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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 equationby 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 equationby 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 threestage 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 IVGLS 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 variancecovariance 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
FullInformation Maximum Likelihood (FIML)
The
fullinformation
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 reducedform 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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