Download Simultaneous Equation Modeling - Econometric Modeling - Lecture Notes and more Study notes Econometrics and Mathematical Economics in PDF only on Docsity!
2
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?
- METHODS OF SIMULTANEOUS EQUATION SYSTEM
- 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.
3
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
5
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 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
6
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 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
8
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-
An equation in the is under identified if K-M = G-
An equation in the model is over identified if K-M > G-
9
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:
- Write all the parameters of the model in the separate table. That is matrix of structural parameters.
- Strike all the rows of coefficients of that equation to be identified.
- Strike all the columns of non-zero coefficients.
- Set the determinant of the order (G-1) from the left items.
- At least one determinants of order (G-1) should have positive vale.
