Economics 241B
Estimation with Instruments
Measurement Error
a variable. At some level, every variable is measured with error. For example, if
we measure a person±s height as 6 feet 2 inches, it is unlikely that they are exactly
6 feet 2 inches tall. It is simply that they are closer to 6 feet 2 inches than to
any other easily measured height. How does mismeasurement of variables a/ect
estimation of a linear regression model? If a variable is mismeasured, then the
resulting error is a component of the regression error.
Consider &rst the dependent variable. In our earliest discussion of the regres
sion error, we said that the best possible outcome is for the regression error to
arise from random mismeasurement of the dependent variable. Mismeasurement
of the dependent variable does not lead to correlation between the regressors and
the error and induces no bias in our OLS estimators. In fact, measurement error of
the dependent variable a/ects only the variance of the regression error. Of course,
we would like the error to have as small a variance as possible, so we would like
measurement error in the dependent variable to be small.
Mismeasurement of the independent variable is another matter. In general,
mismeasurement of the independent variable will lead to correlation between the
regressor and the error and so lead to bias in the OLS coe¢ cient estimators. As
all regressors may have some measurement error, we treat the problem only when
we feel the measurement error in the regressors is large enough relative to the
approximation error made in the model under study. For example, survey data is
often subject to such large errors that it is wise to treat the regressors as though
measured with error. Formally, let the population model (in deviationfrommeans
form) be
Y
t
=
t
+
U
t
:
The regressor is measured with error, so that we observe
X
t
, where
X
t
=
X
t
+
V
t
and
V
t
is measurement error. The estimated model is
Y
t
=
t
+ [
U
t
+
(
X
t
X
t
)]
=
t
+ [
U
t
t
]
;
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View Full Documentwhere the error is the term in
[
]
. Because
V
t
is a component of
X
t
, the regressor
and error are correlated in the estimated model.
For the OLS estimator
B
=
+
P
n
t
=1
X
t
[
U
t
±
t
]
P
n
t
=1
X
2
t
:
Because
X
t
is uncorrelated with
U
t
, we have
EB
=
1
±
E
±P
n
t
=1
X
t
V
t
P
n
t
=1
X
2
t
²³
:
It is di¢ cult to proceed further with exact results, as one cannot condition on
X
t
while examining
V
t
. Rather, we have as an approximation
EB
²
1
±
C
(
X
t
;V
t
)
V
(
X
t
)
³
;
where
V
(
X
t
) =
V
(
X
t
) +
V
(
V
t
)
and
C
(
X
t
;V
t
) =
V
(
V
t
)
. Thus
EB
²
V
(
X
t
)
V
(
X
t
) +
V
(
V
t
)
³
:
Measurement error in the regressor biases the estimator toward zero and the degree
of the bias depends upon the variance of the measurement error. The result is
intuitive, as the measurement error in the regressor becomes more pronounced, the
e/ect of the regressor is obscured and the estimated coe¢ cient is biased toward
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 Fall '08
 Staff
 Economics, Econometrics, Regression Analysis, Xt

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