Prediction: One use of a fitted regression equation is to predict response-variable values for particular ‘‘future’’ combinations of explanatory-variable scores. Suppose, therefore, that we fit the model y ¼ Xfl þ ", obtaining the least-squares estimate b of fl. Let x0 0 ¼ ½1; x01; ... ; x0k % represent a set of explanatory-variable scores for which a prediction is desired, and let Y0 be the (generally unknown, or not yet known) corresponding value of Y.

The explanatory-variable vector x0 0 does not necessarily correspond to an observation in the sample for which the model was fit.

(a) *If we use Yb0 ¼ x0 0b to estimate EðY0Þ, then the error in estimation is δ [ Yb0 ( EðY0Þ.

Show that if the model is correct, then EðδÞ ¼ 0 [i.e., Yb0 is an unbiased estimator of EðY0Þ] and that VðδÞ ¼ σ2

εx0 0ðX0 XÞ

(1 x0.

(b) *We may be interested not in estimating the expected value of Y0 but in predicting or forecasting the actual value Y0 ¼ x0 0fl þ ε0 that will be observed. The error in the forecast is then D [ Yb0 ( Y0 ¼ x0 0b ( ðx0 0fl þ ε0Þ ¼ x0 0ðb ( flÞ ( ε0 Show that EðDÞ ¼ 0 and that VðDÞ ¼ σ2

ε ½1 þ x0 0ðX0 XÞ

(1 x0%. Why is the variance of the forecast error D greater than the variance of δ found in part (a)?

(c) Use the results in parts

(a) and (b), along with the Canadian occupational prestige regression (see Section 5.2.2), to predict the prestige score for an occupation with an average income of $12,000, an average education of 13 years, and 50% women. Place a 90% confidence interval around the prediction, assuming (i) that you wish to estimate EðY0Þ and (ii) that you wish to forecast an actual Y0 score. (Because σ2

ε is not known, you will need to use S2 E and the t-distribution.)

(d) Suppose that the methods of this problem are used to forecast a value of Y for a combination of Xs very different from the X values in the data to which the model was fit.

For example, calculate the estimated variance of the forecast error for an occupation with an average income of $50,000, an average education of 0 years, and 100%

women. Is the estimated variance of the forecast error large or small? Does the variance of the forecast error adequately capture the uncertainty in using the regression equation to predict Y in this circumstance?