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

Part Two
Relaxing the Assumptions of the Classical Model
Illustration of the
BG Test: The
Wages–
Productivity
Relation
To illustrate the test, we will apply it to our illustrative example. Using an AR(6) scheme,
we obtained the results shown in Exercise 12.25. From the regression results given there,
it can be seen that (
n
−
p
)
=
40 and 
R
2
=
0.7498. Therefore, multiplying these two, we
obtain a chi-square value of 29.992. For 6 df (why?), the probability of obtaining a chi-
square value of as much as 29.992 or greater is extremely small; the chi-square table in
Appendix D.4 shows that the probability of obtaining a chi-square value of as much as
18.5476 or greater is only 0.005. Therefore, for the same df, the probability of obtaining
a chi-square value of about 30 must be extremely small. As a matter of fact, the actual
p
value is almost zero.
Therefore, the conclusion is that, for our example, at least one of the six autocorrela-
tions must be nonzero.
Trying varying lag lengths from 1 to 6, we find that only the AR(1) coefficient is signifi-
cant, suggesting that there is no need to consider more than one lag. In essence the BG test
in this case turns out to be
Durbin’s
m
test.
Why So Many Tests of Autocorrelation?
The answer to this question is that “. . . no particular test has yet been judged to be un-
equivocally best [i.e., more powerful in the statistical sense], and thus the analyst is still in
the unenviable position of considering a varied collection of test procedures for detecting
the presence or structure, or both, of autocorrelation.”
34
Of course, a similar argument can
be made about the various tests of heteroscedasticity discussed in the previous chapter.