University of California
D. Steigerwald
Department of Economics
Economics 241B
Exercise 2
1.
Consider the bivariate regression model
,
t
t
t
U
X
Y
in which the regressor is exogenous and all observations are measured as deviations from
means.
You wish to test a hypothesis about the value of
β
with the OLS estimator
B
.
Notation: There are
n
observations and
n
I
is the identity matrix of dimension
n
.
a.
A colleague tells you: “Always form the null hypothesis in such a way that
rejecting the null hypothesis is of interest.”
Do you think this advice is sensible?
b.
To test hypotheses about
β
, the quantity of interest is the statistic
.
/
2
t
t
t
X
U
X
B
If we assume
n
I
N
X
U
2
,
0
~

, then what is the distribution of
(B
β
)X
?
Does
the distribution depend on
X
?
Is the distribution exact, or does it hold only
asymptotically?
c.
If the variance of
(B
β
)X
depends on parameters, construct a statistic (a function
of
B
β
) whose variance does not depend on parameters (and so is a pivotal
statistic).
For the pivotal statistic you construct, give both the distribution
conditional on
X
and the unconditional distribution.
If
σ
²
is unknown, we must
use an estimator of
σ
²
to form the statistic.
Call the resulting statistic the
estimated statistic.
Does the distribution of the estimated statistic depend on
X
?
For the estimated statistic, give both the conditional (on
X
) and unconditional
distributions.
d.
The DurbinWatson statistic is
.
ˆ
ˆ
ˆ
1
2
2
2
1
n
t
t
n
t
t
t
U
U
U
DW
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View Full DocumentFrom your results in the preceding parts, what can you infer about the
distribution, conditional on
X
, of
DW
?
If the distribution of
DW
conditional on
X
depends on
X
, how does this affect the resultant critical values from the
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 Fall '08
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 Economics, Normal Distribution, consultant, estimated standard errors

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