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

Part One
Single-Equation Regression Models
Asymptotically, this is distributed as the chi-square distribution with 1 df (because we have only one
restriction imposed when we omitted the FLR variable from the full model). The 
p
value of obtaining
such a chi-square value for 1 df is almost zero, leading to the conclusion that the FLR variable should
not
be excluded from the model. In other words, the restricted regression in the present instance is not
valid.
Letting RRSS and URSS denote the restricted and unrestricted residual sums of squares, Eq. (4)
can also be expressed as:
−
2 ln
λ
=
n
(ln RRSS
−
ln URSS)
(5)
which is distributed as 
χ
2
with 
r
degrees of freedom, where 
r
is the number of restrictions imposed
on the model (i.e., the number of 
r
coefficients omitted from the original model).
Although we will not go into the details of the Wald and LM tests, these tests can be implemented
as follows:
Wald Statistic (W)
=
(
n
−
k
)(RRSS
−
URSS)
URSS
∼
χ
2
r
(6)
Lagrange Multiplier Statistic (LM)
=
(
n
−
k
+
r
)(RRSS
−
URSS)
RRSS
∼
χ
2
r
(7)
Where 
k
is the number of regressors in the unrestricted model and 
r
is the number of restrictions.
As you can see from the preceding equations, all three tests are asymptotically (i.e., in large sam-
ples) equivalent, that is, they give similar answers. However, in small samples the answers can differ.
There is an interesting relationship among these statistics in that it can be shown that:
W
≥
LR
≥
LM
Therefore, in small samples, a hypothesis can be rejected by the Wald statistic but 
not
rejected by the
LM statistic.
*
As noted in the text, for most of our purposes the 
t
and 
F
tests will suffice. But the three tests dis-
cussed above are of general applicability in that they can be applied to testing nonlinear hypotheses
in linear models, or testing restrictions on variance-covariance matrices. They also can be applied in
situations where the assumption that the errors are normally distributed is not tenable.
Because of the mathematical complexity of the Wald and LM tests, we will not go into more de-
tail here. But as noted, asymptotically, the LR, Wald, and LM tests give identical answers, the choice
of the test depending on computational convenience.
*
For an explanation, see G. S. Maddala, 
Introduction to Econometrics, 
3d ed., John Wiley & Sons, New
York, 2001, p. 177.
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Chapter
1
We will discuss ordinal scale variables in Chapter 15.
2
For a review of the evidence on this subject, see Bruce E. Kaufman and Julie L. Hotchkiss, 
The
Economics of Labor Markets,
5th ed., Dryden Press, New York, 2000.
9
Dummy Variable
Regression Models
In Chapter 1 we discussed briefly the four types of variables that one generally encounters
in empirical analysis: These are: