# 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 ib = ei + x 0 i x 0 ib = 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