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FIGURE 12.9 Current residuals versus lagged residuals.

FIGURE 12.9 Current residuals versus lagged residuals.

19Actually, it is the so-called Studentized residuals that have a unit variance. But in practice the standardized residuals will give the same picture, and hence we may rely on them. On this, see Norman Draper and Harry Smith, Applied Regression Analysis, 3d ed., John Wiley & Sons, New York, 1998, pp. 207-208.

Table 12.5. As this figure reveals, most of the residuals are bunched in the second (northeast) and the fourth (southwest) quadrants, suggesting a strong positive correlation in the residuals.

The graphical method we have just discussed, although powerful and suggestive, is subjective or qualitative in nature. But there are several quantitative tests that one can use to supplement the purely qualitative approach. We now consider some of these tests.

If we carefully examine Figure 12.8, we notice a peculiar feature: Initially, we have several residuals that are negative, then there is a series of positive residuals, and then there are several residuals that are negative. If these residuals were purely random, could we observe such a pattern? Intuitively, it seems unlikely. This intuition can be checked by the so-called runs test, sometimes also know as the Geary test, a nonparametric test.20

To explain the runs test, let us simply note down the signs (+ or —) of the residuals obtained from the wages-productivity regression, which are given in the first column of Table 12.5.

Thus there are 9 negative residuals, followed by 21 positive residuals, followed by 10 negative residuals, for a total of 40 observations.

We now define a run as an uninterrupted sequence of one symbol or attribute, such as + or —. We further define the length of a run as the number of elements in it. In the sequence shown in (12.6.1), there are 3 runs: a run of 9 minuses (i.e., of length 9), a run of 21 pluses (i.e., of length 21) and a run of 10 minuses (i.e., of length 10). For a better visual effect, we have presented the various runs in parentheses.

By examining how runs behave in a strictly random sequence of observations, one can derive a test of randomness of runs. We ask this question: Are the 3 runs observed in our illustrative example consisting of 40 observations too many or too few compared with the number of runs expected in a strictly random sequence of 40 observations? If there are too many runs, it would mean that in our example the residuals change sign frequently, thus indicating negative serial correlation (cf. Figure 12.3b). Similarly, if there are too few runs, they may suggest positive autocorrelation, as in Figure 12.3a. A priori, then, Figure 12.8 would indicate positive correlation in the residuals.

20In nonparametric tests we make no assumptions about the (probability) distribution from which the observations are drawn. On the Geary test, see R. C. Geary, "Relative Efficiency of Count Sign Changes for Assessing Residual Autoregression in Least Squares Regression," Biometrika, vol. 57, 1970, pp. 123-127.

II. The Runs Test 