Thus, given X0 = 100, in repeated sampling, 95 out of 100 intervals like (5.10.5) will include the true mean value; the single best estimate of the true mean value is of course the point estimate 75.3645.
If we obtain 95% confidence intervals like (5.10.5) for each of the X values given in Table 3.2, we obtain what is known as the confidence interval, or confidence band, for the population regression function, which is shown in Figure 5.6.
If our interest lies in predicting an individual Y value, Y0, corresponding to a given X value, say, X0, then, as shown in Appendix 5, Section 5A.3, a best linear unbiased estimator of Y0 is also given by (5.10.1), but its variance is as follows:
It can be shown further that Y0 also follows the normal distribution with mean and variance given by (5.10.1) and (5.10.6), respectively. Substituting o2
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Edition Estimation and Hypothesis
CHAPTER FIVE: TWO VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING 145
for the unknown a2, it follows that t = Y — se (Y — YY0)
also follows the t distribution. Therefore, the t distribution can be used to draw inferences about the true Y0. Continuing with our consumption-income example, we see that the point prediction of Y0 is 75.3645, the same as that of Y0, and its variance is 52.6349 (the reader should verify this calculation). Therefore, the 95% confidence interval for Y0 corresponding to X0 = 100 is seen to be
Comparing this interval with (5.10.5), we see that the confidence interval for individual Y0 is wider than that for the mean value of Y0. (Why?) Computing confidence intervals like (5.10.7) conditional upon the X values given in Table 3.2, we obtain the 95% confidence band for the individual Y values corresponding to these X values. This confidence band along with the confidence band for Y0 associated with the same Xs is shown in Figure 5.6.
Notice an important feature of the confidence bands shown in Figure 5.6. The width of these bands is smallest when X0 = X. (Why?) However, the width widens sharply as Xo moves away from X. (Why?) This change would suggest that the predictive ability of the historical sample regression line falls markedly as X0 departs progressively from X. Therefore, one should exercise great caution in "extrapolating" the historical regression line to predict E(Y | X0) or Yo associated with a given X0 that is far removed from the sample mean X.
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