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47
CHAPTER 9
SOLUTIONS TO PROBLEMS
9.1
There is functional form misspecification if
6
β
≠
0 or
7
≠
0, where these are the population
parameters on
ceoten
2
and
comten
2
, respectively.
Therefore, we test the joint significance of
these variables using the
R
-squared form of the
F
test:
F
= [(.375
−
.353)/(1
−
.375)][(177 –
8)/2]
≈
2.97.
With 2 and
∞
df
, the 10% critical value is 2.30 awhile the 5% critical value is 3.00.
Thus, the
p
-value is slightly above .05, which is reasonable evidence of functional form
misspecification.
(Of course, whether this has a practical impact on the estimated partial effects
for various levels of the explanatory variables is a different matter.)
9.3
(i) Eligibility for the federally funded school lunch program is very tightly linked to being
economically disadvantaged. Therefore, the percentage of students eligible for the lunch program
is very similar to the percentage of students living in poverty.
(ii) We can use our usual reasoning on omitting important variables from a regression
equation.
The variables log(
expend
) and
lnchprg
are negatively correlated:
school districts with
poorer children spend, on average, less on schools.
Further,
3
< 0.
From Table 3.2, omitting
lnchprg
(the proxy for
poverty
) from the regression produces an upward biased estimator of
1
[ignoring the presence of log(
enroll
) in the model].
So when we control for the poverty rate, the
effect of spending falls.
(iii) Once we control for
lnchprg
, the coefficient on log(
enroll
) becomes negative and has a
t
of about –2.17, which is significant at the 5% level against a two-sided alternative.
The
coefficient implies that
n
10
math
Δ
≈
−
(1.26/100)(%
Δ
enroll
) =
−
.0126(%
Δ
enroll
).
Therefore, a
10% increase in enrollment leads to a drop in
math10
of .126 percentage points.
(iv) Both
math10
and
lnchprg
are percentages.
Therefore, a ten percentage point increase in
lnchprg
leads to about a 3.23 percentage point fall in
math10
, a sizeable effect.
(v) In column (1) we are explaining very little of the variation in pass rates on the MEAP
math test:
less than 3%.
In column (2), we are explaining almost 19% (which still leaves much
variation unexplained).
Clearly most of the variation in
math10
is explained by variation in
lnchprg
.
This is a common finding in studies of school performance: family income (or related
factors, such as living in poverty) are much more important in explaining student performance
than are spending per student or other school characteristics.
9.5
The sample selection in this case is arguably endogenous.
Because prospective students may
look at campus crime as one factor in deciding where to attend college, colleges with high crime
rates have an incentive not to report crime statistics. If this is the case, then the chance of