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65
CHAPTER 12
SOLUTIONS TO PROBLEMS
12.1
We can reason this from equation (12.4) because the usual OLS standard error is an
estimate of
/
x
SST
σ
.
When the dependent and independent variables are in level (or log) form,
the AR(1) parameter,
ρ
, tends to be positive in time series regression models.
Further, the
independent variables tend to be positive correlated, so (
x
t
−
x
)(
x
t
+
j
−
x
) – which is what
generally appears in (12.4) when the {
x
t
} do not have zero sample average – tends to be positive
for most
t
and
j
.
With multiple explanatory variables the formulas are more complicated but
have similar features.
If
< 0, or if the {
x
t
} is negatively autocorrelated, the second term in the last line of (12.4)
could be negative, in which case the true standard deviation of
1
ˆ
β
is actually less than
/
x
SST
.
12.3
(i) Because U.S. presidential elections occur only every four years, it seems reasonable to
think the unobserved shocks – that is, elements in
u
t
– in one election have pretty much
dissipated four years later.
This would imply that {
u
t
} is roughly serially uncorrelated.
(ii) The
t
statistic for H
0
:
= 0 is
−
.068/.240
≈
−
.28, which is very small.
Further, the
estimate ˆ
=
−
.068 is small in a practical sense, too.
There is no reason to worry about serial
correlation in this example.
(iii) Because the test based on
ˆ
t
is only justified asymptotically, we would generally be
concerned about using the usual critical values with
n
= 20 in the original regression.
But any
kind of adjustment, either to obtain valid standard errors for OLS as in Section 12.5 or a feasible
GLS procedure as in Section 12.3, relies on large sample sizes, too.
(Remember, FGLS is not
even unbiased, whereas OLS is under TS.1 through TS.3.)
Most importantly, the estimate of
is
practically
small, too.
With ˆ
so close to zero, FGLS or adjusting the standard errors would
yield similar results to OLS with the usual standard errors.
12.5
(i) There is substantial serial correlation in the errors of the equation, and the OLS standard
errors almost certainly underestimate the true standard deviation in
ˆ
EZ
.
This makes the usual
confidence interval for
EZ
and
t
statistics invalid.
(ii) We can use the method in Section 12.5 to obtain an approximately valid standard error.
[See equation (12.43).]
While we might use
g
= 2 in equation (12.42), with monthly data we
might want to try a somewhat longer lag, maybe even up to
g
= 12.