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Model in Matrix Notation

Regression Intervals In the linear regression model the conditional mean of yi given xi = x is m(x) = E (yi j xi = x) = x 0 : In some cases, we want to estimate m(x) at a particular point x: Notice that this is a linear function of : Letting r( ) = x 0 and  = r( ); we see that mb (x) = b = x 0 b and R = x; so s(b) = q x0Vb bx: Thus an asymptotic 95% conÖdence interval for m(x) is  x 0 b  1:96q x0Vb bx  : It is interesting to observe that if this is viewed as a function of x; the width of the conÖdence interval is dependent on x: To illustrate, we return to the log wage regression (3.13) of Section 3.7. The estimated regression equation is log(\W age) = x 0 b = 0:155x + 0:698 where x = education. The covariance matrix estimate from (4.38) is Vb b =  0:001 0:015 0:015 0:243  : Thus the 95% conÖdence interval for the regression is 0:155x + 0:698  1:96p 0:001x 2 0:030x + 0:243: The estimated regression and 95% intervals are shown in Figure 7.6. Notice that the conÖdence bands take a hyperbolic shape. This means that the regression line is less precisely estimated for very large and very small values of education. Plots of the estimated regression line and conÖdence intervals are especially useful when the regression includes nonlinear terms. To illustrate, consider the log wage regression (7.31) which includes experience and its square, with covariance matrix (7.32). We are interested in plotting the regression estimate and regression intervals as a function of experience. Since the regression also includes education, to plot the estimates in a simple graph we need to Öx education at a speciÖc value. We select education=12. This only a§ects the level of the estimated regression, since education enters without an interaction. DeÖne the points of evaluation z(x) = 0 BB@ 12 x x 2=100 1 1 CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 243 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Education log(wage) 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Figure 7.6: Wage on Education Regression Intervals where x =experience. Thus the 95% regression interval for education=12, as a function of x =experience is 0:118 12 + 0:016 x 0:022 x 2 =100 + 0:947  1:96 vuuuuut z(x) 0 0 BB@ 0:632 0:131 0:143 11:1 0:131 0:390 0:731 6:25 0:143 0:731 1:48 9:43 11:1 6:25 9:43 246 1 CCA z(x) 104 = 0:016 x :00022 x 2 + 2:36  0:0196p 70:608 9:356 x + 0:54428 x 2 0:01462 x 3 + 0:000148 x 4: The estimated regression and 95% intervals are shown in Figure 7.7. The regression interval widens greatly for small and large values of experience, indicating considerable uncertainty about the e§ect of experience on mean wages for this population. The conÖdence bands take a more complicated shape than in Figure 7.6 due to the nonlinear