Variance: aR =
If the null hypothesis of randomness is sustainable, following the properties of the normal distribution, we should expect that
Prob [E(R) — 1.96aR < R < E(R) + 1.96aR = 0.95 (12.6.3)
That is, the probability is 95 percent that the preceding interval will include R. Therefore we have this rule:
Decision Rule. Do not reject the null hypothesis of randomness with 95% confidence if R, the number of runs, lies in the preceding confidence interval; reject the null hypothesis if the estimated R lies outside these limits. (Note: You can choose any level of confidence you want.)
Returning to our example, we know that N1 , the number of minuses, is 19 and N2, the number of pluses, is 21 and R = 3. Using the formulas given in (12.6.2), we obtain:
The 95% confidence interval for R in our example is thus:
Obviously, this interval does not include 3. Hence, we can reject the hypothesis that the residuals in our wages-productivity regression are random
CHAPTER TWELVE: AUTOCORRELATION 467
with 95% confidence. In other words, the residuals exhibit autocorrelation. As a general rule, if there is positive autocorrelation, the number of runs will be few, whereas if there is negative autocorrelation, the number of runs will be many. Of course, from (12.6.2) we can find out whether we have too many runs or too few runs.
Swed and Eisenhart have developed special tables that give critical values of the runs expected in a random sequence of N observations if N1 or N2 is smaller than 20. These tables are given in Appendix D, Table D.6. Using these tables, the reader can verify that the residuals in our wages-productivity regression are indeed nonrandom; actually they are positively correlated.
The most celebrated test for detecting serial correlation is that developed by statisticians Durbin and Watson. It is popularly known as the Durbin-Watson d statistic, which is defined as d = ^L-t^ (12.6.5)
t=1 ut which is simply the ratio of the sum of squared differences in successive residuals to the RSS. Note that in the numerator of the d statistic the number of observations is n - 1 because one observation is lost in taking successive differences.
A great advantage of the d statistic is that it is based on the estimated residuals, which are routinely computed in regression analysis. Because of this advantage, it is now a common practice to report the Durbin-Watson d along with summary measures, such as R2, adjusted R2, t, and F. Although it is now routinely used, it is important to note the assumptions underlying the d statistic.
1. The regression model includes the intercept term. If it is not present, as in the case of the regression through the origin, it is essential to rerun the regression including the intercept term to obtain the RSS.22
2. The explanatory variables, the X's, are nonstochastic, or fixed in repeated sampling.
3. The disturbances ut are generated by the first-order autoregressive scheme: ut = put-1 + st. Therefore, it cannot be used to detect higher-order autoregressive schemes.
4. The error term ut is assumed to be normally distributed.
21J. Durbin and G. S. Watson, "Testing for Serial Correlation in Least-Squares Regression," Biometrika, vol. 38, 1951, pp. 159-171.
22However, R. W. Farebrother has calculated d values when the intercept term is absent from the model. See his "The Durbin-Watson Test for Serial Correlation When There Is No Intercept in the Regression," Econometrica, vol. 48, 1980, pp. 1553-1563.
468 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
5. The regression model does not include the lagged value(s) of the dependent variable as one of the explanatory variables. Thus, the test is inapplicable in models of the following type:
where Yt-1 is the one period lagged value of Y. Such models are known as autoregressive models, which we will study in Chapter 17.
6. There are no missing observations in the data. Thus, in our wages-productivity regression for the period 1959-1998, if observations for, say, 1978 and 1982 were missing for some reason, the d statistic makes no allowance for such missing observations.23
The exact sampling or probability distribution of the d statistic given in (12.6.5) is difficult to derive because, as Durbin and Watson have shown, it depends in a complicated way on the X values present in a given sample.24 This difficulty should be understandable because d is computed from ut, which are, of course, dependent on the given X's. Therefore, unlike the t, F, or x2 tests, there is no unique critical value that will lead to the rejection or the acceptance of the null hypothesis that there is no first-order serial correlation in the disturbances u. However, Durbin and Watson were successful in deriving a lower bound dL and an upper bound du such that if the computed d from (12.6.5) lies outside these critical values, a decision can be made regarding the presence of positive or negative serial correlation. Moreover, these limits depend only on the number of observations n and the number of explanatory variables and do not depend on the values taken by these explanatory variables. These limits, for n going from 6 to 200 and up to 20 explanatory variables, have been tabulated by Durbin and Watson and are reproduced in Appendix D, Table D.5 (up to 20 explanatory variables).
The actual test procedure can be explained better with the aid of Figure 12.10, which shows that the limits of d are 0 and 4. These can be established as follows. Expand (12.6.5) to obtain
Since Y. U and Y. U2-1 differ in only one observation, they are approximately equal. Therefore, setting XU-i ~ E U, (12.6.7) may be written as
Yt = 0i + X2t + fa X3t + ■ ■ ■ + pkXkt + y Yt-i + ut (12.6.6)
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