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

(8.4.16)
F
=
Q
2
/
df
Q
4
/
df
=
(ESS
new
−
ESS
old
)
/
number of new regressors
RSS
new
/
df (
=
n
−
number of parameters in the new model)
guj75772_ch08.qxd 12/08/2008 10:03 AM Page 244

---

Chapter 8
Multiple Regression Analysis: The Problem of Inference
245
where ESS
new
=
ESS under the new model (i.e., after adding the new regressors
=
Q
3
),
ESS
old
=
ESS under the old model (
=
Q
1
), and RSS
new
=
RSS under the new model (i.e.,
after taking into account all the regressors
=
Q
4
). For our illustrative example the results
are as shown in Table 8.7.
Now applying Eq. (8.4.16), we obtain:
F
=
196,912
.
9
1742
.
8786
=
112
.
9814
(8.4.17)
Under the usual assumptions, this 
F
value follows the 
F
distribution with 1 and 62 df. The
reader should check that this 
F
value is highly significant, suggesting that the addition of
FLR to the model significantly increases ESS and hence the 
R
2
value. Therefore, FLR
should be added to the model. Again, note that if you square the 
t
-statistic value of the FLR
coefficient in the multiple regression (8.1.4), which is (
−
10.6293)
2
, you will obtain the 
F
value of Eq. (8.4.17), save for the rounding errors.
Incidentally, the 
F
ratio of Eq. (8.4.16) can be recast by using the 
R
2
values only, as we
did in Eq. (8.4.13). As Exercise 8.2 shows, the 
F 
ratio of Eq. (8.4.16) is 
equivalent 
to the
following 
F
ratio:
9
(8.4.18)
F
=
R
2
new
−
R
2
old
df
1
−
R
2
new
df
=
R
2
new
−
R
2
old
number of new regressors
1
−
R
2
new
df (
=
n
−
number of parameters in the new model)