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71
CHAPTER 13
SOLUTIONS TO PROBLEMS
13.1
Without changes in the averages of
any
explanatory variables, the average fertility rate fell
by .545 between 1972 and 1984; this is simply the coefficient on
y84
.
To account for the
increase in average education levels, we obtain an additional effect:
–.128(13.3 – 12.2)
≈
–.141.
So the drop in average fertility if the average education level increased by 1.1 is .545 + .141 =
.686, or roughly two-thirds of a child per woman.
13.3
We do not have repeated observations on the
same
cross-sectional units in each time period,
and so it makes no sense to look for pairs to difference.
For example, in Example 13.1, it is very
unlikely that the same woman appears in more than one year, as new random samples are
obtained in each year.
In Example 13.3, some houses may appear in the sample for both 1978
and 1981, but the overlap is usually too small to do a true panel data analysis.
13.5
No, we cannot include age as an explanatory variable in the original model.
Each person in
the panel data set is exactly two years older on January 31, 1992 than on January 31, 1990.
This
means that
∆
age
i
= 2 for all
i
.
But the equation we would estimate is of the form
Δ
saving
i
=
δ
0
+
β
1
Δ
age
i
+
…
,
where
δ
0
is the coefficient the year dummy for 1992 in the original model.
As we know, when
we have an intercept in the model we cannot include an explanatory variable that is constant
across i; this violates Assumption MLR.3.
Intuitively, since age changes by the same amount for
everyone, we cannot distinguish the effect of age from the aggregate time effect.
13.7
(i) It is not surprising that the coefficient on the interaction term changes little when
afchnge
is dropped from the equation because the coefficient on
afchnge
in (3.12) is only .0077
(and its
t
statistic is very small).
The increase from .191 to .198 is easily explained by sampling
error.
(ii) If
highearn
is dropped from the equation [so that
1
0
β
=
in (3.10)], then we are assuming
that, prior to the change in policy, there is no difference in average duration between high earners
and low earners.
But the very large (.256), highly statistically significant estimate on
highearn
in
(3.12) shows this presumption to be false.
Prior to the policy change, the high earning group
spent about 29.2% [
exp(.256)
1
.292
−
≈
] longer on unemployment compensation than the low
earning group. By dropping
highearn
from the regression, we attribute to the policy change the
difference between the two groups that would be observed without any intervention.

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