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overall significance
of the ob-
served or estimated regression line, that is, whether 
Y
is linearly related to both 
X
2
and 
X
3
.
Can the joint hypothesis in Eq. (8.4.1) be tested by testing the significance of 
ˆ
β
2
and 
ˆ
β
3
individually as in Section 8.3? The answer is no, and the reasoning is as follows.
In testing the individual significance of an observed partial regression coefficient in
Section 8.3, we assumed implicitly that each test of significance was based on a different
(i.e., independent) sample. Thus, in testing the significance of 
ˆ
β
2
under the hypothesis that
β
2
=
0, it was assumed tacitly that the testing was based on a different sample from the one
used in testing the significance of 
ˆ
β
3
under the null hypothesis that 
β
3
=
0. But to test the joint
hypothesis of Eq. (8.4.1), if we use the same sample data, we shall be violating the
assumption underlying the test procedure.
4
The matter can be put differently: In Eq. (8.3.2)
FIGURE 8.2
Histogram of
residuals from
regression (8.1.4).
10
8
6
4
2
0
–80
–40
0
40
80
Series: Residuals
Sample 1 64
Observations 64
Mean –4.95 
x
10
–14
Median
0.709227
Maximum
96.80276
Minimum –84.26686
Std. dev. 
41.07980
Skewness
0.227575
Kurtosis
2.948855
Jarque–Bera
0.559405
Probability
0.756009
3
For our example, the skewness value is 0.2276 and the kurtosis value is 2.9488. Recall that for a
normally distributed variable the skewness and kurtosis values are, respectively, 0 and 3.
4
In any given sample the cov (
ˆ
β
2
,
ˆ
β
3
) may not be zero; that is, 
ˆ
β
2
and 
ˆ
β
3
may be correlated. See
Eq. (7.4.17).
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