'Obtained from Table 12.1.

'Obtained from Table 12.1.

subsequently it is generally smaller than its previous value showing, in general, a positive autocorrelation.

Now suppose the values of X are fixed at 1, 2, 3, ..., 10. Then, given these X's, we can generate a sample of 10 Y values from (12.4.3) and the values of ut given in Table 12.1. The details are given in Table 12.2. Using the data of Table 12.2, if we regress Y on X, we obtain the following (sample) regression:

whereas the true regression line is as given by (12.4.4). Both the regression lines are given in Figure 12.6, which shows clearly how much the fitted regression line distorts the true regression line; it seriously underestimates the true slope coefficient but overestimates the true intercept. (But note that the OLS estimators are still unbiased.)

Figure 12.6 also shows why the true variance of u is likely to be underestimated by the estimator a2, which is computed from the u. The u are generally close to the fitted line (which is due to the OLS procedure) but deviate substantially from the true PRF. Hence, they do not give a correct picture of u. To gain some insight into the extent of underestimation of true a2, suppose we conduct another sampling experiment. Keeping the Xt and et given in Tables 12.1 and 12.2, let us assume p = 0, that is, no autocorrelation. The new sample of Y values thus generated is given in Table 12.3.


FIGURE 12.6 True PRF and the estimated regression line for the data of Table 12.2. TABLE 12.3 SAMPLE OF Y VALUES WITH ZERO SERIAL CORRELATION

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