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

(12.6.13)
that is, in large samples, the square root of the sample size times the estimated first-order
autocorrelation coefficient also follows the standard normal distribution.
As an illustration of the test, for our wages–productivity example, we found that
d
=
0.2176 with 
n
=
46. Therefore, from Eq. (12.6.12) we find that 
√
46
1
−
0
.
2176
2
≈
6
.
0447
Asymptotically, if the null hypothesis of zero (first-order) autocorrelation were true, the
probability of obtaining a
Z
value (i.e., a standardized normal variable) of as much as 6.0447
or greater is extremely small. Recall that for a standard normal distribution, the (two-tail)
critical 5 percent
Z
value is only 1.96 and the 1 percent critical
Z
value is about 2.58. Al-
though our sample size is only 46, for practical purposes it may be large enough to use the
normal approximation. The conclusion remains the same, namely, that the residuals from
the wages–productivity regression suffer from autocorrelation.
But the most serious problem with the 
d
test is the assumption that the regressors are
nonstochastic, that is, their values are fixed in repeated sampling. If this is not the case, then
the 
d
test is not valid either in finite, or small, samples or in large samples.
28
And since this
assumption is usually difficult to maintain in economic models involving time series data,
one author contends that the Durbin–Watson statistic may not be useful in econometrics in-
volving time series data.
29
In his view, more useful tests of autocorrelation are available,
but they are all based on large samples. We discuss one such test below, the 
Breusch–