ECON | 5370 | Advanced Econometric Theor...
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 identication (whether the parameters are even estimable) and endogenei
Showing 10-11 out of 12
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 identication (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 dened 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 identication 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 overidentied if the rank condition
is satised (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 overidentied.
Another way to see that (4) is
overidentied 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

Ace your assessments! Get Better Grades
Browse thousands of Study Materials & Solutions from your Favorite Schools
University of Wyoming
University_of_Wyoming
School:
Advanced_Econometric_Theory_III
Course:
Great resource for chem class. Had all the past labs and assignments
Leland P.
Santa Clara University
Introducing Study Plan
Using AI Tools to Help you understand and remember your course concepts better and faster than any other resource.
Find the best videos to learn every concept in that course from Youtube and Tiktok without searching.
Save All Relavent Videos & Materials and access anytime and anywhere
Prepare Smart and Guarantee better grades