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
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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 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 positive-denite matrix
plim
(
1
T
W
0
i
i
)=0
.
Since plim
(
Z
0
i
i
)
6
=0
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 variance-covariance 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
identied equation where
W
i
=
X
.
4.4
Two-Stage Least Squares (2SLS)
When an equation in the system is overidentied (i.e.,
rows
(
X
0
Z
i
)
>
cols
(
X
0
Z
i
)
), a convenient and intuitive
IV estimator is the
two-stage
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 variance-covariance 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
Limited-Information Maximum Likelihood (LIML)
Limited-information
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 identied,
^
ILS
i
=
^
IV
i
=
^
2
SLS
i
=
^
LIML
i
.
5
Full-Information Estimation
The ve estimators mentioned above are not fully e¢ cient because they ignore cross-equation relationships
between error terms and any omitted endogenous variables.
We consider two fully e¢ cient estimators below.
5.1
Three-Stage 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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