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Adaptive Semiparametric Estimation of the Memory Parameter


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  • Giraitis, Liudas
  • Robinson, Peter M.
  • Samarov, Alexander


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.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 72 (2000)
Issue (Month): 2 (February)
Pages: 183-207

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Handle: RePEc:eee:jmvana:v:72:y:2000:i:2:p:183-207

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Keywords: long range dependence; semiparametric model; rates of convergence; adaptive bandwidth selection;


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  1. Lobato, Ignacio N., 1999. "A semiparametric two-step estimator in a multivariate long memory model," Journal of Econometrics, Elsevier, vol. 90(1), pages 129-153, May.
  2. Velasco, Carlos, 1999. "Non-stationary log-periodogram regression," Journal of Econometrics, Elsevier, vol. 91(2), pages 325-371, August.
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Cited by:
  1. Liudas Giraitis & Peter Robinson, 2002. "Edgeworth expansions for semiparametric Whittle estimation of long memory," LSE Research Online Documents on Economics 2130, London School of Economics and Political Science, LSE Library.
  2. Arteche, Josu & Orbe, Jesus, 2009. "Using the bootstrap for finite sample confidence intervals of the log periodogram regression," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 1940-1953, April.
  3. Hurvich, Clifford M. & Moulines, Eric & Soulier, Philippe, 2002. "The FEXP estimator for potentially non-stationary linear time series," Stochastic Processes and their Applications, Elsevier, vol. 97(2), pages 307-340, February.
  4. Yixiao Sun, 2005. "Adaptive Estimation of the Regression Discontinuity Model," Econometrics 0506003, EconWPA.
  5. L. Giraitis & P.M. Robinson, 2003. "Edgeworth expansions for semiparametric Whittle estimation of long memory," LSE Research Online Documents on Economics 291, London School of Economics and Political Science, LSE Library.
  6. Arteche González, Jesús María, 2005. "Semiparametric estimation in perturbed long memory series," BILTOKI 2005-02, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
  7. Liudas Giraitis & Peter M Robinson, 2002. "Edgeworth Expansions for Semiparametric Whittle Estimation of Long Memory," STICERD - Econometrics Paper Series /2002/438, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  8. Masaki Narukawa & Yasumasa Matsuda, 2008. "Broadband semiparametric estimation of the long-memory parameter by the likelihood-based FEXP approach," TERG Discussion Papers 239, Graduate School of Economics and Management, Tohoku University.


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