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Economics 326
Methods of Empirical Research in Economics
Lecture 21: Multiple linear IV model and twostage
least squares (2SLS)
Vadim Marmer
University of British Columbia
May 5, 2010
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View Full Document Multiple linear IV model
I
In empirical research, we often have to estimate models that
include multiple endogenous and exogenous regressors.
I
Example:
ln
Wage
i
=
γ
0
+
γ
1
Age
i
+
γ
2
Sex
i
+
β
1
Educ
i
+
β
2
Children
i
+
U
i
.
I
Exogenous regressors: age, sex, and a constant.
I
Endogenous regressors: education and children (family size).
1/11
Multiple linear IV model
I
Consider the following model:
y
i
=
γ
0
+
γ
1
X
1
,
i
+
. . .
+
γ
k
X
k
,
i
+
β
1
Y
1
,
i
+
. . .
+
β
m
Y
m
,
i
+
U
i
,
where
I
y
i
is the dependent variable.
I
γ
0
is the coe¢ cient on the constant regressor:
EU
i
=
0
.
I
X
1
,
i
,
. . .
,
X
k
,
i
are the
k
exogenous regressors:
Cov
(
X
1
,
i
,
U
i
) =
. . .
=
Cov
(
X
k
,
i
,
U
i
) =
0
.
I
Y
1
,
i
,
. . .
,
Y
m
,
i
are the
m
endogenous regressors:
Cov
(
Y
1
,
i
,
U
i
)
6
=
0
,
. . .
,
Cov
(
Y
k
,
i
,
U
i
)
6
=
0
.
2/11
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View Full Document
I
There are
k
+
1
+
m
unknown coe¢ cients
y
i
=
γ
0
+
γ
1
X
1
,
i
+
. . .
+
γ
k
X
k
,
i
+
β
1
Y
1
,
i
+
. . .
+
β
m
Y
m
,
i
+
U
i
.
I
The exogeneity conditions
EU
i
=
0 and
Cov
(
X
1
,
i
,
U
i
) =
. . .
=
Cov
(
X
k
,
i
,
U
i
) =
0 give us only
k
+
1
equations:
E
[
y
i
γ
0
γ
1
X
1
,
i
. . .
γ
k
X
k
,
i
β
1
Y
1
,
i
. . .
β
m
Y
m
,
i
] =
0
,
E
[
X
1
,
i
(
y
i
γ
0
γ
1
X
1
,
i
. . .
γ
k
X
k
,
i
β
1
Y
1
,
i
. . .
β
m
Y
m
,
i
)] =
0
,
. . .
E
[
X
k
,
i
(
y
i
γ
0
γ
1
X
1
,
i
. . .
γ
k
X
k
,
i
β
1
Y
1
,
i
. . .
β
m
Y
m
,
i
)] =
0
.
I
There are more unknowns than equations. Thus, the
knowledge of the true covariances between
X
±s,
Y
±s and
y
is
not su¢ cient to recover the unknown coe¢ cients
γ
0
,
γ
1
,
. . .
,
γ
k
,
β
1
,
. . .
,
β
m
.
I
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This note was uploaded on 03/24/2012 for the course ECON 326 taught by Professor Whisler during the Spring '10 term at The University of British Columbia.
 Spring '10
 whisler
 Economics

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