D 412611

These are the bounds of d; any estimated d value must lie within these limits.

It is apparent from Eq. (12.6.10) that if p = 0, d = 2; that is, if there is no serial correlation (of the first-order), d is expected to be about 2. Therefore, as a rule of thumb, if d is found to be 2 in an application, one may assume that there is no first-order autocorrelation, either positive or negative. If p = +1, indicating perfect positive correlation in the residuals, d ~ 0. Therefore, the closer d is to 0, the greater the evidence of positive serial correlation. This relationship should be evident from (12.6.5) because if there is positive autocorrelation, the ut's will be bunched together and their differences will therefore tend to be small. As a result, the numerator sum of squares will be smaller in comparison with the denominator sum of squares, which remains a unique value for any given regression.

470 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

If p = — 1, that is, there is perfect negative correlation among successive residuals, d ~ 4. Hence, the closer d is to 4, the greater the evidence of negative serial correlation. Again, looking at (12.6.5), this is understandable. For if there is negative autocorrelation, a positive ut will tend to be followed by a negative ut and vice versa so that |ut — ut—11 will usually be greater than |ut |. Therefore, the numerator of d will be comparatively larger than the denominator.

The mechanics of the Durbin-Watson test are as follows, assuming that the assumptions underlying the test are fulfilled:

1. Run the OLS regression and obtain the residuals.

2. Compute d from (12.6.5). (Most computer programs now do this routinely.)

3. For the given sample size and given number of explanatory variables, find out the critical dL and dv values.

4. Now follow the decision rules given in Table 12.6. For ease of reference, these decision rules are also depicted in Figure 12.10.

To illustrate the mechanics, let us return to our wages-productivity regression. From the data given in Table 12.5 the estimated d value can be shown to be 0.1229, suggesting that there is positive serial correlation in the residuals. From the Durbin-Watson tables, we find that for 40 observations and one explanatory variable, dL = 1.44 and du = 1. 54 at the 5 percent level. Since the computed d of 0.1229 lies below dL, we cannot reject the hypothesis that there is positive serial correlations in the residuals.

Although extremely popular, the d test has one great drawback in that, if it falls in the indecisive zone, one cannot conclude that (first-order) autocorrelation does or does not exist. To solve this problem, several authors have proposed modifications of the d test but they are rather involved and beyond the scope of this book.25 In many situations, however, it has been found that the upper limit du is approximately the true significance limit and therefore in case d lies in the indecisive zone, one can use the following modified d test: Given the level of significance a,

1. H0: p = 0 versus H1: p > 0. Reject H0 at a level if d < du. That is, there is statistically significant positive autocorrelation.

TABLE 12.6 DURBIN-WATSON d TEST: DECISION RULES

Null hypothesis

Decision

If

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