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Nonparametric inference of discretely sampled stable Lévy processes

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  • Zhao, Zhibiao
  • Wu, Wei Biao

Abstract

We study nonparametric inference of stochastic models driven by stable Lévy processes. We introduce a nonparametric estimator of the stable index that achieves the parametric rate of convergence. For the volatility function, due to the heavy-tailedness, the classical least-squares method is not applicable. We then propose a nonparametric least-absolute-deviation or median-quantile estimator and study its asymptotic behavior, including asymptotic normality and maximal deviations, by establishing a representation of Bahadur-Kiefer type. The result is applied to several major foreign exchange rates.

Suggested Citation

  • Zhao, Zhibiao & Wu, Wei Biao, 2009. "Nonparametric inference of discretely sampled stable Lévy processes," Journal of Econometrics, Elsevier, vol. 153(1), pages 83-92, November.
    • Handle: RePEc:eee:econom:v:153:y:2009:i:1:p:83-92
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    References listed on IDEAS

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    1. Wei Biao Wu & Zhibiao Zhao, 2007. "Inference of trends in time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(3), pages 391-410, June.
    2. Press, S. J., 1972. "Multivariate stable distributions," Journal of Multivariate Analysis, Elsevier, vol. 2(4), pages 444-462, December.
    3. Zhang, Lan & Mykland, Per A. & Ait-Sahalia, Yacine, 2005. "A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1394-1411, December.
    4. Yacine Aït-Sahalia & Jean Jacod, 2008. "Fisher's Information for Discretely Sampled Lévy Processes," Econometrica, Econometric Society, vol. 76(4), pages 727-761, July.
    5. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731.
    6. de Haan, L. & Pereira, T. Themido, 1999. "Estimating the index of a stable distribution," Statistics & Probability Letters, Elsevier, vol. 41(1), pages 39-55, January.
    7. Foster, Dean P & Nelson, Daniel B, 1996. "Continuous Record Asymptotics for Rolling Sample Variance Estimators," Econometrica, Econometric Society, vol. 64(1), pages 139-174, January.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    9. Stanton, Richard, 1997. "A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk," Journal of Finance, American Finance Association, vol. 52(5), pages 1973-2002, December.
    10. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    11. Westerfield, Janice Moulton, 1977. "An examination of foreign exchange risk under fixed and floating rate regimes," Journal of International Economics, Elsevier, vol. 7(2), pages 181-200, May.
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    Cited by:

    1. Stefan Birr & Stanislav Volgushev & Tobias Kley & Holger Dette & Marc Hallin, 2017. "Quantile spectral analysis for locally stationary time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1619-1643, November.
    2. Hounyo, Ulrich & Varneskov, Rasmus T., 2017. "A local stable bootstrap for power variations of pure-jump semimartingales and activity index estimation," Journal of Econometrics, Elsevier, vol. 198(1), pages 10-28.
    3. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "A data-driven framework for consistent financial valuation and risk measurement," European Journal of Operational Research, Elsevier, vol. 289(1), pages 381-398.
    4. Jing, Bing-Yi & Kong, Xin-Bing & Liu, Zhi & Mykland, Per, 2012. "On the jump activity index for semimartingales," Journal of Econometrics, Elsevier, vol. 166(2), pages 213-223.
    5. Adam D. Bull, 2014. "Near-optimal estimation of jump activity in semimartingales," Papers 1409.8150, arXiv.org, revised Jan 2016.
    6. Fabian Mies & Ansgar Steland, 2019. "Nonparametric Gaussian inference for stable processes," Statistical Inference for Stochastic Processes, Springer, vol. 22(3), pages 525-555, October.
    7. repec:wyi:journl:002150 is not listed on IDEAS
    8. Fabienne Comte & Céline Duval & Valentine Genon-Catalot, 2014. "Nonparametric density estimation in compound Poisson processes using convolution power estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 163-183, January.

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