Chapter 15
Simultaneous Equation
Models
? Single equation regression models:
—— The dependent variable,Y,is
expressed as a linear function of one or
more explanatory variables,the Xs.
Assumption the cause-and-effect
relationship,if any,between Y and the
Xs is unidirectional:
·explanatory variables are the cause；
·the dependent variable is the effect.
? Simultaneous equation regression models:
—— Regression models in which there is
more than one equation in which there
are feedback relationships among
variables
15.1 The Nature of Simultaneous
Equation Models
Ct=B1+B2Yt+ut Yt=Ct+It
Endogenous variable:
Variable that is an inherent part of the system
being studied and that is determined within the
system.
Variable that is caused by other variables in a
causal system
Exogenous variable/predetermined variable:
Variable entering from and determined from
outside the system being studied.
◆ If there are more endogenous variables,there
will be more equations.
15.2 The Simultaneous Equation Bias,
Inconsistency of OLS Estimators
Yt=Ct+It
=(B0+B1Yt+ut)+It
=B0+B1Yt+ut+It
? The explanatory variable in a regression
equation is correlated with the error term,this
explanatory variable becomes a random,or
stochastic variable.
ttt uBIBB
BY
111
0
1
1
1
1
1 ?
?
?
?
?
?
In the presence of simultaneous problem,
the OLS estimators are generally not BLUE.
They are biased （ in small sample ） and
inconsistent（ in large sample）
Inconsistent estimator is the estimator
which does not approach the true parameter
value even if the sample size increases
indefinitely.
15.3,The Method of Indirect Least
Squares（ ILS）
1,Simplify the original model,and get the
reduced form regression model
Ct=B1+B2Yt+ut
Ct=A1+A2I2+vt
A1=B1/(1-B2),A2=B2/(1-B2),andυt=ut/(1-B2).
ttt uBIB
B
B
BC
22
2
2
1
1
1
11 ?
?
?
?
?
?
2,Applying OLS to the reduced form of the
model,get the OLS estimators of the reduced
form model.
3.According to the relationship between the
parameters of the reduced form model and
the parameters of the original model,obtain
the estimators of the original parameters,
these estimators are the indirect least squares
estimators.
2
1
1 1 A
AB
?
?
2
2
2 1 A
AB
?
?
The ILS estimators are consistent
estimators,as the sample size increases
indefinitely,there estimators converge
to their true population values,In small
samples,the ILS estimators may be
biased,In contrast,the OLS estimators
are biased as well as inconsistent.
Whether we can use the method of
indirect least squares to estimate the
parameters of simultaneous equation
models,depends on whether we can
retrieve the original structural parameters
from the reduced form estimates,the
answer depends on the so-called
identification problem.
15.5 The Identification Problem
Identification problem,whether we can
estimate the parameters of the particular
equation.
? （ 1 ） The particular equation is exactly
identified,we can estimate the parameters of
the particular equation.
? （ 2） The equation is unidentified or under
identified,we cannot estimate the parameters.
? （ 3） The equation is over identified, there is
more than one numerical value for one or
more parameters of that equation.
Finding out whether an equation is identified:
1,Obtain the reduced form equations of the model.
2,Make a judgement:
? If number of the structured coefficients is more than
the number of the equations,the equation is
unidentified.
? If number of the structured coefficients is equal to
the number of the equations,the equation is
identified.
? If number of the structured coefficients is less than
the number of the equations,the equation is
overidentified.
15.6 Rules For Identification,The
Order Condition of Identification
? order condition of identification
m=number of endogenous (or jointly
dependent) variables in the model
k=total number of variables (endogenous
and exogenous) excluded form the equation
under consideration
1,If k =m-1,the equation is exactly identified.
2,If k＞ m-1,the equation is over identified.
3,If k＜ m-1,the equation is under identified.
? If the equation is exactly identified.
We can estimate it by ILS,and then
retrieve the original coefficient from
the reduced form coefficient.
The ILS estimators are consistent,As
the sample size increases indefinitely,
the estimators converge to their true
values.
? If the equation is over identified,we
can estimate it by 2SLS,To replace the
explanatory variable that is correlated
with the error term of the equation in
which that
The 2SLS estimators are also consistent,.
? If the equation is under identified,we
just can change the specification of the
model,such as developing another
model.
15.7 Estimation Of An Overidentification
Equation,The Method of Two-Stage Least
Squares（ 2SLS）
Two-stage least squares (2SLS).
? Stage 1,To get rid of the likely
correlation between income Y and the
error term u2,first regress Yon all
predetermined variables in the whole
model,not just that equation,Then you
get the relationship between Y and Y?
? Stage 2,Using the relationship between Y
and to rewrite the original model.
Apply OLS to estimate the model.
is uncorrelated withυ t asymptotically,
that is,in large sample (or,more accurately,as
the sample size increases indefinitely).
OLS estimators are consistent estimates of
the parameters.
Y?
Y?
Simultaneous Equation
Models
? Single equation regression models:
—— The dependent variable,Y,is
expressed as a linear function of one or
more explanatory variables,the Xs.
Assumption the cause-and-effect
relationship,if any,between Y and the
Xs is unidirectional:
·explanatory variables are the cause；
·the dependent variable is the effect.
? Simultaneous equation regression models:
—— Regression models in which there is
more than one equation in which there
are feedback relationships among
variables
15.1 The Nature of Simultaneous
Equation Models
Ct=B1+B2Yt+ut Yt=Ct+It
Endogenous variable:
Variable that is an inherent part of the system
being studied and that is determined within the
system.
Variable that is caused by other variables in a
causal system
Exogenous variable/predetermined variable:
Variable entering from and determined from
outside the system being studied.
◆ If there are more endogenous variables,there
will be more equations.
15.2 The Simultaneous Equation Bias,
Inconsistency of OLS Estimators
Yt=Ct+It
=(B0+B1Yt+ut)+It
=B0+B1Yt+ut+It
? The explanatory variable in a regression
equation is correlated with the error term,this
explanatory variable becomes a random,or
stochastic variable.
ttt uBIBB
BY
111
0
1
1
1
1
1 ?
?
?
?
?
?
In the presence of simultaneous problem,
the OLS estimators are generally not BLUE.
They are biased （ in small sample ） and
inconsistent（ in large sample）
Inconsistent estimator is the estimator
which does not approach the true parameter
value even if the sample size increases
indefinitely.
15.3,The Method of Indirect Least
Squares（ ILS）
1,Simplify the original model,and get the
reduced form regression model
Ct=B1+B2Yt+ut
Ct=A1+A2I2+vt
A1=B1/(1-B2),A2=B2/(1-B2),andυt=ut/(1-B2).
ttt uBIB
B
B
BC
22
2
2
1
1
1
11 ?
?
?
?
?
?
2,Applying OLS to the reduced form of the
model,get the OLS estimators of the reduced
form model.
3.According to the relationship between the
parameters of the reduced form model and
the parameters of the original model,obtain
the estimators of the original parameters,
these estimators are the indirect least squares
estimators.
2
1
1 1 A
AB
?
?
2
2
2 1 A
AB
?
?
The ILS estimators are consistent
estimators,as the sample size increases
indefinitely,there estimators converge
to their true population values,In small
samples,the ILS estimators may be
biased,In contrast,the OLS estimators
are biased as well as inconsistent.
Whether we can use the method of
indirect least squares to estimate the
parameters of simultaneous equation
models,depends on whether we can
retrieve the original structural parameters
from the reduced form estimates,the
answer depends on the so-called
identification problem.
15.5 The Identification Problem
Identification problem,whether we can
estimate the parameters of the particular
equation.
? （ 1 ） The particular equation is exactly
identified,we can estimate the parameters of
the particular equation.
? （ 2） The equation is unidentified or under
identified,we cannot estimate the parameters.
? （ 3） The equation is over identified, there is
more than one numerical value for one or
more parameters of that equation.
Finding out whether an equation is identified:
1,Obtain the reduced form equations of the model.
2,Make a judgement:
? If number of the structured coefficients is more than
the number of the equations,the equation is
unidentified.
? If number of the structured coefficients is equal to
the number of the equations,the equation is
identified.
? If number of the structured coefficients is less than
the number of the equations,the equation is
overidentified.
15.6 Rules For Identification,The
Order Condition of Identification
? order condition of identification
m=number of endogenous (or jointly
dependent) variables in the model
k=total number of variables (endogenous
and exogenous) excluded form the equation
under consideration
1,If k =m-1,the equation is exactly identified.
2,If k＞ m-1,the equation is over identified.
3,If k＜ m-1,the equation is under identified.
? If the equation is exactly identified.
We can estimate it by ILS,and then
retrieve the original coefficient from
the reduced form coefficient.
The ILS estimators are consistent,As
the sample size increases indefinitely,
the estimators converge to their true
values.
? If the equation is over identified,we
can estimate it by 2SLS,To replace the
explanatory variable that is correlated
with the error term of the equation in
which that
The 2SLS estimators are also consistent,.
? If the equation is under identified,we
just can change the specification of the
model,such as developing another
model.
15.7 Estimation Of An Overidentification
Equation,The Method of Two-Stage Least
Squares（ 2SLS）
Two-stage least squares (2SLS).
? Stage 1,To get rid of the likely
correlation between income Y and the
error term u2,first regress Yon all
predetermined variables in the whole
model,not just that equation,Then you
get the relationship between Y and Y?
? Stage 2,Using the relationship between Y
and to rewrite the original model.
Apply OLS to estimate the model.
is uncorrelated withυ t asymptotically,
that is,in large sample (or,more accurately,as
the sample size increases indefinitely).
OLS estimators are consistent estimates of
the parameters.
Y?
Y?
