CHAPTER
6
SOLUTIONS TO PROBLEMS
6.1
The generality is not necessary.
The
t
statistic on
roe
2
is only about
−
.30, which shows that
roe
2
is very statistically insignificant.
Plus, having the squared term has only a minor effect on
the slope even for large values of
roe
.
(The approximate slope is .0215
−
.00016
roe
, and even
when
roe
= 25 – about one standard deviation above the average
roe
in the sample – the slope is
.211, as compared with .215 at
roe
= 0.)
6.3
(i) The turnaround point is given by
1
ˆ
β
/(2
2
ˆ
), or .0003/(.000000014)
≈
21,428.57;
remember, this is sales in millions of dollars.
(ii) Probably.
Its
t
statistic is about –1.89, which is significant against the onesided
alternative H
0
:
1
< 0 at the 5% level (
cv
≈
–1.70 with
df
= 29).
In fact, the
p
value is about
.036.
(iii) Because
sales
gets divided by 1,000 to obtain
salesbil
, the corresponding coefficient gets
multiplied by 1,000:
(1,000)(.00030) = .30.
The standard error gets multiplied by the same
factor.
As stated in the hint,
salesbil
2
=
sales
/1,000,000, and so the coefficient on the quadratic
gets multiplied by one million:
(1,000,000)(.0000000070) = .0070; its standard error also gets
multiplied by one million.
Nothing happens to the intercept (because
rdintens
has not been
rescaled) or to the
R
2
:
n
rdintens
=
2.613
+ .30
salesbil
– .0070
salesbil
2
(0.429)
(.14)
(.0037)
n
= 32,
R
2
= .1484.
(iv) The equation in part (iii) is easier to read because it contains fewer zeros to the right of
the decimal.
Of course the interpretation of the two equations is identical once the different
scales are accounted for.
6.5
This would make little sense.
Performances on math and science exams are measures of
outputs of the educational process, and we would like to know how various educational inputs
and school characteristics affect math and science scores.
For example, if the stafftopupil ratio
has an effect on both exam scores, why would we want to hold performance on the science test
fixed while studying the effects of
staff
on the math pass rate?
This would be an example of
controlling for too many factors in a regression equation.
The variable
scill
could be a dependent
variable in an identical regression equation.
6.7
The second equation is clearly preferred, as its adjusted
R
squared is notably larger than that
in the other two equations.
The second equation contains the same number of estimated
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View Full Documentparameters as the first, and the one fewer than the third.
The second equation is also easier to
interpret than the third.
6.9
(i) Because
ˆ
exp( 1.96 )
1
σ
−<
2
ˆ
/ 2)
exp(1.96
and
, the point prediction is always above the
lower bound. The only issue is whether the point prediction is below the upper bound. This is the
case when
2
ˆ
exp(
/ 2)
1
>
ˆ
exp(
)
≤
or, taking logs,
2
ˆˆ
/ 2 1.96
≤
, or
ˆ
2(1.96)
3.92
≤
=
ˆ
3.92
.
Therefore, the point prediction is in the approximate 95% prediction interval for
≤
.
Because ˆ
is the estimated standard deviation in the regression with log(
y
) as the dependent
variable, 3.92 is a very large value for the estimated standard deviation of the error, which is on
the order of 400 percent. Most of the time, the estimated SER is well below that.
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 Spring '08
 BREMAN
 Statistics, Econometrics, Regression Analysis

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