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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 1
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 endogeneity bias
are the primary topics in this chapter.
2
The Model
The simultaneous system can be written as
Y
+
XB
=
E
(1)
where the variable matrices are
Y
T
M
=
2
6
6
6
6
6
6
6
4
Y
11
Y
12
Y
1
M
Y
21
Y
22
Y
2
M
.
.
.
.
.
.
.
.
.
Y
T
1
Y
T
2
Y
TM
3
7
7
7
7
7
7
7
5
;
X
T
K
=
2
6
6
6
6
6
6
6
4
X
11
X
12
X
1
K
X
21
X
22
X
2
K
.
.
.
.
.
.
.
.
.
X
T
1
X
T
2
X
TK
3
7
7
7
7
7
7
7
5
;
E
T
M
=
2
6
6
6
6
6
6
6
4
11
12
1
M
21
22
2
M
.
.
.
.
.
.
.
.
.
T
1
T
2
TM
3
7
7
7
7
7
7
7
5
and the coe¢ cient matrices are
M
M
=
2
6
6
6
6
6
6
6
4
11
21
M
1
12
22
M
2
.
.
.
.
.
.
.
.
.
1
M
2
M
MM
3
7
7
7
7
7
7
7
5
;
B
K
M
=
2
6
6
6
6
6
6
6
4
11
21
M
1
12
22
M
2
.
.
.
.
.
.
.
.
.
1
K
2
K
MK
3
7
7
7
7
7
7
7
5
:
Some denitions.
Y
t;j
is the
jth
endogenous variable.
X
t;j
is the
jth
exogenous or predetermined variable
Equations (1) are referred to as
structural
equations.
and
B
are the
structural
parameters.
To examine the assumptions about the error terms, rewrite the
E
matrix as
~
E
=
vec
(
E
)=(
11
;
21
; :::;
T
1
;
12
;
22
; :::;
T
2
; :::;
1
M
;
2
M
; :::;
TM
)
0
:
1
Page 2
We assume
E
(
~
E
)
=
0
E
(
~
E
~
E
0
)
=
I
T
where the variance-covariance matrix for
t
=(
t
1
;
t
2
; :::;
tM
)
0
is
=
2
6
6
6
6
6
6
6
4
11
21
M
1
12
22
M
2
.
.
.
.
.
.
.
.
.
1
M
2
M
MM
3
7
7
7
7
7
7
7
5
:
2.1
Reduced Form
The
reduced-form solution to (1) is
Y
=
XB
1
+
E
1
=
X
+
V
where
=
B
1
,
V
=
E
1
and the error vector
~
V
=
vec
(
V
)
satises
E
(
~
V
)
=
0
E
(
~
V
~
V
0
)
=
(
1
0
1
I
T
) = (
I
T
)
where
=
0
.
2.2
Demand and Supply Example
Consider the following demand and supply equations
Q
s
t
=
0
+
1
P
t
+
2
W
t
+
3
Z
t
+
s
t
Q
d
t
=
0
+
1
P
t
+
3
Z
t
+
d
t
Q
s
t
=
Q
d
t
2
Page 3
where
Q
s
t
,
Q
d
t
and
P
t
are endogenous variables and
W
t
and
Z
t
are exogenous variables.
Let
Q
=
Q
s
t
=
Q
d
t
.
In matrix form, the system can be written as
Y
=
2
6
6
6
6
6
6
6
4
Q
1
P
1
Q
2
P
2
.
.
.
.
.
.
Q
T
P
T
3
7
7
7
7
7
7
7
5
;
X
=
2
6
6
6
6
6
6
6
4
1
W
1
Z
1
1
W
2
Z
2
.
.
.
.
.
.
.
.
.
1
W
T
Z
T
3
7
7
7
7
7
7
7
5
;
E
=
2
6
6
6
6
6
6
6
4
s
1
d
1
s
2
d
2
.
.
.
.
.
.
s
T
d
T
3
7
7
7
7
7
7
7
5
and
=
2
6
4
1
1
1
1
3
7
5
;
B
=
2
6
6
6
6
4
0
0
2
0
3
3
3
7
7
7
7
5
3
Identication
Identication
Question. Given data on
X
and
Y
, can we identify
,
B
and
?
3.1
Estimation of
and
Begin by making the standard assumptions about the reduced form
Y
=
X
+
V
:
plim
(
1
T
X
0
X
)=
Q
plim
(
1
T
X
0
V
)=0
plim
(
1
T
V
0
V
)=
:
These assumptions imply that the equation-by-equation OLS estimates of
and
will be consistent.
3.2
Relationship Between (
;
) and (
;B;
)
With these estimates (
^
and
^
) in hand, the question is whether we can map back to
,
B
and
?
We
know the following
1.
=
B
1
and
2.
=
1
0
1
.
To see if identication is possible, we can count the number of known elements on the left-hand side and
compare with the number of unknown elements on the right-hand side.
Number
of
Known
Elements
KM
elements in
3
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