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Adaptive Semiparametric Estimation of the Memory Parameter

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

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.

Suggested Citation

  • Giraitis, Liudas & Robinson, Peter M. & Samarov, Alexander, 2000. "Adaptive Semiparametric Estimation of the Memory Parameter," Journal of Multivariate Analysis, Elsevier, vol. 72(2), pages 183-207, February.
    • Handle: RePEc:eee:jmvana:v:72:y:2000:i:2:p:183-207
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    References listed on IDEAS

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    1. Clifford M. Hurvich & Rohit Deo & Julia Brodsky, 1998. "The mean squared error of Geweke and Porter‐Hudak's estimator of the memory parameter of a long‐memory time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(1), pages 19-46, January.
    2. Velasco, Carlos, 1999. "Non-stationary log-periodogram regression," Journal of Econometrics, Elsevier, vol. 91(2), pages 325-371, August.
    3. 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.
    4. Clifford M. Hurvich & Bonnie K. Ray, 1995. "Estimation Of The Memory Parameter For Nonstationary Or Noninvertible Fractionally Integrated Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(1), pages 17-41, January.
    5. Liudas Giraitis & Peter M. Robinson & Alexander Samarov, 1997. "Rate Optimal Semiparametric Estimation Of The Memory Parameter Of The Gaussian Time Series With Long‐Range Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 18(1), pages 49-60, January.
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    Citations

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    Cited by:

    1. Arteche González, Jesús María & Orbe Lizundia, Jesús María, 2008. "Selection of the number of frequencies using bootstrap techniques in log-periodogram regression," BILTOKI 1134-8984, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    2. 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.
    3. 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.
    4. Liudas Giraitis & Peter M Robinson, 2002. "Edgeworth Expansions for Semiparametric Whittle Estimation of Long Memory," STICERD - Econometrics Paper Series 438, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    5. Yixiao Sun, 2005. "Adaptive Estimation of the Regression Discontinuity Model," Econometrics 0506003, University Library of Munich, Germany.
    6. Grace Yap & Wen Cheong Chin, 2016. "Spectral bandwidth selection for long memory," Modern Applied Science, Canadian Center of Science and Education, vol. 10(8), pages 1-63, August.
    7. Giraitis, Liudas & Robinson, Peter, 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.
    8. J. Arteche, 2012. "Semiparametric Inference in Correlated Long Memory Signal Plus Noise Models," Econometric Reviews, Taylor & Francis Journals, vol. 31(4), pages 440-474.
    9. Duncan A J Blythe & Vadim V Nikulin, 2017. "Long-range temporal correlations in neural narrowband time-series arise due to critical dynamics," PLOS ONE, Public Library of Science, vol. 12(5), pages 1-28, May.
    10. Arteche, J., 2006. "Semiparametric estimation in perturbed long memory series," Computational Statistics & Data Analysis, Elsevier, vol. 51(4), pages 2118-2141, December.
    11. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2007. "Nonstationarity-extended local Whittle estimation," Journal of Econometrics, Elsevier, vol. 141(2), pages 1353-1384, December.
    12. Valdério A. Reisen & Eric Moulines & Philippe Soulier & Glaura C. Franco, 2010. "On the properties of the periodogram of a stationary long‐memory process over different epochs with applications," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(1), pages 20-36, January.
    13. Giraitis, L. & Robinson, P.M., 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.
    14. Bardet Jean-Marc & Dola Béchir, 2016. "Semiparametric Stationarity and Fractional Unit Roots Tests Based on Data-Driven Multidimensional Increment Ratio Statistics," Journal of Time Series Econometrics, De Gruyter, vol. 8(2), pages 115-153, July.
    15. 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.
    16. Josu Arteche & Jesus Orbe, 2009. "Bootstrap‐based bandwidth choice for log‐periodogram regression," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(6), pages 591-617, November.
    17. Saeed Heravi & Kerry Patterson, 2005. "Optimal And Adaptive Semi‐Parametric Narrowband And Broadband And Maximum Likelihood Estimation Of The Long‐Memory Parameter For Real Exchange Rates," Manchester School, University of Manchester, vol. 73(2), pages 165-213, March.

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