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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 4
1
2
M
(
M
+ 1)
elements in
Total =
M
(
K
+
1
2
(
M
+ 1))
:
Number
of
Unknown
Elements
M
2
elements in
1
2
M
(
M
+ 1)
elements in
B
=
KM
Total =
M
(
M
+
K
+
1
2
(
M
+ 1))
:
Therefore, we are
M
2
pieces of information shy of identifying the structural parameters.
In other words,
there is more than one set of structural parameters that are consistent with the reduced form.
We say the
model is
underidentied.
3.3
Identication Conditions
There are several possibilities for obtaining identication:
1. Normalization (i.e., set
ii
=
1
in
for
i
=1
; :::; m
).
2. Identities (e.g., national income accounting identity).
3. Exclusion restrictions (e.g., demand and supply shift factors).
4. Other linear (and nonlinear) restrictions (e.g., Blanchard-Quah long-run restriction).
3.3.1
Rank and Order Conditions
Begin by rewriting the
i
th
equation from
=
B
1
in matrix form as
I
K
2
6
4
i
B
i
3
7
5
=0
(2)
where
i
and
B
i
represent the
i
th
columns of
and
B
, respectively.
Since the rank of
[
I
K
]
equals
K
, (2)
represents a system of
K
equations in
M
+
K
1
unknowns (after normalization).
In achieving identication,
we will introduce linear restrictions as follows
R
i
2
6
4
i
B
i
3
7
5
=0
(3)
4
Page 5
where
Rank
(
R
i
)=
J
.
Putting equations (2) and (3) together and redening
i
= (
i
;B
i
)
0
gives
2
6
4
(
.
.
.
I
K
)
R
i
3
7
5
i
=0
:
From this discussion, it is clear that
R
i
must provide at least
M
1
new pieces of information.
Here are
the formal rank and order conditions.
1.
Order
Condition.
The order condition states that
Rank
(
R
i
)=
J
M
1
is a necessary but not
su¢ cient condition for identication.
A situation where the order condition is not su¢ cient is when
R
i
j
=0
.
More details on the order condition below.
2.
Rank
Condition.
The rank condition states that
Rank
(
R
i
) =
M
1
is a necessary and su¢ cient
condition for identication.
We can now summarize all possible identication outcomes.
Under
Identication.
If either
Rank
(
R
i
)
<M
1
or
Rank
(
R
i
)
<M
1
, the
i
th
equation is
underidentied.
Exact
Identication.
If
Rank
(
R
i
)=
M
1
and
Rank
(
R
i
) =
M
1
, the
i
th
equation is exactly
identied.
Over
Identication.
If
Rank
(
R
i
)
>M
1
and
Rank
(
R
i
) =
M
1
, the
i
th
equation is overidentied.
3.3.2
Identication Conditions in the Demand and Supply Example
Begin with supply and note that
M
=2
.
The order condition is simple.
Since all the variables are in
the supply equation, there is no restriction matrix
R
s
so that
Rank
(
R
s
)=0
<
1
.
The supply equation is
underidentied.
There is no need to look at the rank condition.
Next, consider demand.
The relevant matrix equations are
2
6
4
(
.
.
.
I
K
)
R
d
3
7
5
d
=
2
6
6
6
6
6
6
6
6
6
6
4
11
12
1
0
0
21
22
0
1
0
31
32
0
0
1
0
0
0
1
0
3
7
7
7
7
7
7
7
7
7
7
5
2
6
6
6
6
6
6
6
6
6
6
4
1
1
0
2
3
3
7
7
7
7
7
7
7
7
7
7
5
=0
;
for which the order condition is clearly satised (i.e.,
Rank
(
R
d
)=1=
M
1
).
For the rank condition, we
5
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