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CHAPTER
8
MULTIPLE REGRESSION:
ESTIMATION AND HYPOTHESIS TESTING
QUESTIONS
8.1.
(
a
)
It measures the change in the
mean
value of the dependent
variable (
Y
)
for a unit change in the value of an explanatory variable (
X
), holding the
values of all other explanatory
variables constant.
Mathematically, it is the
partial derivative of (mean)
Y
with respect to the given explanatory variable.
(
b
)
It measures the proportion, or percentage, of the total variation in the
dependent variable,
∑

2
)
(
Y
Y
i
, explained by
all
the explanatory variables
included in the model.
(
c
)
Exact
linear
relationship among the explanatory variables.
(
d
)
More than one exact
linear
relationship among the explanatory variables.
(
e
)
Testing the hypothesis about a single (partial) regression coefficient.
(
f
)
Testing the hypothesis about two or more partial regression coefficients
simultaneously.
(
g
)
An
2
R
value that is adjusted for degrees of freedom.
8.2.
(
a
)
(1) State the null and alternative hypotheses.
(2) Choose the level of significance.
(3) Find the
t
value of the coefficient under the null hypothesis,
0
H
.
(4) Compare this 
t
 value with the critical value at the chosen level of
significance and the given d.f.
(5) If the computed
t
value exceeds the critical
t
value,
we reject the null
hypothesis.
Make sure that you use the appropriate onetailed or
twotailed test.
(
b
)
Here the null hypothesis is:
0
...
:
H
3
2
0
=
=
=
=
k
B
B
B
56
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View Full Documentthat is, all partial slopes are zero.
The alternative hypothesis is that this is
not so, that is, one or more partial slope coefficients are nonzero.
Here, we
use the ANOVA technique and the
F
test. If the computed
F
value under the
null hypothesis exceeds the critical
F
value at the chosen level of
significance, we reject the null hypothesis.
Otherwise, we do not reject it.
Make sure that the numerator and denominator d.f. are properly counted.
Note
: In both (
a
) and (
b
), instead of choosing the level of significance in
advance, obtain the
p
value of the estimated test statistic.
If it is reasonably
low, you can reject the null hypothesis.
8.3.
(
a
)
True
.
This is obvious from the formula relating the two
2
R
s.
(
b
)
False
. Use the
F
test.
(
c
)
False
.
When
2
R
= 1, the value of
F
is infinite.
But when it is zero, the
F
value is also zero.
(
d
)
True
, which can be seen from the normal and
t
distribution tables.
(
e
)
True
.
It can be shown that
32
3
2
12
)
(
b
B
B
b
E
+
=
, where
32
b
is the slope
coefficient in the regression of
3
X
on
2
X
. From this relationship, the
conclusion follows.
(
f
)
False
.
It is statistically different from zero, not 1.
(
g
)
False
.
We also need to know the level of significance.
(
h
)
False
.
By the overall significance we mean that all partial regression
coefficients are not simultaneously equal to zero, or that
2
R
is different
from zero.
(
i
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