Anon

98 The Basics of financial economeTrics

The denominator of the test is simply the sum of the squares of the error
terms; the numerator is the squared difference of the successive residuals.
It can be shown that if the sample size is large, then the Durbin-Watson
d test statistic given by equation (4.25) is approximately related to the auto-
correlation given by equation (4.24) as

d ≈ 2 (1 − ρauto) (4.26)

Since ρauto can vary between −1 and 1, this means that d can vary from 0 to
4 as shown:

Value of ρauto Interpretation of ρauto Approximate Value of d

−1 Perfect negative autocorrelation 4

0 No autocorrelation 2

1 Perfect positive autocorrelation 0

From the above table we see that if d is close to 2 there is no autocorrela-
tion. A d value less than 2 means there is potentially positive autocorrelation;
the closer the value to 0 the greater the likelihood of positive autocorre-
lation. There is potentially negative autocorrelation if the computed d
exceeds 2 and the closer the value is to 4, the greater the likelihood of nega-
tive autocorrelation.
In previous hypothesis tests discussed in this book, we stated that there
is a critical value that a test statistic had to exceed in order to reject the
null hypothesis. In the case of the Durbin-Watson d statistic, there is not
a single critical value but two critical values, which are denoted by dL and
dU. Moreover, there are ranges for the value of d where no decision can be
made about the presence of autocorrelation. The general decision rule given
the null hypothesis and the computed value for d is summarized in the fol-
lowing table:

Null Hypothesis Range for Computed d Decision Rule

No positive autocorrelation 0 < d < dL Reject the null hypothesis

No positive autocorrelation dL ≤ d ≤ dU No decision

No negative autocorrelation 4 − dL < d < 4 Reject the null hypothesis

No negative autocorrelation 4 – dU ≤ d ≤ 4 – dL No decision

No autocorrelation dU < d < 4 – dU Accept the null hypothesis

Where does one obtain the critical values dL and dU? There are tables
that report those values for the 5% and 1% levels of significance. The critical