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Limitations of the Best Linear Projection

ConÖdence Regions A conÖdence region Cb is a set estimator for  2 R q when q > 1: A conÖdence region Cb is a set in R q intended to cover the true parameter value with a pre-selected probability 1 : Thus an ideal conÖdence region has the coverage probability P( 2 Cb) = 1 . In practice it is typically not possible to construct a region with exact coverage, but we can calculate its asymptotic coverage. When the parameter estimator satisÖes the conditions of Theorem 7.13, a good choice for a conÖdence region is the ellipse Cb = f : W()  c1 g with c1 the 1 quantile of the  2 q distribution. (Thus Fq(c1 ) = 1 :) It can be computed by, for example, chi2inv(1- ,q)in MATLAB. Theorem 7.13 implies P   2 Cb  ! P  2 q  c1  = 1 which shows that Cb has asymptotic coverage 1 : To illustrate the construction of a conÖdence region, consider the estimated regression (7.31) of the model log(\W age) = 1 education + 2 experience + 3 experience2 =100 + 4: Suppose that the two parameters of interest are the percentage return to education 1 = 100 1 and the percentage return to experience for individuals with 10 years experience 2 = 100 2 + 20 3. These two parameters are a linear transformation of the regression parameters with point estimates b =  100 0 0 0 0 100 20 0  b =  11:8 1:2  ; and have the covariance matrix estimate Vb b =  0 100 0 0 0 0 100 20  Vb b 0 BB@ 0 0 100 0 0 100 0 20 1 CCA =  0:632 0:103 0:103 0:157  with inverse Vb 1 b =  1:77 1:16 1:16 7:13  : Thus the Wald statistic is W() =  b  0 Vb 1 b  b   =  11:8 1 1:2 2 0  1:77 1:16 1:16 7:13   11:8 1 1:2 2  = 1:77 (11:8 1) 2 2:32 (11:8 1) (1:2 2) + 7:13 (1:2 2) 2 : The 90% quantile of the  2 2 distribution is 4.605 (we use the  2 2 distribution as the dimension of  is two), so an asymptotic 90% conÖdence region for the two parameters is the interior of the ellipse W() = 4:605 which is displayed in Figure 7.8. Since the estimated correlation of the two coe¢ cient estimates is modest (about 0.3) the re CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 247 Return to Education (%) Return to Experience (%) 10 11 12 13 14 0.0 0.5 1.0 1.5 2.0 2.5 Figure 7.8: ConÖdence Region for Return to Experience and Return to Education