Download Simultaneous Equation Modeling

1 | P a g e NPTEL Project Econometric Modelling Vinod Gupta School of Management Module‐ 18: Simultaneous Equation Modeling
Lecture24: Simultaneous Equation Modeling
Rudra P. Pradhan Vinod Gupta School of Management Indian Institute of Technology Kharagpur, India Email: [email protected] 1 2 | P a g e MODULE OBJECTIVE
This module attempts to explore the possibilities of interdependence among the variables.
General economic activity depends on several economics forces acting simultaneously. So,
understanding this structure is very important to solve various socio-economic problems. In this
section, we highlight the followings for the simultaneous equation modeling:
1. WHAT IS SIMULTANEOUS EQUATION SYSTEM?
2. METHODS OF SIMULTANEOUS EQUATION SYSTEM
3. IDENTIFICATION PROBLEM
WHAT ARE SIMULTANEOUS EQUATION MODELS?
It is a system that describes the joint dependence of variables. Regression modeling has two
structures: Single equation modelling; structural equation modelling.
SINGLE EQUATION MODELLING:

One dependent variable with one or more independent variables;

It is a case of multiple regression analysis;

One way causality is allowed.
2 3 | P a g e STRUCTURAL EQUATION MODELLING:

Joint dependence of variables;

All are interdependent;

Identification is required for estimation process;

Two ways causality is allowed.
The above two cases can be analyzed by this model:
In case of the single equation models where we dealt with only unidirectional cause and effect
relationship, in simultaneous equation models we lump together a set of variables that can be
solved simultaneously by the remaining set of variables. Simple market model: consists of
demand equation, supply equation and equilibrium condition

Qs is the supply equation and Qd is the demand equation and the third equation is the
equilibrium equation.
3 4 | P a g e 
Qs ,Qd are the endogenous variables while let R and Y be the exogenous variables
FEATURES OF SIMULTANEOUS EQUATION MODELS
 There is more than one equation in the system
 Variables are classified into two groups: endogenous and exogenous
 Parameters cannot be estimated in a particular equation without knowing the information
about other equations.
 The direct OLS is not feasible to estimate the parameters.
DIFFERENT TYPES OF SIMULTANEOUS EQUATION MODELS
STRUCTURAL EQUATION MODELS:
Structural equation models (SEMs), also called simultaneous equation models, are multivariate
(i.e., multiple equation) regression models. Unlike the more traditional multivariate linear model,
however, the response variable in one regression equation in an SEM may appear as a predictor
in another equation; indeed, variables in an SEM may influence one-another reciprocally, either
directly or through other variables as intermediaries. These structural equations are meant to
represent causal relationships among the variables in the model.
REDUCED FORM MODELS:
In statistics, and particularly in econometrics, the reduced form of a system of equations is the
result of solving the system for the endogenous variables. This gives the latter as a function of
the exogenous variables, if any. In econometrics, "structural form" models begin from deductive
4 5 | P a g e theories of the economy, while "reduced form" models begin by identifying particular
relationships between variables.
Let Y and X be random vectors. Y is the vector of the variables to be explained (endogenous
variables) by a statistical model and X is the vector of explanatory (exogenous) variables. In
addition let be a vector of error terms.
Then the general expression of a structural form is,
Where, f is a function, possibly from vectors to vectors in the case of a multiple-equation model.
The reduced form of this model is given by,
with g a function.
RECURSIVE MODELS:
A model is recursive if its structural equations are can be ordered in such a way that the first
includes only predetermined variables in the right hand side and the second consists of
predetermined variables and the endogenous variable on the right hand side. This system of
equations can be estimated by OLS which will yield unbiased and consistent estimates
VARIOUS ISSUES UNDER SIMULTANEOUS EQUATION SYSTEM
ENDOGENOUS VARIABLE:
A variable included in the model is said to endogenous if it determines the model and is also
determined by it. When there are two or more than two such variables, there is a feedback
among these variables and hence
they are jointly-determined by the system of equations.
Consequently the error terms and the endogenous variables are always assumed to be
5 6 | P a g e correlated. Statistically it means that endogenous variables are stochastic variables and their
distribution can be derived from that of the error component.
EXOGENOUS VARIABLE:
A variable included in the model is said to exogenous if it determines the model and is not
determined by it. Consequently the error terms and the exogenous variables are always assumed
to be independent of each other or uncorrelated. Statistically it means that exogenous
variables are non-stochastic variables. Lagged dependent variables are also exogenous
variables and also referred to as predetermined variables.
MAJOR PROBLEMS:
Simultaneous equation has two most important issues; Problem identification and problem of
estimation
PROBLEMS OF IDENTIFICATION
A model is identifiable if coefficients of all its equations can be uniquely determined from those
of its reduced form model. Even if coefficients of some of its equations cannot be determined
uniquely, still the model is not identifiable. An equation in a model is said to be over identified
if the information in the regression function is more than sufficient to determine the structural
coefficients uniquely. Any equation which is not identifiable is said to be under identified.
Alternatively, an equation in a model is said to be under identified if the information in the RF is
not sufficient to determine the structural coefficients uniquely.
6 7 | P a g e ASSESSING IDENTIFICATION PROBLEMS:
There are three possible ways of identification:
Over-identified model
Under-identified model
Exactly- identified model
It can be noted that a simultaneous equation system is consistent, if it satisfies the followings:
1. Model must be uniquely in statistical form
2. All equations are identified one
3. No of equations is equal to no of parameters
4. All parameters are indentified
There are two approaches to assess the identification of the equations/ model. They are

ORDER CONDITION FOR IDENTIFICATION

RANK CONDITION FOR IDENTIFICATION
7 8 | P a g e ORDER CONDITION FOR IDENTIFICATION:
It is a necessary condition for the identification problem. The test is s based on the number of
exogenous variables to be excluded in each equation to make that equation identifiable. For an
equation to be identified the total number of variables excluded from it must be equal to or
greater than the number of endogenous variables in the model less one. It can also define that
“for an equation to be identified the total number of variables excluded from it but included in
other equations must be at least as great as the number of equations in the system less one.” To interpret the same, let us define the followings:
M=Total number of endogenous variables in the system of equations
m=Number of endogenous variables in the equation under consideration
K= Total number of exogenous variables in the system of equations
k= Number of exogenous variables in the equation under consideration
Now, the identification of the models can be tracked by the following conditions:
K  M  G  1 Order Condition for Identification problem:

An equation in the model is under identified if K-M < G-1

An equation in the is under identified if K-M = G-1

An equation in the model is over identified if K-M > G-1
8 9 | P a g e 
Sometimes the order
condition fails to give proper identification so we use the rank
condition

First use the rank condition to find out if the equation is identified or not. If the equation is
identified, then use the order condition to check if it is exactly identified or over identified.
RANK CONDITION FOR IDENTIFICATION PROBLEM:
It is a sufficient condition for the identification problem. The structure defines like this.
“In a system of G equations, any particular equation is identified, if it is possible to construct at
least one non-zero determinant of order (G-1) from the coefficients of the variables excluded
from the particular equation but included in other equation of the model.”
The followings steps to be taken to do the same:
1. Write all the parameters of the model in the separate table. That is matrix of structural
parameters.
2. Strike all the rows of coefficients of that equation to be identified.
3. Strike all the columns of non-zero coefficients.
4. Set the determinant of the order (G-1) from the left items.
5. At least one determinants of order (G-1) should have positive vale.
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