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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 8
plim
(
1
T
W
0
i
Z
i
)=
wz
, a nite invertible matrix
plim
(
1
T
W
0
i
W
i
)=
ww
, a nite positivedenite matrix
plim
(
1
T
W
0
i
i
)=0
.
Since plim
(
Z
0
i
i
)
6
=0
, we can instead examine
plim
(
1
T
W
0
i
y
i
)
=
plim
(
1
T
W
0
i
Z
i
)
+
plim
(
1
T
W
0
i
i
)
=
plim
(
1
T
W
0
i
Z
i
)
.
Naturally, the
instrumental
variable estimator is
^
IV
i
=(
W
0
i
Z
i
)
1
(
W
0
i
y
i
)
which is consistent and has asymptotic variancecovariance matrix
asy:var:
(
^
IV
i
)=
ii
T
[
1
wz
ww
1
zw
]
:
This can be estimated by
est:asy:var:
(
^
IV
i
)=^
ii
(
W
0
i
Z
i
)
1
W
0
i
W
i
(
Z
0
i
W
i
)
1
and
^
ii
=
1
T
(
y
i
Z
i
^
IV
i
)
0
(
y
i
Z
i
^
IV
i
)
:
A degrees of freedom correction is optional.
Notice that ILS is a special case of IV estimation for an exactly
identied equation where
W
i
=
X
.
4.4
TwoStage Least Squares (2SLS)
When an equation in the system is overidentied (i.e.,
rows
(
X
0
Z
i
)
>
cols
(
X
0
Z
i
)
), a convenient and intuitive
IV estimator is the
twostage
least
squares estimator.
The 2SLS estimator works as follows:
Stage
#1. Regress
Y
i
on
X
and form
^
Y
i
=
X
^
OLS
.
Stage
#2. Estimate
i
by an OLS regression of
y
i
on
^
Y
i
and
X
i
.
More formally, let
^
Z
i
=(
^
Y
i
;X
i
)
.
The 2SLS estimator is given by
^
2
SLS
i
=(
^
Z
0
i
^
Z
i
)
1
(
^
Z
0
i
y
i
)
8
Page 9
where the asymptotic variancecovariance matrix for
^
2
SLS
i
can be estimated consistently by
est:asy:var
(
^
2
SLS
i
)=^
ii
(
^
Z
0
i
^
Z
i
)
1
and
^
ii
=
1
T
(
y
i
Z
i
^
2
SLS
i
)
0
(
y
i
Z
i
^
2
SLS
i
)
.
4.5
LimitedInformation Maximum Likelihood (LIML)
Limitedinformation
maximum
likelihood estimation refers to ML estimation of a single equation in the
system.
For example, if we assume normally distributed errors, then we can form the joint probability
distribution function of
(
y
i
;Y
i
)
and maximize it by choosing
i
and the appropriate elements of
.
Since
the LIML estimator is more complex but asymptotically equivalent to the 2SLS estimator, it is not widely
used.
Note.
When the
i
th
equation is exactly identied,
^
ILS
i
=
^
IV
i
=
^
2
SLS
i
=
^
LIML
i
.
5
FullInformation Estimation
The ve estimators mentioned above are not fully e¢ cient because they ignore crossequation relationships
between error terms and any omitted endogenous variables.
We consider two fully e¢ cient estimators below.
5.1
ThreeStage Least Squares (3SLS)
Begin by writing the system (1) as
2
6
6
6
6
6
6
6
4
y
1
y
2
.
.
.
y
M
3
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
4
Z
1
0
0
0
Z
2
0
.
.
.
.
.
.
.
.
.
0
0
Z
M
3
7
7
7
7
7
7
7
5
2
6
6
6
6
6
6
6
4
1
2
.
.
.
M
3
7
7
7
7
7
7
7
5
+
2
6
6
6
6
6
6
6
4
1
2
.
.
.
M
3
7
7
7
7
7
7
7
5
)
y
=
Z
+
where
=
~
E
and
E
(
0
)=
I
T
.
Then, applying the principle from SUR estimation, the fully e¢ cient estimator is
^
=(
W
0
(
1
I
)
Z
)
1
(
W
0
(
1
I
)
y
)
9
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