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A note on conditional covariance matrices for elliptical distributions

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  • Piotr Jaworski
  • Marcin Pitera

Abstract

In this short note we provide an analytical formula for the conditional covariance matrices of the elliptically distributed random vectors, when the conditioning is based on the values of any linear combination of the marginal random variables. We show that one could introduce the univariate invariant depending solely on the conditioning set, which greatly simplifies the calculations. As an application, we show that one could define uniquely defined quantile-based sets on which conditional covariance matrices must be equal to each other if only the vector is multivariate normal. The similar results are obtained for conditional correlation matrices of the general elliptic case.

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  • Piotr Jaworski & Marcin Pitera, 2017. "A note on conditional covariance matrices for elliptical distributions," Papers 1703.00918, arXiv.org.
    • Handle: RePEc:arx:papers:1703.00918
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    References listed on IDEAS

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    1. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    2. Fabrizio Durante & Piotr Jaworski, 2010. "Spatial contagion between financial markets: a copula‐based approach," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 26(5), pages 551-564, September.
    3. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    4. Furman, Edward & Landsman, Zinoviy, 2006. "Tail Variance Premium with Applications for Elliptical Portfolio of Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 433-462, November.
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    Cited by:

    1. Damian Jelito & Marcin Pitera, 2018. "New fat-tail normality test based on conditional second moments with applications to finance," Papers 1811.05464, arXiv.org, revised Apr 2020.
    2. Ansari, Jonathan & Shushi, Tomer & Vanduffel, Steven, 2024. "Up- and down-correlations in normal variance mixture models," Statistics & Probability Letters, Elsevier, vol. 205(C).
    3. Jaworski, Piotr & Pitera, Marcin, 2020. "A note on conditional variance and characterization of probability distributions," Statistics & Probability Letters, Elsevier, vol. 163(C).
    4. Damian Jelito & Marcin Pitera, 2021. "New fat-tail normality test based on conditional second moments with applications to finance," Statistical Papers, Springer, vol. 62(5), pages 2083-2108, October.

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