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Reverse Regression

Edgeworth Expansion* Theorem 7.11 showed that the t-ratio T() is asymptotically normal. In practice this means that we use the normal distribution to approximate the Önite sample distribution of T. How good is this approximation? Some insight into the accuracy of the normal approximation can be obtained by an Edgeworth expansion, which is a higher-order approximation to the distribution of T. The following result is an application of Theorem 6.34. Theorem 7.15 Under Assumptions 7.2, 7.3 and > 0, E kek 16 < 1; E kxk 16 < 1, g (u) has Öve continuous derivatives in a neighborhood of , and E  exp  t  kek 4 + kxk 4   B < 1, as n ! 1 P (T()  x) = (x) + n 1=2 p1(x)(x) + n 1 p2(x)(x) + o n 1  uniformly in x, where p1(x) is an even polynomial of order 2, and p2(x) is an odd polynomial of degree 5, with coe¢ cients depending on the moments of e and x up to order 16. Theorem 7.15 shows that the Önite sample distribution of the t-ratio can be approximated up to o(n 1 ) by the sum of three terms, the Örst being the standard normal distribution, the second a O n 1=2  adjustment and the third a O n 1  ad CHAPTER 7. ASYMPTOTIC THEORY FOR LEAST SQUARES 248 Consider a one-sided conÖdence interval C = h b z1 s(b); 1  where z1 is the 1 th quantile of Z  N (0; 1), thus (z1 ) 1 . Then P ( 2 C) = P (T()  z1 ) = (z1 ) + n 1=2 p1(z1 )(z1 ) + O n 1  = 1 + O  n 1=2  : This means that the actual coverage is within O n 1=2  of the desired 1 level. Now consider a two-sided interval C = h b z1 =2s(b); b+ z1 =2s(b) i . It has coverage P ( 2 C) = P jT()j  z1 =2  = 2(z1 =2 ) 1 + n 1 2p2(z1 =2 )(z1 =2 ) + o n 1  = 1 + O n 1  : This means that the actual coverage is within O n 1  of the desired 1 level. The accuracy is better than the one-sided interval because the O n 1=2  term in the Edgeworth expansion has o§setting e§ect