University of California
D. Steigerwald
Department of Economics
Economics 140A
Exercise 7
1.
Consider the population model (in deviationfrommeans form)
,
*
t
t
t
U
X
Y
in which
*
t
X
is latent.
The observed regressor is
,
*
t
t
t
V
X
X
in which
t
V
is independent of
t
U
.
Further, we know that
t
V
and
t
U
are symmetrically
distributed about 0.
a.
Detail why, in the presence of measurement error,
β
is not identified.
b.
Let
med(X)
be the median of
n
X
X
,
,
1
and define
1
t
Z
if
X
med
X
t
1
t
Z
if
X
med
X
t
.
In certain texts,
t
Z
has been advanced as a valid instrument.
Is
t
Z
a valid
instrument?
c.
Can the assumption that
t
V
is symmetrically distributed about 0 restore
identification?
If so, construct the corresponding estimator of
β
.
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2.
Consider the simple model in deviationfrommeans form
(1)
Y
t
=
X
t
+ U
t
.
Inspection of the distribution of the residuals reveals that
V(U
t
) =
2
2
t
Z
, where
Z
t
is
known, so that you know the error variance up to a multiplicative constant.
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 Fall '08
 Staff
 Economics, Linear Regression, Regression Analysis, ols estimator

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