The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

ANOVA table.
Given the entries
of Table 5.3, we now consider the following variable:
F
=
MSS of ESS
MSS of RSS
=
ˆ
β
2
2
x
2
i
ˆ
u
2
i
(
n
−
2)
(5.9.1)
=
ˆ
β
2
2
x
2
i
ˆ
σ
2
If we assume that the disturbances 
u
i
are normally distributed, which we do under the
CNLRM, and if the null hypothesis (
H
0
) is that 
β
2
=
0, then it can be shown that the 
F
vari-
able of Equation 5.9.1 follows the 
F
distribution with 1 df in the numerator and (
n
−
2) df
in the denominator. (See Appendix 5A, Section 5A.3, for the proof. The general properties
of the 
F
distribution are discussed in 
Appendix A.
)
What use can be made of the preceding 
F
ratio? It can be shown
18
that
E
ˆ
β
2
2
x
2
i
=
σ
2
+
β
2
2
x
2
i
(5.9.2)
and
E
ˆ
u
2
i
n
−
2
=
E
(
ˆ
σ
2
)
=
σ
2
(5.9.3)
(Note that 
β
2
and 
σ
2
appearing on the right sides of these equations are the true parame-
ters.) Therefore, if 
β
2
is in fact zero, Equations 5.9.2 and 5.9.3 both provide us with identi-
cal estimates of true 
σ
2
. In this situation, the explanatory variable 
X
has no linear influence
on 
Y
whatsoever and the entire variation in 
Y
is explained by the random disturbances 
u
i
.
If, on the other hand, 
β
2
is not zero, Eqs. (5.9.2) and (5.9.3) will be different and part of the
variation in
Y
will be ascribable to 
X
. Therefore, the 
F
ratio of Eq. (5.9.1) provides a test of
the null hypothesis 
H
0
:
β
2
=
0. Since all the quantities entering into this equation can be
obtained from the available sample, this 
F
ratio provides a test statistic to test the null
hypothesis that true 
β
2
is zero. All that needs to be done is to compute the 
F
ratio and
compare it with the critical 
F
value obtained from the 
F
tables at the chosen level of
significance, or obtain the