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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 6
need to nd the rank of
R
d
=
0
0
0
1
0
2
6
6
6
6
6
6
6
6
6
6
4
1
1
1
1
0
0
2
0
3
3
3
7
7
7
7
7
7
7
7
7
7
5
=
2
0
:
Clearly,
Rank
(
R
d
) = 1 =
M
1
, so that the demand equation is exactly identied.
4
Limited-Information Estimation
We will consider ve di/erent limited-information estimation techniques OLS, indirect least squares (ILS),
instrumental variable (IV) estimation, two-stage least squares (2SLS) and limited-information maximum
likelihood (LIML).
The term limited information refers to equation-by-equation estimation, as opposed to
full-information estimation which uses the linkages among the di/erent equations.
Begin by writing the
i
th
equation as
Y
i
+
XB
i
=
i
y
i
=
Y
i
i
+
Y
i
i
+
X
i
i
+
X
i
i
+
i
where
Y
i
represents the vector of endogenous variables (other than
y
i
) in the
i
th
equation,
Y
i
represents the
vector of endogenous variables excluded from the
i
th
equation, and similarly for
X
.
Therefore,
i
=0
and
i
=0
so that
y
i
=
Y
i
i
+
X
i
i
+
i
=
Y
i
X
i
2
6
4
i
i
3
7
5
+
i
=
Z
i
i
+
i
:
4.1
Ordinary Least Squares (OLS)
The OLS estimator of
i
is
^
OLS
i
=(
Z
0
i
Z
i
)
1
(
Z
0
i
y
i
)
:
The expected value of
^
i
is
E
(
^
OLS
i
)=
i
+
E
[(
Z
0
i
Z
i
)
1
Z
0
i
i
]
:
6
Page 7
However, since
y
i
and
Y
i
are jointly determined (recall
Z
i
contains
Y
i
), we cannot expect that
E
(
Z
0
i
i
)=0
or plim
(
Z
0
i
i
)=0
.
Therefore, OLS estimates will be biased and inconsistent.
This is commonly known as
simultaneity
or
endogeneity
bias.
4.2
Indirect Least Squares (ILS)
The
indirect
least
squares estimator simply uses the consistent reduced-form estimates (
^
and
^
) and the
relations
=
B
1
and
=
1
0
1
to solve for
,
B
and
.
The ILS estimator is only feasible if the
system is exactly identied.
To see this, consider the
i
th
equation as given in (2)
i
=
B
i
where
^
=(
X
0
X
)
1
X
0
Y
.
Substitution gives
(
X
0
X
)
1
X
0
y
i
Y
i
2
6
4
1
^
i
3
7
5
=
2
6
4
^
i
0
3
7
5
:
Multiplying through by
(
X
0
X
)
gives
X
0
y
i
+
X
0
Y
i
^
i
=
X
0
X
2
6
4
^
i
0
3
7
5
)
X
0
y
i
=
X
0
Y
i
^
i
+
X
0
X
i
^
i
=
X
0
Z
i
^
i
:
Therefore, the ILS estimator for the
i
th
equation can be written as
^
ILS
i
=(
X
0
Z
i
)
1
(
X
0
y
i
)
:
There are three cases:
1. If the
i
th
equation is exactly identied, then
X
0
Z
i
is square and invertible.
2. If the
i
th
equation is underidentied, then
X
0
Z
i
is not square.
3. If the
i
th
equation is overidentied, then
X
0
Z
i
is not square although a subset could be used to obtain
consistent albeit ine¢ cient estimates of
i
.
4.3
Instrumental Variable (IV) Estimation
Let
W
i
be an instrument matrix (dimension
T
(
K
i
+
M
i
)
) satisfying
7
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