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Consistent estimation of the memory parameter for nonlinear time series

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  • Violetta Dalla
  • Liudas Giraitis
  • Javier Hidalgo

Abstract

For linear processes, semiparametric estimation of the memory parameter, based on the log-periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process xi_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite-sample performance of the estimator is investigated in a small Monte Carlo study. Copyright 2005 Blackwell Publishing Ltd.

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Bibliographic Info

Article provided by Wiley Blackwell in its journal Journal of Time Series Analysis.

Volume (Year): 27 (2006)
Issue (Month): 2 (03)
Pages: 211-251

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Handle: RePEc:bla:jtsera:v:27:y:2006:i:2:p:211-251

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  1. Yixiao Sun & Peter C.B. Phillips, 2002. "Nonlinear Log-Periodogram Regression for Perturbed Fractional Processes," Cowles Foundation Discussion Papers 1366, Cowles Foundation for Research in Economics, Yale University.
  2. Ghysels, E. & Harvey, A. & Renault, E., 1996. "Stochastic Volatility," Cahiers de recherche 9613, Universite de Montreal, Departement de sciences economiques.
  3. Donald W.K. Andrews & Yixiao Sun, 2002. "Adaptive Local Polynomial Whittle Estimation of Long-range Dependence," Cowles Foundation Discussion Papers 1384, Cowles Foundation for Research in Economics, Yale University.
  4. Peter M Robinson & Carlos Velasco, 2000. "Whittle Pseudo-Maximum Likelihood Estimation for Nonstationary Time Series - (Now published in Journal of the American Statistical Association, 95, (2000), pp.1229-1243.)," STICERD - Econometrics Paper Series /2000/391, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  5. Breidt, F. Jay & Crato, Nuno & de Lima, Pedro, 1998. "The detection and estimation of long memory in stochastic volatility," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 325-348.
  6. Robinson, P. M., 2001. "The memory of stochastic volatility models," Journal of Econometrics, Elsevier, vol. 101(2), pages 195-218, April.
  7. Arteche, Josu, 2004. "Gaussian semiparametric estimation in long memory in stochastic volatility and signal plus noise models," Journal of Econometrics, Elsevier, vol. 119(1), pages 131-154, March.
  8. Deo, Rohit S. & Hurvich, Clifford M., 2001. "On The Log Periodogram Regression Estimator Of The Memory Parameter In Long Memory Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 17(04), pages 686-710, August.
  9. Harvey, Andrew & Ruiz, Esther & Shephard, Neil, 1994. "Multivariate Stochastic Variance Models," Review of Economic Studies, Wiley Blackwell, vol. 61(2), pages 247-64, April.
  10. Clifford Hurvich & Eric Moulines & Philippe Soulier, 2004. "Estimating Long Memory in Volatility," Econometrics 0412006, EconWPA.
  11. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, vol. 59(2), pages 347-70, March.
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Cited by:
  1. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2007. "Nonstationarity-extended local Whittle estimation," Journal of Econometrics, Elsevier, vol. 141(2), pages 1353-1384, December.
  2. Frank S. Nielsen, 2009. "Local Whittle estimation of multivariate fractionally integrated processes," CREATES Research Papers 2009-38, School of Economics and Management, University of Aarhus.
  3. George Kapetanios & Fotis Papailias, 2011. "Block Bootstrap and Long Memory," Working Papers 679, Queen Mary, University of London, School of Economics and Finance.
  4. Francis W. Ahking, 2004. "Non-Parametric Tests of Real Exchange rates in the Post-Bretton Woods Era," Working papers 2004-05, University of Connecticut, Department of Economics.
  5. Marco Avarucci & Domenico Marinucci, 2005. "Polynomial Cointegration Among Stationary Processes With Long Memory," Economics Working Papers we055123, Universidad Carlos III, Departamento de Economía.
  6. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2009. "Two estimators of the long-run variance: Beyond short memory," Journal of Econometrics, Elsevier, vol. 150(1), pages 56-70, May.
  7. Faÿ, Gilles & Moulines, Eric & Roueff, François & Taqqu, Murad S., 2009. "Estimators of long-memory: Fourier versus wavelets," Journal of Econometrics, Elsevier, vol. 151(2), pages 159-177, August.
  8. Michael Levine & Soledad Torres & Frederi Viens, 2009. "Estimators for the long-memory parameter in LARCH models, and fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 12(3), pages 221-250, October.
  9. Hira Koul & Nao Mimoto & Donatas Surgailis, 2013. "Goodness-of-fit tests for long memory moving average marginal density," Metrika, Springer, vol. 76(2), pages 205-224, February.
  10. Karim M. Abadir & Walter Distaso & Liudas Giraitis, 2011. "An I() model with trend and cycles," Post-Print peer-00834425, HAL.
  11. Richard T. Baillie & George Kapetanios, 2006. "Nonlinear Models with Strongly Dependent Processes and Applications to Forward Premia and Real Exchange Rates," Working Papers 570, Queen Mary, University of London, School of Economics and Finance.
  12. 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.

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