Adaptive Semiparametric Estimation of the Memory Parameter

Liudas Giraitis (), Peter M. Robinson and Alexander Samarov

Journal of Multivariate Analysis, 2000, vol. 72, issue 2, 183-207

Abstract: In Giraitis, Robinson, and Samarov (1997), we have shown that the optimal rate for memory parameter estimators in semiparametric long memory models with degree of "local smoothness" [beta] is n-r([beta]), r([beta])=[beta]/(2[beta]+1), and that a log-periodogram regression estimator (a modified Geweke and Porter-Hudak (1983) estimator) with maximum frequency m=m([beta])[asymptotically equal to]n2r([beta]) is rate optimal. The question which we address in this paper is what is the best obtainable rate when [beta] is unknown, so that estimators cannot depend on [beta]. We obtain a lower bound for the asymptotic quadratic risk of any such adaptive estimator, which turns out to be larger than the optimal nonadaptive rate n-r([beta]) by a logarithmic factor. We then consider a modified log-periodogram regression estimator based on tapered data and with a data-dependent maximum frequency m=m([beta]), which depends on an adaptively chosen estimator [beta] of [beta], and show, using methods proposed by Lepskii (1990) in another context, that this estimator attains the lower bound up to a logarithmic factor. On one hand, this means that this estimator has nearly optimal rate among all adaptive (free from [beta]) estimators, and, on the other hand, it shows near optimality of our data-dependent choice of the rate of the maximum frequency for the modified log-periodogram regression estimator. The proofs contain results which are also of independent interest: one result shows that data tapering gives a significant improvement in asymptotic properties of covariances of discrete Fourier transforms of long memory time series, while another gives an exponential inequality for the modified log-periodogram regression estimator.

Keywords: long range dependence; semiparametric model; rates of convergence; adaptive bandwidth selection (search for similar items in EconPapers)
Date: 2000
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Citations: View citations in EconPapers (17)

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