Econ 104 PS8 Solutions
Molin Zhong
December 17, 2011
1
Problem 1
1.1
Part A
The key conditions for the validity of the instrumental variable is that:
1.
cov
(
z, x
)
6
= 0
2.
cov
(
z, v
) = 0 where v is the error term
We know z is correlated with x.
Here,
v
=
β
2
w
+
u
. Because z and w are uncorrelated, z and v are uncorre
lated because z and u are uncorrelated by assumption. Hence, z is still a valid
instrument and the 2SLS procedure goes through.
1.2
Part B
If z and w are correlated, z is no longer uncorrelated with the error term v so
long as
β
2
6
= 0. Hence, it does not fulfill the exogeneity condition and z cannot
be used as an instrument. Therefore, the 2SLS estimator is no longer consistent.
This is because in the second step, the new regressor is a function of z, which
is correlated with the error term, so the OLS step will lead to an inconsistent
estimate.
2
Problem 2: 12.7
The test of overidentifying restrictions regresses
b
u
T SLS
i
=
δ
0
+
δ
1
z
1
i
+
δ
2
z
2
i
+
i
where
b
u
T SLS
i
is the residuals from the two stage least squares regression. The J
test statistic tests if these
δ
1
and
δ
2
are jointly equal to 0, which should be the
case if the exogeneity condition holds for both of the instrumental variables.
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 Spring '12
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 Regression Analysis, error term, Molin Zhong

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