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

Step III:
Obtain 
Z
1
=
(ln
Y f
−
ln 
f
).
Step IV:
Regress 
Y
on 
X
’s and 
Z
1
obtained in Step III. Reject 
H
0
if the coefficient of
Z
1
is statistically significant by the usual 
t
test.
Step V:
Obtain 
Z
2
=
(antilog of ln
f
−
Y f
).
Step VI:
Regress log of 
Y
on the logs of 
X
’s and 
Z
2
. Reject 
H
1
if the coefficient of 
Z
2
is statistically significant by the usual 
t
test.
Although the MWD test seems involved, the logic of the test is quite simple. If the linear
model is in fact the correct model, the constructed variable
Z
1
should not be statistically sig-
nificant in Step IV, for in that case the estimated
Y
values from the linear model and those
estimated from the log–linear model (after taking their antilog values for comparative pur-
poses) should not be different. The same comment applies to the alternative hypothesis
H
1
.
*
Optional.
19
Russell Davidson and James G. MacKinnon, 
Estimation and Inference in Econometrics,
Oxford Univer-
sity Press, New York, 1993, p. 456.
20
J. MacKinnon, H. White, and R. Davidson, “Tests for Model Specification in the Presence of Alterna-
tive Hypothesis; Some Further Results,” 
Journal of Econometrics,
vol. 21, 1983, pp. 53–70. A similar
test is proposed in A. K. Bera and C. M. Jarque, “Model Specification Tests: A Simultaneous Approach,”
Journal of Econometrics,
vol. 20, 1982, pp. 59–82.
21
This discussion is based on William H. Greene, 
ET. The Econometrics Toolkit Version 3,
Econometric
Software, Bellport, New York, 1992, pp. 245–246.
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Chapter 8
Multiple Regression Analysis: The Problem of Inference