Econometric Method - Simultaneous equation model.pdf

Econometrics of Simultaneous Equation Model

1. SIMULTANEOUS
EQUATION MODEL
RAJEEV MS
II M.A ECONOMICS

2. INTRODUCTIONINTRODUCTIONINTRODUCTION
Simultaneous equations models (SEMs) are used in
econometrics and statistics to analyze systems where
multiple variables are interdependent and influence each
other simultaneously.
These models are crucial when dealing with complex
economic, social, or financial systems where variables are
not independent but rather are mutually influencing.

3. A simultaneous equations model is a type of econometric
model in which several equations are used to represent a
system of relationships between variables.
Each equation in the model represents a different
relationship, but all equations are interconnected
because they share common variables.
DEFINITIONDEFINITIONDEFINITION

4. There was a single dependent variable Y and one or more
explanatory variable X.
But in many situations, such a one-way or unidirectional cause-
and-effect relationship is not meaningful.
This occurs if Y is determined by the X’s, and some of the X’s are,
in turn, determined by Y.
In short, there is a two-way, or simultaneous relationship between
Y and (some of) the X’s, which makes the distinction between
dependent and explanatory variables of dubious value.
NATURE OF SEMNATURE OF SEMNATURE OF SEM

5. It is better to lump together a set of variables that can be
determined simultaneously by the remaining set of variables
—precisely what is done in simultaneous equation models.
In such models there is more than one equation—one for each
of the mutually dependent or endogenous variables.
And unlike the single-equation models, in the simultaneous-
equation models one may not estimate the parameters of a single
equation without taking into account information provided by
other equations in the system.
NATURE OF SEMNATURE OF SEMNATURE OF SEM

6. Consider a situation of an ideal market where transaction of
only one commodity, say wheat.
Assume that the number of buyers and sellers is large so
that the market is a perfectly competitive market.
It is also assumed that the amount of wheat that comes into
the market in a day is completely sold out on the same day.
No seller takes it back.
EXAMPLE
EXAMPLEEXAMPLE

7. Let
D denotes the Demand of the commodity, say wheat, at time t,
Ts denotes the supply of the commodity, say wheat, at time t,
Tq denotes the quantity of the commodity, say wheat,
transacted at time t.
EXAMPLEEXAMPLEEXAMPLE

10. Now consider an additional variable for the model as lagged value of price
P t , denoted as P t-1 .
In a market, generally the price of the commodity depends on the
price of the commodity on previous day.
If the price of commodity today is less than the previous day, then
buyer would like to buy more.
For seller also, today’s price of commodity depends on previous day’s
price and based on which he decides the quantity of commodity
(wheat) to be brought in the market.

11. So the lagged price affects the demand and supply equations
both. Updating both the models, we can now write that.

12. Note that the lagged variables are considered as
exogenous variable.
The updated list of endogenous and exogenous
variables is as follows

13. The price P of a commodity and the quantity Q sold are
determined by the intersection of the demand-and-
supply curves for that commodity.
Thus, assuming for simplicity that the demand-and-
supply curves are linear and adding the stochastic
disturbance .
we may write the empirical demand-and-supply
functions as

14. The system of equations is called the structural form of the
model.

15. The α’s and β’s are the parameters.
α1 is expected to be negative (downward-sloping demand
curve),
β1 is expected to be positive (upward-sloping supply curve).
Now it is not too difficult to see that P and Q are jointly
dependent variables.
If U1t changes because of changes in other variables affecting Qd
t (such as income, wealth, and tastes), the demand curve will shift
upward if U1t is positive and downward if u1t is negative.

16. These shifts are shown in Figure 1. As the figure shows, a shift in the
demand curve changes both P and Q.
Similarly, a change in U2t (because of strikes, weather, import or
export restrictions, etc.) will shift the supply curve, again affecting
both P and Q.
Because of this simultaneous dependence between Q and P, U1t and
U2t cannot be independent.
Therefore, a regression of Q on P would violate an important
assumption of the classical linear regression model, i.e the assumption
of no correlation between the explanatory variable(s) and the
disturbance term

17. FIGURE 1

18. So there are only two structural relationships.
The price is determined by the mechanism of market and not by the
buyer or supplier.
Thus Qt and Pt are the endogenous variables.
Without loss of generality, we can assume that the variables
associated with a1 and a2 are X1 & X2 respectively such that
are predetermined and so they can be regarded as exogenous variables.

19. REFERENCES
BASIC ECONOMETRICS - DAMODAR N. GUJARATI
(5TH EDITION) DAWN C. PORTER.
ECONOMETRICS | CHAPTER 17 | SIMULTANEOUS
EQUATIONS MODELS | SHALABH, IIT KANPUR

20. THANK YOU