observations. Let
c
l
t
.025
se(
ˆ
e
0
) and
c
u
t
.025
se(
ˆ
e
0
) be the lower and upper
bounds of the prediction interval for
logy
0
. That is, P(
c
l
logy
0
c
u
)
.95. Because the expo-
nential function is strictly increasing, it is also true that P[exp(
c
l
)
exp(
logy
0
)
exp(
c
u
)]
.95,
that is, P[exp(
c
l
)
y
0
exp(
c
u
)]
.95. Therefore, we can take exp(
c
l
) and exp(
c
u
) as the lower
and upper bounds, respectively, for a 95% prediction interval for
y
0
. For large
n
,
t
.025
1.96, and
so a 95% prediction interval for
y
0
is exp[−1.96 se(
ê
0
)] exp(
ˆ
0
x
0
ˆ
)
to exp[−1.96 se(
ê
0
)]
exp(
ˆ
0
x
0
ˆ
), where
x
0
ˆ
is shorthand for
ˆ
1
x
0
1
…
ˆ
k
x
0
k
. Remember, the
ˆ
j
and se(
ê
0
) are
obtained from the regression with log(
y
) as the dependent variable. Because we assume normality
of
u
in (6.38), we probably would use (6.40) to obtain a point prediction for
y
0
. Unlike in equa-
tion (6.37), this point prediction will not lie halfway between the upper and lower bounds exp(
c
l
)
and exp(
c
u
). One can obtain different 95% prediction intervalues by choosing different quantiles
in the
t
n
−
k
−1
distribution. If
q
1
and
q
2
are quantiles with
2
1
.95, then we can choose
c
l
q
1
se(
ê
0
) and
c
u
q
2
se(
ê
0
).
As an example, consider the CEO salary regression, where we make the prediction at the
same values of
sales
,
mktval
, and
ceoten
as in Example 6.7. The standard error of the regression
for (6.43) is about .505, and the standard error of
1
logy
0
is about .075. Therefore, using equation
(6.36), se(
ê
0
)
.511; as in the GPA example, the error variance swamps the estimation error in
the parameters, even though here the sample size is only 177. A 95% prediction interval for
salary
0
is exp[−1.96 (.511)] exp(7.013) to exp[1.96 (.511)] exp(7.013), or about 408.071 to 3,024.678,
that is, $408,071 to $3,024,678. This very wide 95% prediction interval for CEO salary at the given
sales, market value, and tenure values shows that there is much else that we have not included in
the regression that determines salary. Incidentally, the point prediction for salary, using (6.40), is
about $1,262,075—higher than the predictions using the other estimates of
0
and closer to the lower
bound than the upper bound of the 95% prediction interval.

Chapter 6
Multiple Regression Analysis: Further Issues
215
S U M M A R Y
In this chapter, we have covered some important multiple regression analysis topics.
Section 6.1 showed that a change in the units of measurement of an independent variable
changes the OLS coefficient in the expected manner: if
x
j
is multiplied by
c
, its coefficient is
divided by
c
. If the dependent variable is multiplied by
c
,
all
OLS coefficients are multiplied by
c
. Neither
t
nor
F
statistics are affected by changing the units of measurement of any variables.
We discussed beta coefficients, which measure the effects of the independent variables on
the dependent variable in standard deviation units. The beta coefficients are obtained from a
standard OLS regression after the dependent and independent variables have been transformed
into
z
-scores.

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