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Case (ii) is the one in which m is chosen to minimize the error in the normal approximation. Note the di erence between (2.22) and (2.18). Case (iii) has the advantage of entailing a smaller con dence interval. However, this is little comfort if the interval is not suitably centred and the normal interpretation appropriate, and Robinson (1995c), Nishiyama and Robinson (2000) suggested that it is m that minimizes the deviation from the normal approximation that is most relevant in normal-based inference on 0, not minimizing the MSE, making (2.18) more relevant than (2.22). We can go further and optimally estimate K in (2.18). As in Nishiyama and Robinson (2000), consider, in view of (2.19), Kopt = arg min max K y2R (y)( 3K +1=2 + K ;1=2 p(y)) choosing Kopt to minimize the maximal deviation from the usual normal approximation. We obtain the simple solution Kopt =(3 3( +1=2)) ;1=( +1) : An alternative to carrying out inference using the Edgeworth approximation is to invert the Edgeworth expansion to get a new statistic whose distribution is closely approximated by the standard normal. From (2.13), uniformly in y, and hence It may beshown that (2.23) implies P (U h m y)= (y + 3qm + m ;1=2 p(y)) + o(qm + m ;1=2 ) (2.23) P (U h m + 3qm + p(y)m ;1=2 P (U h m + 3qm + p(y)m ;1=2 uniformly in y = o(m 1=6 ). Indeed, by (2.13) y) (y): y) = (y)+o(qm + m ;1=2 ) P fU h m yg; y + 3qm + m ;1=2 p(y) = o(qm + m ;1=2 ): Set z = y + 3qm + m ;1=2 p(y) =y + m ;1=2 y 2 =3+a where a = 3qm +2m ;1=2 =3. Then y = ;1+p1+4m ;1=2 (z ; a)=3 2m ;1=2 : =3 Assuming that z = o(m 1=6 ), by Taylor expansion it follows that y = z = a ; m ;1=2 (z ; a) 2 =3+o(m ;1=2 )=z ; 3qm ; m ;1=2 p(z)+o(m ;1=2 ): The CLT (1.7) of Robinson (1995c) was established without Gaussianity, and with only nite moments of order four assumed. The asymptotic variance in (1.7) is una ected by cumulants of order three and more, and thus hypothesis tests and interval estimates based on (1.7) are broadly applicable. Looking only at our formal higher-order expansion, it is immediately appearent that the bias term (in qm) will not be a ected by non-Gaussianity, sonor will be the expansion when m increases so fast that qm dominates (see (2.16), (2.21)). Moreover, preliminary investigations suggest that when Xt is a linear process in iid innovations satisfying suitable moment conditions, the m ;1=2 term in the formal expansion is also generally una ected. (Speci cally, the leading terms in Corollary 7.1 are unchanged.) However as proof of validity of our expansions even in the linear case seems considerably harder and lengthier, we donot pursue the details here, adding that the estimates b bh optimise narrow-band forms of Gaussian likelihoods, and are thus in part motivated by Gaussianity, which inany case is frequently assumed in higher-order asymptotic theory. 8
3 Empirical expansions and bias correction The present section develops our results to provide feasible improved statistical inference on 0. An approximate 100 % con dence interval for 0 based on the CLT isgiven by (bh ; z =2m ;1=2 bh + z =2m ;1=2 ) (3.1) where 1 ; (z )= . From (2.23) a more accurate con dence interval is (bh + 3qmm ;1=2 + p(z =2)m ;1 ; z =2m ;1=2 bh + 3qmm ;1=2 + p(z =2)m ;1 + z =2m ;1=2 ): (3.2) Of course (3.1) and (3.2) correspond to level- hypothesis tests on 0. We reject the null hypothesis 0 = 0 0, for given 0 0 (e.g. 0 0 = 0, corresponding to a test of short memory) if 0 0 falls outside (3.1) or, more accurately, (3.2). An obvious aw in the preceding discussion is that 3 is unknown in practice. However, given an estimate b 3 such that b 3 ! 3 a.s (3.3) we deduce from (2.13) the empirical Edgeworth expansion sup P fU y2R h m yg; (y) ; (y)(b3qm + m ;1=2 p(y)) = o(qm + m ;1=2 ) a:s: (3.4) We can likewise replace 3 by b3 in (2.15), (2.19), (2.21), (2.23) and (3.2). We discuss two alternative estimates of 3. Our rst is b31 =( n m0 ) R(1) m0 (bh) (3.5) where in which and 1 Skm0( )= c0m0 R (1) 0( ) 0 m ( )=S1m (3.6) S0m0( ) m 0 X j=l jm0 =logj ; (m0 ; l +1) ;1 m 0 k jm 0 3j Ih( 3j) k 0 (3.7) X j=l log j (3.8) where m 0 is another bandwidth, increasing faster than m. Note that R (1) m ( )=(d=d )Rm( ) (see (2.4)), and so R (1) m (bh) =0. Our second estimate of 3 is b32 = bc1 3 ( ) (3.9) bc0 ( +1) 2 2 where, as in Henry and Robinson (1996), bc0 and bc1 are given by least squares regression based on (2.6), i.e. bc0 bc1 = 2 4 m0 X j=l 1 3j 3j 2 3j !3;1 0 Xm 5 j=l 1 3j b h 3j Ih( 3j): (3.10) Estimation of 1 is relevant in connection with (2.16). We de ne b 11 by replacing 3j by 1j, Ih by I and 3 by 1 in (3.7), and then bh by b in (3.5). Likewise we can de ne b 12 by (3.9) with 3 replaced by 1in(3.9)and3 3jIh and bh by 1 jI and b in (3.10). 9
Page 1 and 2: EDGEWORTH EXPANSION FOR SEMIPARAMET
Page 3 and 4: 1 Introduction First-order asymptot
Page 5 and 6: and Fox andTaqqu's (1986) extension
Page 7 and 8: We now describe our regularity cond
Page 9: where If m Kn 2 =(2 +1) , K 2 (0 1)
Page 13 and 14: Theorem 3.2 Let =2and Assumptions
Page 15 and 16: By Lemma 5.6 we have where and Um =
Page 17 and 18: To prove (4.18), note that j d d R(
Page 19 and 20: Lemma 5.4 applied to S0m 0( 0) ;1 i
Page 21 and 22: Thus if jm ;1=2Zjj =2 foranyj =0:::
Page 23 and 24: and S1(2 ; S0) = m m S1(1 ; S0) = 2
Page 25 and 26: y Pm j=1 log j = m log m ; m + O(lo
Page 27 and 28: Lemma 5.9 Let the random process (t
Page 29 and 30: Using Lemma 8.1 of Bhattacharya and
Page 31 and 32: For large m, by (6.12), Tr(S 2 )=Va
Page 33 and 34: Denote v = v(x1x2). Then x1 =(;v +
Page 35 and 36: Proof of Lemma 7.1. We show that ES
Page 37 and 38: set of =(V1:::Vk) such thatVs 2V0,
Page 39 and 40: Proof. Note rst that Cum(Zq1Zq2) =
Page 41 and 42: REFERENCES Andrews, D.W.K. and Gugg

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