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

likelihood, Wald, and Lagrange multiplier tests
can ac-
complish this purpose. The interesting thing to note is that 
asymptotically
(i.e., in large
*Optional.
17
For a discussion of the Chow test under heteroscedasticity, see William H. Greene, 
Econometric
Analysis,
4th ed., Prentice Hall, Englewood Cliffs, NJ, 2000, pp. 292–293, and Adrian C. Darnell,
A Dictionary of Econometrics,
Edward Elgar, U.K., 1994, p. 51.
18
For an accessible discussion, see A. Buse, “The Likelihood Ratio, Wald and Lagrange Multiplier Tests:
An Expository Note,’’ 
American Statistician,
vol. 36, 1982, pp. 153–157.
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Part One
Single-Equation Regression Models
samples) all three tests are equivalent in that the test statistic associated with each of these
tests follows the chi-square distribution.
Although we will discuss the 
likelihood ratio test
in the appendix to this chapter, in
general we will not use these tests in this textbook for the pragmatic reason that in small, or
finite, samples, which is unfortunately what most researchers deal with, the 
F
test that we
have used so far will suffice. As Davidson and MacKinnon note:
For linear regression models, with or without normal errors, there is of course no need to look
at LM, W and LR at all, since no information is gained from doing so over and above what is
already contained in F.
19
*
8.10
Testing the Functional Form of Regression: Choosing
between Linear and Log–Linear Regression Models
The choice between a linear regression model (the regressand is a linear function of the
regressors) or a log–linear regression model (the log of the regressand is a function of the
logs of the regressors) is a perennial question in empirical analysis. We can use a test pro-
posed by MacKinnon, White, and Davidson, which for brevity we call the 
MWD test,
to
choose between the two models.
20
To illustrate this test, assume the following
H
0
: 
Linear Model: Y 
is a linear function of regressors, the 
X
’s.
H
1
: 
Log–Linear Model:
ln
Y
is a linear function of logs of regressors, the logs of 
X
’s.
where, as usual, 
H
0
and 
H
1
denote the null and alternative hypotheses.
The MWD test involves the following steps:
21
Step I:
Estimate the linear model and obtain the estimated
Y
values. Call them
Yf
(i.e.,
ˆ
Y
).
Step: II:
Estimate the log–linear model and obtain the estimated ln
Y
values; call them
ln 
f
(i.e., 
ln
Y
).