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Economics 140A
Estimation with Instruments
In our previous lecture, we discussed the problems arising from measurement
error. In particular, we found that measurement error in a regressor leads to
correlation between a regressor and the error. Correlation between a regressor
and the error in turn leads to bias in the OLS coe¢ cient estimator.
If measurement error is present, how can we correct the problem and reduce
(or remove) the correlation between the regressor and the error? The answer is
through the use of an additional variable termed an instrument. An instrument
is a variable that theory does not suggest belongs in the regression, but that
is correlated with the problem regressor. Because theory does not suggest the
instrument as a regressor, the instrument is uncorrelated with the regression error.
Formally, consider the twovariable regression model in deviationfrommeans
form
Y
t
=
t
+
U
t
:
As we have seen, the OLS estimator is a methodofmoments estimator that uses
the population moment
E
(
X
t
U
t
) = 0
:
That is, the OLS estimator is the value
B
such that
n
X
t
=1
X
t
(
Y
t
BX
t
) = 0
:
If the regressor and the error are correlated, then
E
(
X
t
U
t
)
6
= 0
. As a result, the
OLS estimator does not equate the population moment and the sample analog
(because the sample analog is set to zero) and so is biased.
An improved estimator is obtained by again equating population moments
and their sample analogs. Although the regressor and the error are correlated,
the regressor and the instrument
Z
t
are uncorrelated
E
(
Z
t
U
t
) = 0
:
The instrumental variable estimator is the methodofmoments estimator
B
IV
for
which
n
X
t
=1
Z
t
(
Y
t
B
IV
X
t
) = 0
:
(0.1)
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View Full DocumentFrom (0.1) it follows that
B
IV
=
P
n
t
=1
Z
t
Y
t
P
n
t
=1
Z
t
X
t
;
for which
EB
IV
=
+
E
n
t
=1
Z
t
U
t
P
n
t
=1
Z
t
X
t
±
6
=
Despite the fact that
E
(
U
t
j
Z
t
) = 0
, which follows from
E
(
Z
t
U
t
) = 0
, the estima
tor is biased because we cannot condition on
X
t
when treating
U
t
as random. (If
X
t
and
U
t
to be random.) The bias of
B
IV
does diminish as the sample grows, so
B
IV
is a
consistent estimator of
, while the OLSE is inconsistent. The two requirements
for an instrument are clearly seen. The instrument must be uncorrelated with the
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 Fall '08
 Staff
 Economics

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