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Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case

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  • CLAUDIA KLÜPPELBERG
  • GABRIEL KUHN
  • LIANG PENG

Abstract

. In general, the risk of joint extreme outcomes in financial markets can be expressed as a function of the tail dependence function of a high‐dimensional vector after standardizing marginals. Hence, it is of importance to model and estimate tail dependence functions. Even for moderate dimension, non‐parametrically estimating a tail dependence function is very inefficient and fitting a parametric model to tail dependence functions is not robust. In this paper, we propose a semi‐parametric model for (asymptotically dependent) tail dependence functions via an elliptical copula. Under this model assumption, we propose a novel estimator for the tail dependence function, which proves favourable compared to the empirical tail dependence function estimator, both theoretically and empirically.

Suggested Citation

  • Claudia Klüppelberg & Gabriel Kuhn & Liang Peng, 2008. "Semi‐Parametric Models for the Multivariate Tail Dependence Function – the Asymptotically Dependent Case," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 701-718, December.
    • Handle: RePEc:bla:scjsta:v:35:y:2008:i:4:p:701-718
    DOI: 10.1111/j.1467-9469.2008.00602.x
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    References listed on IDEAS

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    1. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    2. Einmahl, J.H.J. & Krajina, A. & Segers, J.J.J., 2007. "A Method of Moments Estimator of Tail Dependence," Other publications TiSEM 6ee60ab8-3c01-4bd9-aa5e-7, Tilburg University, School of Economics and Management.
    3. Ferreira, Helena & Ferreira, Marta, 2012. "Tail dependence between order statistics," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 176-192.
    4. Opitz, T., 2013. "Extremal t processes: Elliptical domain of attraction and a spectral representation," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 409-413.
    5. Joe, Harry & Li, Haijun & Nikoloulopoulos, Aristidis K., 2010. "Tail dependence functions and vine copulas," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 252-270, January.
    6. Antonio Dalessandro & Gareth W. Peters, 2020. "Efficient and Accurate Evaluation Methods for Concordance Measures via Functional Tensor Characterizations of Copulas," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1089-1124, September.
    7. Tankov, Peter, 2016. "Tails of weakly dependent random vectors," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 73-86.
    8. V'eronique Maume-Deschamps & Didier Rulli`ere & Khalil Said, 2017. "Asymptotic multivariate expectiles," Papers 1704.07152, arXiv.org, revised Jan 2018.
    9. Falk, Michael & Padoan, Simone A. & Wisheckel, Florian, 2019. "Generalized Pareto copulas: A key to multivariate extremes," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    10. Krajina, A., 2010. "An M-estimator of multivariate tail dependence," Other publications TiSEM 66518e07-db9a-4446-81be-c, Tilburg University, School of Economics and Management.
    11. Hua, Lei & Joe, Harry, 2011. "Tail order and intermediate tail dependence of multivariate copulas," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1454-1471, November.
    12. Li, Deyuan & Peng, Liang, 2009. "Goodness-of-fit test for tail copulas modeled by elliptical copulas," Statistics & Probability Letters, Elsevier, vol. 79(8), pages 1097-1104, April.
    13. Lorenzo Ricci & David Veredas, 2012. "TailCoR," Working Papers 1227, Banco de España.
      • Sla{dj}ana Babi'c & Christophe Ley & Lorenzo Ricci & David Veredas, 2020. "TailCoR," Papers 2011.14817, arXiv.org.
    14. Packham, Natalie & Kalkbrener, Michael & Overbeck, Ludger, 2014. "Default probabilities and default correlations under stress," Frankfurt School - Working Paper Series 211, Frankfurt School of Finance and Management.
    15. Krajina, A., 2009. "A Method of Moments Estimator of Tail Dependence in Elliptical Copula Models," Discussion Paper 2009-42, Tilburg University, Center for Economic Research.
    16. Derumigny, A. & Fermanian, J.-D., 2022. "Identifiability and estimation of meta-elliptical copula generators," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    17. Asimit, Alexandru V. & Gerrard, Russell & Hou, Yanxi & Peng, Liang, 2016. "Tail dependence measure for examining financial extreme co-movements," Journal of Econometrics, Elsevier, vol. 194(2), pages 330-348.
    18. Mendes, Beatriz Vaz de Melo & Accioly, Victor Bello, 2012. "On the dependence structure of realized volatilities," International Review of Financial Analysis, Elsevier, vol. 22(C), pages 1-9.
    19. Durante, Fabrizio & Jaworski, Piotr & Mesiar, Radko, 2011. "Invariant dependence structures and Archimedean copulas," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1995-2003.
    20. Jaser Miriam & Haug Stephan & Min Aleksey, 2017. "A simple non-parametric goodness-of-fit test for elliptical copulas," Dependence Modeling, De Gruyter, vol. 5(1), pages 330-353, December.
    21. Choroś-Tomczyk, Barbara & Härdle, Wolfgang Karl & Okhrin, Ostap, 2013. "Valuation of collateralized debt obligations with hierarchical Archimedean copulae," Journal of Empirical Finance, Elsevier, vol. 24(C), pages 42-62.
    22. Krajina, A., 2009. "A Method of Moments Estimator of Tail Dependence in Elliptical Copula Models," Other publications TiSEM f3f5a961-02ff-4a2b-ab93-4, Tilburg University, School of Economics and Management.

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