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Expectation: Mathematical Details

ation. 7.15 Forecast Intervals Suppose we are given a value of the regressor vector xn+1 for an individual outside the sample, and we want to forecast (guess) yn+1 for this individual. This is equivalent to forecasting yn+1 given xn+1 = x; which will generally be a function of x. A reasonable forecasting rule is the conditional mean m(x) as it is the mean-square-minimizing forecast. A point forecast is the estimated conditional mean mb (x) = x 0 b. We would also like a measure of uncertainty for the forecast. The forecast error is ebn+1 = yn+1 mb (x) = en+1 x 0  b  . As the out-of-s CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 244 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Experience log(wage) 0 5 10 15 20 25 30 35 40 45 50 55 60 Figure 7.7: Wage on Experience Regression Intervals is independent of the in-sample estimate b; this has conditional variance E eb 2 n+1jxn+1 = x  = E  e 2 n+1 2x 0  b  en+1 + x 0  b   b 0 xjxn+1 = x  = E e 2 n+1 j xn+1 = x  + x 0E  b   b 0 x =  2 (x) + x 0V bx: (7.36) Under homoskedasticity E e 2 n+1 j xn+1 =  2 . In this case a simple estimator of (7.36) is b 2 + x 0Vb bx; so a standard error for the forecast is sb(x) = q b 2 + x0Vb bx: Notice that this is di§erent from the standard error for the conditional mean. The conventional 95% forecast interval for yn+1 uses a normal approximation and sets h x 0 b  2sb(x) i : It is di¢ cult, however, to fully justify this choice. It would be correct if we have a normal approximation to the ratio en+1 x 0  b  sb(x) : The di¢ culty is that the equation error en+1 is generally non-normal, and asymptotic theory cannot be applied to a single observation. The only special exception is the case where en+1 has the exact distribution N(0; 2 ); which is generally invalid. To get an accurate forecast interval, we need to estimate the conditional distribution of en+1 given xn+1 = x; which is a much more di¢ cult task. Perhaps due to this di¢ culty, many applied forecasters use the simple approximate interval h x 0 b  2sb(x) i despite the lack of a convincing ju