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SimultaneousEquations.pdf
SimultaneousEquations.pdf
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ECON | 5370 | Advanced Econometric Theory III | le...
ECON | 5370 | Advanced Econometric Theory III | lecture_notes |Simultaneous Equation Models 1 Introduction Many economic problems involve the interaction of multiple endogenous variables within a system of equa- tions. Estimating the parameters of such as system is typically not as simple as doing OLS equation-by- equation. Issues such as identication (whether the parameters are even estimable) and endogenei
ECON | 5370 | Advanced Econometric ...
ECON | 5370 | Advanced Econometric Theory III | lecture_notes |Simultaneous Equation Models 1 Introduction Many economic problems involve the interaction of multiple endogenous variables within a system of equa- tions. Estimating the parameters of such as system is typically not as simple as doing OLS equation-by- equation. Issues such as identication (whether the parameters are even estimable) and endogenei
Page 10
where
W
indicates an instrumental variable matrix in the form of
Z
.
Zellner and Theil (1962) suggest the
following three-stage procedure for estimating
.
Stage
#1. Calculate
^
Y
i
for each equation (
i
=1
; :::; M
) using OLS and the reduced form.
Stage
#2. Use
^
Y
i
to calculate
^
2
SLS
i
and
^
ij
=
1
T
(
y
i
Z
i
^
2
SLS
i
)
0
(
y
j
Z
j
^
2
SLS
j
)
.
Stage
#3. Calculate the IV-GLS estimator
^
3
SLS
=
[
Z
0
(
^
1
X
(
X
0
X
)
1
0
)
Z
]
1
[
Z
0
(
^
1
X
(
X
0
X
)
1
0
)
y
]
=
[
^
Z
0
(
^
1
I
)
^
Z
]
1
[
^
Z
0
(
^
1
I
)
y
]
:
The asymptotic variance-covariance matrix can be estimated by
est:asy:var
(
^
3
SLS
)
=
[
Z
0
(
^
1
X
(
X
0
X
)
1
0
)
Z
]
1
=
[
^
Z
0
(
^
1
I
)
^
Z
]
1
:
5.2
Full-Information Maximum Likelihood (FIML)
The
full-information
maximum
likelihood estimator is asymptotically e¢ cient.
Assuming multivariate nor-
mally distributed errors, we maximize
ln
L
(
;
j
y;Z
)=
MT
2
ln(2
)+
T
2
ln
j
j
1
+
T
ln
j
j
1
2
(
y
Z
)
0
(
1
I
T
)(
y
Z
)
by choosing
,
B
and
.
The FIML estimator can be computationally burdensome and has the same
asymptotic distribution as the 3SLS estimator.
As a result, most researchers use 3SLS.
6
Application
Consider estimating a traditional Keynesian consumption function using quarterly data between 1947 and
2003.
The simultaneous system is
C
t
=
0
+
1
DI
t
+
c
t
(4)
DI
t
=
2
+
C
t
+
I
t
+
G
t
+
NX
t
+
y
t
(5)
where the variables are dened as follows:
Endogenous
Variables
C
t
Consumption.
10
Page 11
DI
t
Disposable Income.
Exogenous
Variables
I
t
Investment.
G
t
Government Spending.
NX
t
Net Exports.
The conditions for identication of (4), the more interesting equation to be estimated, are shown below.
Begin by writing the system as
Y
+
XB
=
Y
2
6
4
1
1
1
1
3
7
5
+
X
2
6
6
6
6
6
6
6
4
0
2
0
1
0
1
0
1
3
7
7
7
7
7
7
7
5
=
E
where
Y
t
=(
C
t
; DI
t
)
and
X
t
= (1
;I
t
;G
t
;NX
t
)
.
The order condition depends on the rank of the restriction
matrix for (4),
R
1
=
2
6
6
6
6
4
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
3
7
7
7
7
5
;
which is obviously
Rank
(
R
1
)=3
>M
1=1
:
Therefore, the model is overidentied if the rank condition
is satised (i.e.,
Rank
(
R
1
) =
M
1
).
The relevant matrix for the rank condition is
R
1
=
2
6
6
6
6
4
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
3
7
7
7
7
5
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1
1
1
1
0
2
0
1
0
1
0
1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
4
0
1
0
1
0
1
3
7
7
7
7
5
;
which has
Rank
(
R
1
) = 1
.
Therefore, equation (4) is overidentied.
Another way to see that (4) is
overidentied is to solve for the reduced-form representation of
C
t
C
t
=
0
+
1
[
2
+
C
t
+
I
t
+
G
t
+
NX
t
+
y
t
]+
c
t
=
1
1
1
[(
0
+
1
2
)+
1
I
t
+
1
G
t
+
1
NX
t
+(
c
t
+
1
y
t
)]
C
t
=
0
+
1
I
t
+
1
G
t
+
1
NX
t
+
v
t
:
(6)
11
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