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Multiple Tests and Bonferroni Corrections

Consistency of Error Variance Estimators Using the methods of Section 7.2 we can show that the estimators b 2 = 1 n Pn i=1 eb 2 i and s 2 = 1 nk Pn i=1 eb 2 i are consistent for  2 : The trick is to write the residual ebi as equal to the error ei plus a deviation term ebi = yi x 0 i b = ei + x 0 i x 0 i b = ei x 0 i  b  : Thus the squared residual equals the squared error plus a deviation eb 2 i = e 2 i 2eix 0 i  b  +  b 0 xix 0 i  b  : (7.17) So when we take the average of the squared residuals we obtain the average of the squared errors, plus two terms which are (hopefully) asymptotically negligible. b 2 = 1 n Xn i=1 e 2 i 2 1 n Xn i=1 eix 0 i !  b  (7.18) +  b 0 1 n Xn i=1 xix 0 i !  CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 230 Indeed, the WLLN shows that 1 n Xn i=1 e 2 i p !  2 1 n Xn i=1 eix 0 i p ! E eix 0 i  = 0 1 n Xn i=1 xix 0 i p ! E xix 0 i  = Qxx and Theorem 7.1 shows that b p ! . Hence (7.18) converges in probability to  2 ; as desired. Finally, since n=(n k) ! 1 as n ! 1; it follows that s 2 =  n n k  b 2 p !  2 : Thus both estimators are consistent. Theorem 7.4 Under Assumption 7.1, b 2 p !  2 and s 2 p !  2