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Local Polynomial Whittle Estimation of Long-range Dependence

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Abstract

The local Whittle (or Gaussian semiparametric) estimator of long range dependence, proposed by Kunsch (1987) and analyzed by Robinson (1995a), has a relatively slow rate of convergence and a finite sample bias that can be large. In this paper, we generalize the local Whittle estimator to circumvent those problems. Instead of approximating the short-run component of the spectrum, phi(lambda), by a constant in a shrinking neighborhood of frequency zero, we approximate its logarithm by a polynomial. This leads to a "local polynomial Whittle" (LPW) estimator. Following the work of Robinson (1995a), we establish the asymptotic bias, variance, mean-squared error (MSE), and normality of the LPW estimator. We determine the asymptotically MSE-optimal bandwidth, and specify a plug-in selection method for its practical implementation. When phi(lambda) is smooth enough near the origin, we find that the bias of the LPW estimator goes to zero at a faster rate than that of the local Whittle estimator, and its variance is only inflated by a multiplicative constant. In consequence, the rate of convergence of the LPW estimator is faster than that of the local Whittle estimator, given an appropriate choice of the bandwidth m. We show that the LPW estimator attains the optimal rate of convergence for a class of spectra containing those for which varphi(lambda) is smooth of order s > 1 near zero. When phi(lambda) is infinitely smooth near zero, the rate of convergence of the LPW estimator based on a polynomial of high degree is arbitrarily close to n^{-1/2}.

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File URL: http://cowles.econ.yale.edu/P/cd/d12b/d1293.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1293.

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Length: 36 pages
Date of creation: Feb 2001
Date of revision:
Handle: RePEc:cwl:cwldpp:1293

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Asymptotic bias; asymptotic normality; bias reduction; long memory; minimax rate; optimal bandwidth; Whittle likelihood;

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Cited by:
  1. Sun, Yixiao & Phillips, Peter C. B., 2003. "Nonlinear log-periodogram regression for perturbed fractional processes," Journal of Econometrics, Elsevier, vol. 115(2), pages 355-389, August.
  2. Robinson, Peter M. & Henry, Marc, 2003. "Higher-order kernel semiparametric M-estimation of long memory," Journal of Econometrics, Elsevier, vol. 114(1), pages 1-27, May.
  3. 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.
  4. Frank S. Nielsen, 2008. "Local polynomial Whittle estimation covering non-stationary fractional processes," CREATES Research Papers 2008-28, School of Economics and Management, University of Aarhus.
  5. Katsumi Shimotsu, 2006. "Simple (but effective) tests of long memory versus structural breaks," Working Papers 1101, Queen's University, Department of Economics.
  6. Clifford M. Hurvich & Eric Moulines & Philippe Soulier, 2005. "Estimating Long Memory in Volatility," Econometrica, Econometric Society, vol. 73(4), pages 1283-1328, 07.
  7. Katsumi Shimotsu & Peter C.B. Phillips, 2000. "Local Whittle Estimation in Nonstationary and Unit Root Cases," Cowles Foundation Discussion Papers 1266, Cowles Foundation for Research in Economics, Yale University, revised Sep 2003.
  8. 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.
  9. 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.

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