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Estimating the Marginal Law of a Time Series with Applications to Heavy Tailed Distributions

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Author Info

  • Christian Francq

    ()
    (CREST)

  • Jean-Michel Zakoïan

    ()
    (CREST)

Abstract

This article addresses estimating parametric marginal densities of stationary time series in the absence of precise information on the dynamics of the underlying process. We propose using an estimator obtained by maximization of the "quasi-marginal" likelihood, which is a likelihood written as if the observations were independent. We study the effect of the (neglected) dynamics on the asymptotic behavior of this estimator. The consistency and asymptotic normality of the estimator are established under mild assumptions on the dependence structure. Applications of the asymptotic results to the estimation of stable, generalized extreme value and generalized Pareto distributions are proposed. The theoretical results are illustrated on financial index returns. Supplementary materials for this article are available online.

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Bibliographic Info

Paper provided by Centre de Recherche en Economie et Statistique in its series Working Papers with number 2011-30.

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Length: 50
Date of creation: 2011
Date of revision:
Handle: RePEc:crs:wpaper:2011-30

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Related research

Keywords: alpha-stable distribution; composite likelihood; GEV distribution; GPD; pseudo-likelihood; quasi-marginal maximum likelihood; stock returns distributions;

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References

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  1. Shiqing Ling & Michael McAleer, 2010. "A general asymptotic theory for time-series models," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(1), pages 97-111.
  2. Cotter, John, 2007. "Varying the VaR for unconditional and conditional environments," Journal of International Money and Finance, Elsevier, vol. 26(8), pages 1338-1354, December.
  3. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
  4. Francq, Christian & Roy, Roch & Zakoian, Jean-Michel, 2005. "Diagnostic Checking in ARMA Models With Uncorrelated Errors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 532-544, June.
  5. Boubacar Mainassara, Y. & Carbon, M. & Francq, C., 2012. "Computing and estimating information matrices of weak ARMA models," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 345-361.
  6. Dennis Jansen & Casper de Vries, 1988. "On the frequency of large stock returns: putting booms and busts into perspective," Working Papers 1989-006, Federal Reserve Bank of St. Louis.
  7. Gamini Premaratne, 2005. "A Test for Symmetry with Leptokurtic Financial Data," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 3(2), pages 169-187.
  8. Stephen J. Taylor, 2007. "Introduction to Asset Price Dynamics, Volatility, and Prediction
    [Asset Price Dynamics, Volatility, and Prediction]
    ," Introductory Chapters, Princeton University Press.
  9. Einmahl, John H. J. & Li, Jun & Liu, Regina Y., 2009. "Thresholding Events of Extreme in Simultaneous Monitoring of Multiple Risks," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 982-992.
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Cited by:
  1. Auray, Stéphane & Eyquem, Aurélien & Jouneau-Sion, Frédéric, 2014. "Modeling tails of aggregate economic processes in a stochastic growth model," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 76-94.

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