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

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  • Christian Francq
  • Jean-Michel Zakoïan

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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  • Christian Francq & Jean-Michel Zakoïan, 2013. "Estimating the Marginal Law of a Time Series With Applications to Heavy-Tailed Distributions," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(4), pages 412-425, October.
    • Handle: RePEc:taf:jnlbes:v:31:y:2013:i:4:p:412-425
    DOI: 10.1080/07350015.2013.801776
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    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Echaust Krzysztof, 2014. "A Comparison of Tail Behaviour of Stock Market Returns," Folia Oeconomica Stetinensia, Sciendo, vol. 14(1), pages 22-34, June.
    9. 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.

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