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42
CHAPTER 8
SOLUTIONS TO PROBLEMS
8.1
Parts (ii) and (iii).
The homoskedasticity assumption played no role in Chapter 5 in showing
that OLS is consistent.
But we know that heteroskedasticity causes statistical inference based on
the usual
t
and
F
statistics to be invalid, even in large samples.
As heteroskedasticity is a
violation of the Gauss-Markov assumptions, OLS is no longer BLUE.
8.3
False.
The unbiasedness of WLS and OLS hinges crucially on Assumption MLR.4, and, as
we know from Chapter 4, this assumption is often violated when an important variable is
omitted.
When MLR.4 does not hold, both WLS and OLS are biased.
Without specific
information on how the omitted variable is correlated with the included explanatory variables, it
is not possible to determine which estimator has a small bias.
It is possible that WLS would
have more bias than OLS or less bias.
Because we cannot know, we should not claim to use
WLS in order to solve “biases” associated with OLS.
8.5
(i) No.
For each coefficient, the usual standard errors and the heteroskedasticity-robust ones
are practically very similar.
(ii) The effect is
−
.029(4) =
−
.116, so the probability of smoking falls by about .116.
(iii) As usual, we compute the turning point in the quadratic:
.020/[2(.00026)]
≈
38.46, so
about 38 and one-half years.
(iv) Holding other factors in the equation fixed, a person in a state with restaurant smoking
restrictions has a .101 lower chance of smoking.
This is similar to the effect of having four more
years of education.
(v) We just plug the values of the independent variables into the OLS regression line:
2
ˆ
.656 .069 log(67.44) .012 log(6,500) .029(16) .020(77) .00026(77 )
.0052.
smokes
=−
⋅
+
⋅
−
+
−
≈
Thus, the estimated probability of smoking for this person is close to zero.
(In fact, this person is
not a smoker, so the equation predicts well for this particular observation.)
8.7
(i) This follows from the simple fact that, for uncorrelated random variables, the variance of
the sum is the sum of the variances:
22
,,
Var(
)
Var( )
Var(
)
ii
e
i
i
e
f
v
fv
f
v
σ
+
=+
=
+
.
(ii) We compute the covariance between any two of the composite errors as
,
,
,
,
2
Cov(
,
)
Cov(
,
)
Cov( ,
) Cov( ,
) Cov(
,
) Cov(
,
)
Var( ) 0 0 0
,
i
ei
g
i i
ei i
g
g
i
i
g
if
uu
fvfv
f
f
f
v
vf
vv
f
+
=
+
+
+
=
+++=
where we use the fact that the covariance of a random variable with itself is its variance and the
assumptions that
,
, and
e
i
g
f
are pairwise uncorrelated.