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Edgeworth expansions for spectral density estimates and studentized sample mean

By Carlos Velasco and Peter M. Robinson

Abstract

We establish valid Edgeworth expansions for the distribution of smoothed nonparametric spectral estimates, and of studentized versions of linear statistics such as the sample mean, where the studentization employs such a nonparametric spectral estimate. Particular attention is paid to the spectral estimate at zero frequency and, correspondingly, the studentized sample mean, to reflect econometric interest in autocorrelation-consistent or long-run variance estimation. Our main focus is on stationary Gaussian series, though we discuss relaxation of the Gaussianity assumption. Only smoothness conditions on the spectral density that are local to the frequency of interest are imposed. We deduce empirical expansions from our Edgeworth expansions designed to improve on the normal approximation in practice and also deduce a feasible rule of bandwidth choice

Topics: HB Economic Theory
Publisher: Cambridge University Press
Year: 2001
OAI identifier: oai:eprints.lse.ac.uk:315
Provided by: LSE Research Online
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    Citations

    1. 1+ Also, from the continuity of YN, we can obtain, for some c . 0,
    2. (1972). 1958! An Introduction to Multivariate Statistical Analysis.
    3. are fixed constants independent of N or M, we can write for some constants Jj~s!
    4. (1987). B~0,rN! we have by Fourier inversion 6~PN 2QN ~2!! ,
    5. eN, where eN 5 O~M22 1 eN~2!!+ Therefore we can write
    6. following Bentkus and Rudzkis ~1982! we study the characteristic function of the spectral density estimate, which itself appears in the joint characteristic function+ Define t~t2!5E@exp$it2u2%# 5 t'~t2!exp$2it2E%, where t'~t2!
    7. HN~m!AN ~2s!~m!FN ~4s12!~m!dm, (B.33) say, where we have changed variables as
    8. M ND (B.21) for some polynomial F+ Now to study g, we first bound
    9. (1975). O~MlogN!, and the integrals with respect to the remaining variables can be bounded in the same way, ~B+18! being of order O~N21Ms11log2s11N! again+ Therefore, from ~B+16!, ~B+17! and the previous discussion for ~B+18!, the proposition follows+
    10. on choosing db sufficiently small+ Now for a we can choose a da . 0 so small that, for 7t7 #
    11. Proof of Lemma 14. Similarly to Feller ~1971, p+ 535! we have for complex a and b that 6ea 21 2
    12. sN 2VN 2MN~2c1 2qN 2!21, and ~noting that NM0qN 2 r `, for all p .
    13. where we include in BN , the expansions for the corresponding cumulants up to the order M2d, , but in O BN , only the leading terms are kept, so

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