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

TABLE 8.5
ANOVA Table for
Regression
Equation (8.4.14)
Source of Variation
SS
df
MSS
ESS (due to PGNP)
60,449.5
1
60,449.5
RSS
303,228.5
62
4890.7822
Total
363,678
63
244
Part One
Single-Equation Regression Models
ESS. But how does one decide whether an 
X
variable significantly reduces RSS? The analy-
sis of variance technique can be easily extended to answer this question.
Suppose we first regress child mortality on PGNP and obtain the following regression:
CM
i
=
157.4244
−
0.0114 PGNP
(8.4.14)
t
=
(15.9894)
(
−
3.5156)
r
2
=
0.1662
p
value
=
(0.0000)
(0.0008)
adj
r
2
=
0.1528
As these results show, PGNP has a significant effect on CM. The ANOVA table corre-
sponding to the preceding regression is given in Table 8.5.
Assuming the disturbances 
u
i
are normally distributed and the hypothesis that PGNP
has no effect on CM, we obtain the 
F
value of
F
=
60,449
.
5
4890
.
7822
=
12
.
3598
(8.4.15)
which follows the
F
distribution with 1 and 62 df. This
F
value is highly significant, as the
computed
p
value is 0.0008. Thus, as before, we reject the hypothesis that PGNP has no
effect on CM. Incidentally, note that
t
2
=
(
−
3.5156)
2
=
12.3594, which is approximately
the same as the
F
value of Eq. (8.4.15), where the
t
value is obtained from Eq. (8.4.14). But
this should not be surprising in view of the fact that the square of the
t
statistic with
n
df is
equal to the
F
value with 1 df in the numerator and
n
df in the denominator, a relationship first
established in Chapter 5. Note that in the present example,
n
=
64.
Having run the regression (8.4.14), let us suppose we decide to add FLR to the model
and obtain the multiple regression (8.1.4). The questions we want to answer are:
1. What is the marginal, or incremental, contribution of FLR, knowing that PGNP is
already in the model and that it is significantly related to CM?
2. Is the incremental contribution of FLR statistically significant?
3. What is the criterion for adding variables to the model?
The preceding questions can be answered by the ANOVA technique. To see this, let us con-
struct Table 8.6. In this table 
X
2
refers to PGNP and 
X
3
refers to FLR.
To assess the 
incremental
contribution of 
X
3
after allowing for the contribution of 
X
2
, we
form
=
Q
2
/
1
Q
4
/
61
for our example