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

(13.4.7)
3. Let the
R
2
obtained from Eq. (13.4.7) be
R
2
new
and that obtained from Eq. (13.4.6) be
R
2
old
. Then we can use the 
F
test first introduced in Eq. (8.4.18), namely,
F
=
R
2
new
−
R
2
old
number of new regressors
1
−
R
2
new
(
n
−
number of parameters in the new model)
(8.4.18)
to find out if the increase in 
R
2
from using Eq. (13.4.7) is statistically significant.
4. If the computed 
F
value is significant, say, at the 5 percent level, one can accept the
hypothesis that the model (13.4.6) is mis-specified.
Returning to our illustrative example, we have the following results (standard errors in
parentheses):
ˆ
Y
i
=
166.467
+
19.933
X
i
(13.4.8)
(19.021)
(3.066)
R
2
=
0.8409
0
u
i
Y
150
200
300
400
250
350
FIGURE 13.2
Residuals 
ˆ
u
i
and
estimated 
Y
from the
linear cost function: 
Y
i
=
λ
1 
+
λ
2
X
i
+
u
i
.
guj75772_ch13.qxd 16/08/2008 03:24 PM Page 480

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Chapter 13
Econometric Modeling: Model Specification and Diagnostic Testing
481
ˆ
Y
i
=
2140.7223
+
476.6557
X
i
−
0.09187
ˆ
Y
2
i
+
0.000119
ˆ
Y
3
i
(132.0044)
(33.3951)
(0.00620)
(0.0000074)
(13.4.9)
R
2
=
0.9983
Note:
ˆ
Y
2
i
and 
ˆ
Y
3
i
in Eq. (13.4.9) are obtained from Eq. (13.4.8).
Now applying the 
F
test we find
F
=
(0
.
9983
−
0
.
8409)
/
2
(1
−
0
.
9983)
/
(10
−
4)
=
284
.
4035
(13.4.10)
The reader can easily verify that this 
F
value is highly significant, indicating that the model
(13.4.8) is mis-specified. Of course, we have reached the same conclusion on the basis of
the visual examination of the residuals as well as the Durbin–Watson 
d
value. It should be
added that, since 
ˆ
Y
i
is estimated, it is a random variable and, therefore, the usual tests of
significance apply if the sample is reasonably large.
One advantage of RESET is that it is easy to apply, for it does not require one to specify
what the alternative model is. But that is also its disadvantage because knowing that a
model is mis-specified does not help us necessarily in choosing a better alternative.
As one author notes:
In practice, the RESET test may not be particularly good at detecting any specific alternative
to a proposed model, and its usefulness lies in acting as a general indicator that something is
wrong. For this reason, a test such as RESET is sometimes described as a test of 
misspecifica-
tion,
as opposed to a test of specification. This distinction is rather subtle, but the basic idea is
that a specification test looks at some particular aspect of a given equation, with clear null and
alternative hypotheses in mind. A 
misspecification
test, on the other hand, can detect a range of
alternatives and indicate that something is wrong under the null, without necessarily giving
clear guidance as to what alternative hypothesis is appropriate.
25
Lagrange Multiplier (LM) Test for Adding Variables
This is an alternative to Ramsey’s RESET test. To illustrate this test, we will continue with
the preceding illustrative example.
If we compare the linear cost function (13.4.6) with the cubic cost function (13.4.4), the
former is a 
restricted version
of the latter (recall our discussion of