Chapter 17
Simultaneous Equations Models
In any regression modeling, generally, an equation is considered to represent a relationship describing a
phenomenon. Many situations involve a set of relationships which explain the behaviour of certain
variables. For example, in analyzing the market conditions for a particular commodity, there can be a
demand equation and a supply equation which explain the price and quantity of commodity exchanged in the
market at market equilibrium. So there are two equations to explain the whole phenomenon - one for demand
and another for supply. In such cases, it is not necessary that all the variables should appear in all the
equations. So estimation of parameters under this type of situation has those features that are not present
when a model involves only a single relationship. In particular, when a relationship is a part of a system, then
some explanatory variables are stochastic and are correlated with the disturbances. So the basic assumption
of a linear regression model that the explanatory variable and disturbance are uncorrelated or explanatory
variables are fixed is violated and consequently ordinary least squares estimator becomes inconsistent.
Similar to the classification of variables as explanatory variable and study variable in linear regression
model, the variables in simultaneous equation models are classified as endogenous variables and exogenous
variables.
Endogenous variables (Jointly determined variables)
The variables which are explained by the functioning of system and values of which are determined by the
simultaneous interaction of the relations in the model are endogenous variables or jointly determined
variables.
Exogenous variables (Predetermined variables)
The variables that contribute to provide explanations for the endogenous variables and values of which are
determined from outside the model are exogenous variables or predetermined variables.
Exogenous variables help is explaining the variations in endogenous variables. It is customary to include past
values of endogenous variables in the predetermined group. Since exogenous variables are predetermined, so
they are independent of disturbance term in the model. They satisfy those assumptions which explanatory
variables satisfy in the usual regression model. Exogenous variables influence the endogenous variables but
Econometrics | Chapter 17 | Simultaneous Equations Models | Shalabh, IIT Kanpur
1
,are not themselves influenced by them. One variable which is endogenous for one model can be exogenous
variable for the other model.
Note that in the linear regression model, the explanatory variables influence the study variable but not vice
versa. So relationship is one sided.
The classification of variables as endogenous and exogenous is important because a necessary condition for
uniquely estimating all the parameters is that the number of endogenous variables is equal to the number of
independent equations in the system. Moreover, the main distinction of predetermined variable in estimation
of parameters is that they are uncorrelated with disturbance term in the equations in which they appear.
Simultaneous equation systems:
A model constitutes a system of simultaneous equations if all the relationships involved are needed for
determining the value of at least one of the endogenous variables included in the model. This implies that at
least one of the relationships includes more them one endogenous variable.
Example 1:
Now we consider the following example in detail and introduce various concepts and terminologies used in
describing the simultaneous equations models.
Consider a situation of an ideal market where transaction of only one commodity, say wheat, takes place.
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. Now we develop a model for such mechanism.
Let
dt denotes the demand of the commodity, say wheat, at time t ,
st denotes the supply of the commodity, say wheat, at time t , and
qt denotes the quantity of the commodity, say wheat, transacted at time t.
By economic theory about the ideal market, we have the following condition:
d t st , t 1, 2,..., n .
Econometrics | Chapter 17 | Simultaneous Equations Models | Shalabh, IIT Kanpur
2
, Observe that
the demand of wheat depends on
- price of wheat ( pt ) at time t.
- income of buyer (it ) at time t.
the supply of wheat depends on
- price of wheat ( pt ) at time t.
- rainfall ( rt ) at time t .
From market conditions, we have
qt d t st .
Demand, supply and price are determined from each other.
Note that
income can influence demand and supply, but demand and supply cannot influence the income.
supply is influenced by rainfall, but rainfall is not influenced by the supply of wheat.
Our aim is to study the behaviour of st , pt and rt which are determined by the simultaneous equation
model.
Since endogenous variables are influenced by exogenous variables but not vice versa, so
st , pt and rt are endogenous variables.
it and rt are exogenous variables.
Now consider an additional variable for the model as lagged value of price pt , denoted as pt 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.
Econometrics | Chapter 17 | Simultaneous Equations Models | Shalabh, IIT Kanpur
3
Simultaneous Equations Models
In any regression modeling, generally, an equation is considered to represent a relationship describing a
phenomenon. Many situations involve a set of relationships which explain the behaviour of certain
variables. For example, in analyzing the market conditions for a particular commodity, there can be a
demand equation and a supply equation which explain the price and quantity of commodity exchanged in the
market at market equilibrium. So there are two equations to explain the whole phenomenon - one for demand
and another for supply. In such cases, it is not necessary that all the variables should appear in all the
equations. So estimation of parameters under this type of situation has those features that are not present
when a model involves only a single relationship. In particular, when a relationship is a part of a system, then
some explanatory variables are stochastic and are correlated with the disturbances. So the basic assumption
of a linear regression model that the explanatory variable and disturbance are uncorrelated or explanatory
variables are fixed is violated and consequently ordinary least squares estimator becomes inconsistent.
Similar to the classification of variables as explanatory variable and study variable in linear regression
model, the variables in simultaneous equation models are classified as endogenous variables and exogenous
variables.
Endogenous variables (Jointly determined variables)
The variables which are explained by the functioning of system and values of which are determined by the
simultaneous interaction of the relations in the model are endogenous variables or jointly determined
variables.
Exogenous variables (Predetermined variables)
The variables that contribute to provide explanations for the endogenous variables and values of which are
determined from outside the model are exogenous variables or predetermined variables.
Exogenous variables help is explaining the variations in endogenous variables. It is customary to include past
values of endogenous variables in the predetermined group. Since exogenous variables are predetermined, so
they are independent of disturbance term in the model. They satisfy those assumptions which explanatory
variables satisfy in the usual regression model. Exogenous variables influence the endogenous variables but
Econometrics | Chapter 17 | Simultaneous Equations Models | Shalabh, IIT Kanpur
1
,are not themselves influenced by them. One variable which is endogenous for one model can be exogenous
variable for the other model.
Note that in the linear regression model, the explanatory variables influence the study variable but not vice
versa. So relationship is one sided.
The classification of variables as endogenous and exogenous is important because a necessary condition for
uniquely estimating all the parameters is that the number of endogenous variables is equal to the number of
independent equations in the system. Moreover, the main distinction of predetermined variable in estimation
of parameters is that they are uncorrelated with disturbance term in the equations in which they appear.
Simultaneous equation systems:
A model constitutes a system of simultaneous equations if all the relationships involved are needed for
determining the value of at least one of the endogenous variables included in the model. This implies that at
least one of the relationships includes more them one endogenous variable.
Example 1:
Now we consider the following example in detail and introduce various concepts and terminologies used in
describing the simultaneous equations models.
Consider a situation of an ideal market where transaction of only one commodity, say wheat, takes place.
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. Now we develop a model for such mechanism.
Let
dt denotes the demand of the commodity, say wheat, at time t ,
st denotes the supply of the commodity, say wheat, at time t , and
qt denotes the quantity of the commodity, say wheat, transacted at time t.
By economic theory about the ideal market, we have the following condition:
d t st , t 1, 2,..., n .
Econometrics | Chapter 17 | Simultaneous Equations Models | Shalabh, IIT Kanpur
2
, Observe that
the demand of wheat depends on
- price of wheat ( pt ) at time t.
- income of buyer (it ) at time t.
the supply of wheat depends on
- price of wheat ( pt ) at time t.
- rainfall ( rt ) at time t .
From market conditions, we have
qt d t st .
Demand, supply and price are determined from each other.
Note that
income can influence demand and supply, but demand and supply cannot influence the income.
supply is influenced by rainfall, but rainfall is not influenced by the supply of wheat.
Our aim is to study the behaviour of st , pt and rt which are determined by the simultaneous equation
model.
Since endogenous variables are influenced by exogenous variables but not vice versa, so
st , pt and rt are endogenous variables.
it and rt are exogenous variables.
Now consider an additional variable for the model as lagged value of price pt , denoted as pt 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.
Econometrics | Chapter 17 | Simultaneous Equations Models | Shalabh, IIT Kanpur
3
