Economics 245A
Instruments
Linear Models
Consider the scalar regression model in deviationfrommeans form
Y
t
=
t
+
U
t
:
One can always consider the problem of endogeneity (
Cov
(
X
t
;U
t
)
6
= 0
) to be one
of a mismeasured regressor. If we only had the correct measure of the regressor,
then OLS could be used. It may be helpful to think of the measurement error as
falling into two cases.
Case 1  The mismeasurement in
X
record the correct value at each observation, without any systematic tendency to
misrecord the value based on the value of
Y
or of other regressors. For case 1,
statistical assumptions can restore identi±cation.
components are correlated with observed components. In such cases, instruments
are needed to restore identi±cation.
Statistical Measurement Error
As discussed above, the classic treatment of measurement error allows statis
tical assumptions to restore identi±cation. Consider ±rst
Y
t
=
t
+
U
t
;
in which
X
t
is measured without error. We have
C
(
X
t
;Y
t
) =
(
X
t
)
:
Both
C
(
X
t
;Y
t
)
and
V
(
X
t
)
can be consistently estimated from their sample mo
ments, hence the methodofmoments (OLS) estimator is
B
=
P
X
t
Y
t
P
X
2
t
:
Now suppose
X
t
is measured with error
X
t
=
X
t
+
W
t
:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentWe assume that
W
t
is not correlated with
U
t
nor with
X
t
. The observed regression
model is
Y
t
=
t
+ (
U
t
t
)
:
We have
C
(
X
t
;Y
t
) =
(
X
t
)
(
W
t
)
:
Because the variance of the measurement error cannot be consistently estimated
from the observed data, the OLS estimator is not consistent
C
(
X
t
;Y
t
)
V
(
X
t
)
=
1
V
(
W
t
)
V
(
X
t
)
±
:
In essence, the parameter
error.
If we assume
that the measurement error is symmetric around zero, then we can construct a
consistent MoM estimator. Note
E
²
X
2
t
Y
t
³
=
²
X
3
t
³
²
W
3
t
³
:
Hence the assumption that
E
(
W
3
t
) = 0
implies
=
E
(
X
2
t
Y
t
)
E
(
X
3
t
)
.
The resulting consistent MoM estimator is
~
B
=
P
X
2
t
Y
t
P
X
3
t
:
For the more common case, in which the regressor is considered endogenous,
the simple assumption of symmetric measurement error will not work (and may
not make sense). In essence, the ±measurement error"
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Economics, Econometrics, Normal Distribution, Regression Analysis, Xt, Generalized method of moments

Click to edit the document details