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17
CHAPTER 4
SOLUTIONS TO PROBLEMS
4.2
(i) and (iii) generally cause the
t
statistics not to have a
t
distribution under H
0
.
Homoskedasticity is one of the CLM assumptions.
An important omitted variable violates
Assumption MLR.3.
The CLM assumptions contain no mention of the sample correlations
among independent variables, except to rule out the case where the correlation is one.
4.3
(i) While the standard error on
hrsemp
has not changed, the magnitude of the coefficient has
increased by half.
The
t
statistic on
hrsemp
has gone from about –1.47 to –2.21, so now the
coefficient is statistically less than zero at the 5% level.
(From Table G.2 the 5% critical value
with 40
df
is –1.684.
The 1% critical value is –2.423, so the
p
value is between .01 and .05.)
(ii) If we add and subtract
2
β
log(
employ
) from the righthandside and collect terms, we
have
log(
scrap
) =
0
+
1
hrsemp +
[
2
log(sales) –
2
log(
employ
)]
+ [
2
log(
employ
) +
3
log(
employ
)] +
u
=
0
+
1
hrsemp
+
2
log(
sales
/
employ
)
+ (
2
+
3
)log(
employ
) +
u
,
where the second equality follows from the fact that log(
sales
/
employ
) = log(
sales
) –
log(
employ
).
Defining
3
θ
≡
2
+
3
gives the result.
(iii) No.
We are interested in the coefficient on log(
employ
), which has a
t
statistic of .2,
which is very small.
Therefore, we conclude that the size of the firm, as measured by
employees, does not matter, once we control for training
and
sales per employee (in a
logarithmic functional form).
(iv) The null hypothesis in the model from part (ii) is H
0
:
2
= –1.
The
t
statistic is [–.951 –
(–1)]/.37 = (1 – .951)/.37
≈
.132; this is very small, and we fail to reject whether we specify a
one or twosided alternative.
4.4
(i) In columns (2) and (3), the coefficient on
profmarg
is actually negative, although its
t
statistic is only about –1.
It appears that, once firm sales and market value have been controlled
for, profit margin has no effect on CEO salary.
(ii) We use column (3), which controls for the most factors affecting salary.
The
t
statistic on
log(
mktval
) is about 2.05, which is just significant at the 5% level against a twosided alternative.
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(We can use the standard normal critical value, 1.96.)
So log(
mktval
) is statistically significant.
Because the coefficient is an elasticity, a ceteris paribus 10% increase in market value is
predicted to increase
salary
by 1%.
This is not a huge effect, but it is not negligible, either.
(iii) These variables are individually significant at low significance levels, with
t
ceoten
≈
3.11
and
t
comten
≈
–2.79.
Other factors fixed, another year as CEO with the company increases salary
by about 1.71%.
On the other hand, another year with the company, but not as CEO, lowers
salary by about .92%.
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 Spring '11
 Suzuki
 Statistics, Econometrics, Regression Analysis, Standard Deviation, Statistical hypothesis testing, Statistical significance

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