Chapter 11 联立方程模型
Simultaneous Equation Models
Simultaneous Equation Models
Example(供给 — 需求模型）
Demanding equation:
Supplying equation:
Simultaneous equation model:
10 11 12 1dQ b b P b Y
20 21 22 1 2sQ b b P b P
10 11 12 1
20 21 22 1 2
d
s
ds
Q b b P b Y
Q b b P b P
QQ
P— 价格,Y— 收入,Qs—供给,Qd—需求平衡方程：
Variables的类型,
(1) Endogenous variable(内生变量,P,Qs,Qd)
(2) Exogenous variable(外生变量,Y)
(3) Predetermined variable(前定变量,Y,P-1)
Equations:
(1) Behavior (structural) equations
(2) Relation identities (Given)
dsQQ?
Simultaneity Bias(联立性偏误）,
Some endogenous variables are both
dependent variables and independent variables
at the same time and some endogenous variables
are relevant to the error terms,So the OLS
method does not hold good,That is
Simultaneity Bias.
内生变量与误差项相关。
Structural form:
Structural forms are models which describe the
structural relations of economic variables.
The equations of a structural form are called the
structural equations and the parameters are
called the structural parameters.
In a structural form,if the number of the endo-
genous variables is the same as the number
of the equations,the structural form is
complete.
Complete and linear structural form:
In matrix form,
Or
Y— endogenous variable sample matrix
X— predetermined variable sample matrix
11
,1,2,,
g k
ij jt ij jt it
jj
b Y r X i g?
BY Γ X ε
Y
B Γ ε
X
Reduced form(简化式方程）,
By supposing,from
We have
Approach to estimating a structural form:
(1) Estimate the reduced form:
12(,,,; ),1,2,,i i k iY f X X X i g
BY Γ X ε| | 0?B
11
11,
YB Γ XB ε Π X ν
Π B Γ ν B ε
1 1 2 2
1,,a n d 1,,
i t i t i t i k k t tY X X X
t n i g
(2) Use
to estimate the structural parameters.
That is Indirect least squares method(ILS)
The ILSE are inconsistent estimators.
Unfortunately,sometimes the ILS does not work.
1Π B Γ
Identification of Simultaneous Equations Models
什么是模型识别,
First review
the Complete and linear structural form:
In matrix form,
Reduced form:
Parameter relation system,
11
,1,2,,
g k
ij jt ij jt it
jj
b Y r X i g?
BY Γ X ε
11
11,
YB Γ XB ε Π X ν
Π B Γ ν B ε
由已知的简化式模型去确定其结构式模型的问题就称为 模型识别问题若一个随机结构方程的系 (参 )数可由参数关系体系
(方程组 )解出,则称此 随机结构方程可识别,否则,称此随机结构方程不可识别 。
若一个随机结构方程可识别，且系 (参 )数的解唯一，
则称此 随机结构方程恰好识别 ；如解不唯一，则称此 随机结构方程过度识别 。
若结构式模型中的每一个随机结构方程都可识别，则称此 结构式模型可识别,否则,称此 结构式模型不可识别 。
模型识别的条件
结构式模型识别的条件：
Suppose that there are endogenous
variables and predetermined variables in
the structural equation.
11
,1,2,,
g k
ij jt ij jt it
jj
b Y r X i g?
BY Γ X ε
ig
Y
B Γ ε
X
ik
thi
将结构参数矩阵 的第 i行去掉，再去掉此行中非零元素所对应的列而形成的矩阵记为
1) 秩条件 (rank condition):
如 则第 i个结构方程可识别，
否则，不可识别。
2) 阶条件 (order condition),
当第 i个结构方程可识别时，
如,则第 i个结构方程 恰好 识别如,则第 i个结构方程 过度 识别所以，若,则第 i个结构方程不可 识别
( )B Γ
( ) ( )( )iiB Γ
( ) ( )r a n k ( ) 1ii gB Γ
1iik k g
1iik k g
1iik k g
简化式模型识别的条件：
将简化式参数矩阵 中去掉第 i个结构方程不含的内生变量所对应的行，并去掉此结构方程所含的前定变量对应的列后，
剩下的元素按原次序构成的矩阵记为 。
1) 秩条件：
如,则第 i个结构方程可识别，
否则，不可识别。
2) 阶条件,与结构式相同。
11
11,
YB Γ XB ε Π X ν
Π B Γ ν B ε
Π
()iΠ
()r a n k ( ) 1ii gΠ
两阶段最小二乘法
（ Two-stage least squares method,TSLS)：
ILS与 IV的综合运用！
Step 1,estimate the reduced form by OLS;
Step 2,use the fitted value for the endogenous
variable as the value of the IV and then
estimate the structural equation,
Simultaneous Equation Models
Simultaneous Equation Models
Example(供给 — 需求模型）
Demanding equation:
Supplying equation:
Simultaneous equation model:
10 11 12 1dQ b b P b Y
20 21 22 1 2sQ b b P b P
10 11 12 1
20 21 22 1 2
d
s
ds
Q b b P b Y
Q b b P b P
P— 价格,Y— 收入,Qs—供给,Qd—需求平衡方程：
Variables的类型,
(1) Endogenous variable(内生变量,P,Qs,Qd)
(2) Exogenous variable(外生变量,Y)
(3) Predetermined variable(前定变量,Y,P-1)
Equations:
(1) Behavior (structural) equations
(2) Relation identities (Given)
dsQQ?
Simultaneity Bias(联立性偏误）,
Some endogenous variables are both
dependent variables and independent variables
at the same time and some endogenous variables
are relevant to the error terms,So the OLS
method does not hold good,That is
Simultaneity Bias.
内生变量与误差项相关。
Structural form:
Structural forms are models which describe the
structural relations of economic variables.
The equations of a structural form are called the
structural equations and the parameters are
called the structural parameters.
In a structural form,if the number of the endo-
genous variables is the same as the number
of the equations,the structural form is
complete.
Complete and linear structural form:
In matrix form,
Or
Y— endogenous variable sample matrix
X— predetermined variable sample matrix
11
,1,2,,
g k
ij jt ij jt it
jj
b Y r X i g?
BY Γ X ε
Y
B Γ ε
X
Reduced form(简化式方程）,
By supposing,from
We have
Approach to estimating a structural form:
(1) Estimate the reduced form:
12(,,,; ),1,2,,i i k iY f X X X i g
BY Γ X ε| | 0?B
11
11,
YB Γ XB ε Π X ν
Π B Γ ν B ε
1 1 2 2
1,,a n d 1,,
i t i t i t i k k t tY X X X
t n i g
(2) Use
to estimate the structural parameters.
That is Indirect least squares method(ILS)
The ILSE are inconsistent estimators.
Unfortunately,sometimes the ILS does not work.
1Π B Γ
Identification of Simultaneous Equations Models
什么是模型识别,
First review
the Complete and linear structural form:
In matrix form,
Reduced form:
Parameter relation system,
11
,1,2,,
g k
ij jt ij jt it
jj
b Y r X i g?
BY Γ X ε
11
11,
YB Γ XB ε Π X ν
Π B Γ ν B ε
由已知的简化式模型去确定其结构式模型的问题就称为 模型识别问题若一个随机结构方程的系 (参 )数可由参数关系体系
(方程组 )解出,则称此 随机结构方程可识别,否则,称此随机结构方程不可识别 。
若一个随机结构方程可识别，且系 (参 )数的解唯一，
则称此 随机结构方程恰好识别 ；如解不唯一，则称此 随机结构方程过度识别 。
若结构式模型中的每一个随机结构方程都可识别，则称此 结构式模型可识别,否则,称此 结构式模型不可识别 。
模型识别的条件
结构式模型识别的条件：
Suppose that there are endogenous
variables and predetermined variables in
the structural equation.
11
,1,2,,
g k
ij jt ij jt it
jj
b Y r X i g?
BY Γ X ε
ig
Y
B Γ ε
X
ik
thi
将结构参数矩阵 的第 i行去掉，再去掉此行中非零元素所对应的列而形成的矩阵记为
1) 秩条件 (rank condition):
如 则第 i个结构方程可识别，
否则，不可识别。
2) 阶条件 (order condition),
当第 i个结构方程可识别时，
如,则第 i个结构方程 恰好 识别如,则第 i个结构方程 过度 识别所以，若,则第 i个结构方程不可 识别
( )B Γ
( ) ( )( )iiB Γ
( ) ( )r a n k ( ) 1ii gB Γ
1iik k g
1iik k g
1iik k g
简化式模型识别的条件：
将简化式参数矩阵 中去掉第 i个结构方程不含的内生变量所对应的行，并去掉此结构方程所含的前定变量对应的列后，
剩下的元素按原次序构成的矩阵记为 。
1) 秩条件：
如,则第 i个结构方程可识别，
否则，不可识别。
2) 阶条件,与结构式相同。
11
11,
YB Γ XB ε Π X ν
Π B Γ ν B ε
Π
()iΠ
()r a n k ( ) 1ii gΠ
两阶段最小二乘法
（ Two-stage least squares method,TSLS)：
ILS与 IV的综合运用！
Step 1,estimate the reduced form by OLS;
Step 2,use the fitted value for the endogenous
variable as the value of the IV and then
estimate the structural equation,
