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

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  • Christian Francq

    (CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, IP Paris - Institut Polytechnique de Paris)

  • Jean-Michel Zakoïan

    (LFA - Laboratoire de Finance Assurance - Centre de Recherche en Économie et Statistique (CREST) - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique, EQUIPPE - Economie Quantitative, Intégration, Politiques Publiques et Econométrie - Université de Lille, Sciences et Technologies - Université de Lille, Sciences Humaines et Sociales - PRES Université Lille Nord de France - Université de Lille, Droit et Santé)

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.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Christian Francq & Jean-Michel Zakoïan, 2013. "Estimating the Marginal Law of a Time Series With Applications to Heavy-Tailed Distributions," Post-Print hal-05417510, HAL.
    • Handle: RePEc:hal:journl:hal-05417510
    DOI: 10.1080/07350015.2013.801776
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    2. Lin Zhang & Harry Joe & Natalia Nolde, 2024. "Margin‐closed vector autoregressive time series models," Journal of Time Series Analysis, Wiley Blackwell, vol. 45(2), pages 269-297, March.
    3. 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.
    4. Fries, Sébastien & Zakoian, Jean-Michel, 2019. "Mixed Causal-Noncausal Ar Processes And The Modelling Of Explosive Bubbles," Econometric Theory, Cambridge University Press, vol. 35(6), pages 1234-1270, December.
    5. Delaigle, Aurore & Meister, Alexander & Rombouts, Jeroen, 2016. "Root-T consistent density estimation in GARCH models," Journal of Econometrics, Elsevier, vol. 192(1), pages 55-63.
    6. Echaust Krzysztof, 2014. "A Comparison of Tail Behaviour of Stock Market Returns," Folia Oeconomica Stetinensia, Sciendo, vol. 14(1), pages 22-34, June.
    7. Christian H. Weiß, 2018. "Goodness-of-fit testing of a count time series’ marginal distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(6), pages 619-651, August.
    8. Valentin Courgeau & Almut E.D. Veraart, 2022. "Asymptotic theory for the inference of the latent trawl model for extreme values," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(4), pages 1448-1495, December.
    9. Zhang, Xingfa & Zhang, Rongmao & Li, Yuan & Ling, Shiqing, 2022. "LADE-based inferences for autoregressive models with heavy-tailed G-GARCH(1, 1) noise," Journal of Econometrics, Elsevier, vol. 227(1), pages 228-240.
    10. Yang, Yaxing & Ling, Shiqing, 2017. "Self-weighted LAD-based inference for heavy-tailed threshold autoregressive models," Journal of Econometrics, Elsevier, vol. 197(2), pages 368-381.
    11. Preminger, Arie & Storti, Giuseppe, 2014. "Least squares estimation for GARCH (1,1) model with heavy tailed errors," MPRA Paper 59082, University Library of Munich, Germany.

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