: Qt=b1+b2Pt+b3Yt+ut
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.
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
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:
– 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.
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
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?
, 22
a2b2 a2 b2
, 23
a2a3 a2 b2
, v2t
a2 ut et a2 b2
.
The Identification Problem
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
We can see the endogeneity of the Pt and Qt variables graphically in the figure.
SEM: structural model
Considering following supply-demand system
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.
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.
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.
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)
The Identification Problem
Considering following supply-demand system
– Supply: Qt=a1+a2Pt+et – Demand: Qt=b1+b2Pt+ut
The reduced form
Pt
11
+ v1t ,
11
b1 a2
The reduced form
Pt 11 + 12Yt + 13Tt + v1t ,
11
b1 a2
a1 b2
, 12
b2 a2 b2
, 13
a3 a2 b2
, v1t
ut
a2
et b2
.
Qt 21 + 22Yt + 23Tt + v2t ,
21
a 2 b1 a2
a1b2 b2
The reduced form
Pt 11 + 12Yt + v1t ,
Βιβλιοθήκη Baidu11
b1 a2
a1 b2
, 12
b3 a2 b2
, v1t
ut a2
et b2
Qt 21 + 22Yt + v2t ,
21
a 2 b1 a2
a1b2 b2
, 22
a2b3 a2 b2
,v2t
a 2ut a2
An equation is exactly identified, if a unique parameter value exists and is overidentified if more than one value is obtainable for some parameters.
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.
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.
b2et b2
a2
22 12
,a1
21
a 2 11,
therefore,
supply
function
is
identified.
The Identification Problem
Considering following supply-demand system
– Supply: Qt=a1+a2Pt+a3Tt+et – Demand: Qt=b1+b2Pt+b3Yt+ut
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.