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Estimators of long-memory: Fourier versus wavelets


  • Faÿ, Gilles
  • Moulines, Eric
  • Roueff, François
  • Taqqu, Murad S.


Semi-parametric estimation methods of the long-memory exponent of a time series have been studied in several papers, some applied, others theoretical, some using Fourier methods, others using a wavelet-based technique. In this paper, we compare the Fourier and wavelet approaches to the local regression method and to the local Whittle method. We provide an overview of these methods, describe what has been done and indicate the available results and the conditions under which they hold. We discuss their relative strengths and weaknesses both from a practical and a theoretical perspective. We also include a simulation-based comparison. The software written to support this work is available on demand and we illustrate its use at the end of the paper.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:econom:v:151:y:2009:i:2:p:159-177

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    References listed on IDEAS

    1. Deo, Rohit & Hurvich, Clifford & Lu, Yi, 2006. "Forecasting realized volatility using a long-memory stochastic volatility model: estimation, prediction and seasonal adjustment," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 29-58.
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    6. Violetta Dalla & Liudas Giraitis & Javier Hidalgo, 2006. "Consistent estimation of the memory parameter for nonlinear time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(2), pages 211-251, March.
    7. Hurvich, Clifford & Lang, Gabriel & Soulier, Philippe, 2005. "Estimation of Long Memory in the Presence of a Smooth Nonparametric Trend," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 853-871, September.
    8. J. Bardet & G. Lang & E. Moulines & P. Soulier, 2000. "Wavelet Estimator of Long-Range Dependent Processes," Statistical Inference for Stochastic Processes, Springer, vol. 3(1), pages 85-99, January.
    9. Donald W. K. Andrews & Yixiao Sun, 2004. "Adaptive Local Polynomial Whittle Estimation of Long-range Dependence," Econometrica, Econometric Society, vol. 72(2), pages 569-614, March.
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    11. E. Moulines & F. Roueff & M. S. Taqqu, 2007. "On the Spectral Density of the Wavelet Coefficients of Long-Memory Time Series with Application to the Log-Regression Estimation of the Memory Parameter," Journal of Time Series Analysis, Wiley Blackwell, vol. 28(2), pages 155-187, March.
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    Cited by:

    1. Hailin Sang & Yongli Sang, 2017. "Memory properties of transformations of linear processes," Statistical Inference for Stochastic Processes, Springer, vol. 20(1), pages 79-103, April.
    2. Jean-Christophe Breton & Jean-François Coeurjolly, 2012. "Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size," Statistical Inference for Stochastic Processes, Springer, vol. 15(1), pages 1-26, April.
    3. Barunik, Jozef & Barunikova, Michaela, 2015. "Revisiting the long memory dynamics of implied-realized volatility relation: A new evidence from wavelet band spectrum regression," FinMaP-Working Papers 43, Collaborative EU Project FinMaP - Financial Distortions and Macroeconomic Performance: Expectations, Constraints and Interaction of Agents.
    4. repec:gam:jecnmx:v:6:y:2018:i:1:p:13-:d:135826 is not listed on IDEAS
    5. Michis, Antonis A., 2014. "Time scale evaluation of economic forecasts," Economics Letters, Elsevier, vol. 123(3), pages 279-281.
    6. Baruník, Jozef & Kočenda, Evžen & Vácha, Lukáš, 2016. "Gold, oil, and stocks: Dynamic correlations," International Review of Economics & Finance, Elsevier, vol. 42(C), pages 186-201.
    7. Roueff, François & von Sachs, Rainer, 2011. "Locally stationary long memory estimation," Stochastic Processes and their Applications, Elsevier, vol. 121(4), pages 813-844, April.
    8. Kraicová Lucie & Baruník Jozef, 2017. "Estimation of long memory in volatility using wavelets," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 21(3), pages 1-22, June.
    9. Heni Boubaker & Anne Péguin-Feissolle, 2013. "Estimating the Long-Memory Parameter in Nonstationary Processes Using Wavelets," Computational Economics, Springer;Society for Computational Economics, vol. 42(3), pages 291-306, October.
    10. Jozef Barunik & Michaela Barunikova, 2012. "Revisiting the fractional cointegrating dynamics of implied-realized volatility relation with wavelet band spectrum regression," Papers 1208.4831,, revised Feb 2013.
    11. Zhang, Pu & Sun, Qi & Xiao, Wei-Lin, 2014. "Parameter identification in mixed Brownian–fractional Brownian motions using Powell's optimization algorithm," Economic Modelling, Elsevier, vol. 40(C), pages 314-319.
    12. Roueff, F. & Taqqu, M.S., 2009. "Central limit theorems for arrays of decimated linear processes," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 3006-3041, September.
    13. M. Ege Yazgan & Hakan Yilmazkuday, 2016. "High versus low inflation: implications for price-level convergence," Empirical Economics, Springer, vol. 50(4), pages 1527-1563, June.
    14. Baruník, Jozef & Hlínková, Michaela, 2016. "Revisiting the long memory dynamics of the implied–realized volatility relationship: New evidence from the wavelet regression," Economic Modelling, Elsevier, vol. 54(C), pages 503-514.
    15. Kei Nanamiya, 2014. "Modelling For The Wavelet Coefficients Of Arfima Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(4), pages 341-356, July.
    16. Dominique Guegan & Zhiping Lu & Beijia Zhu, 2012. "Comparaison of Several Estimation Procedures for Long Term Behavior," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00673934, HAL.
    17. Linyuan Li & Kewei Lu, 2013. "On rate-optimal nonparametric wavelet regression with long memory moving average errors," Statistical Inference for Stochastic Processes, Springer, vol. 16(2), pages 127-145, July.
    18. Mohamed Boutahar & Rabeh Khalfaoui2, 2011. "Estimation of the long memory parameter in non stationary models: A Simulation Study," Working Papers halshs-00595057, HAL.
    19. Sophie Achard & Irène Gannaz, 2016. "Multivariate Wavelet Whittle Estimation in Long-range Dependence," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(4), pages 476-512, July.
    20. Marie Busch & Philipp Sibbertsen, 2018. "An Overview of Modified Semiparametric Memory Estimation Methods," Econometrics, MDPI, Open Access Journal, vol. 6(1), pages 1-21, March.
    21. F. Roueff & M. S. Taqqu, 2009. "Asymptotic normality of wavelet estimators of the memory parameter for linear processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(5), pages 534-558, September.
    22. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xi-Li & Wang, Ying-Luo, 2010. "Pricing currency options in a fractional Brownian motion with jumps," Economic Modelling, Elsevier, vol. 27(5), pages 935-942, September.
    23. Thomas Conlon & John Cotter & Ramazan Gençay, 2015. "Long-run international diversification," Working Papers 201502, Geary Institute, University College Dublin.


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