Gaussian copula of stable random vectors and application

Year 2020, , 887 - 901, 02.04.2020

Phuc Ho Dang , Truc Giang Vo Thi

https://doi.org/10.15672/hujms.621919

Abstract

In this paper, we present a new method to investigate data of multivariate heavy-tailed distributions. We show that for any given number $\alpha \in (0;2]$, each Gaussian copula is also the copula of an $\alpha$-stable random vector. Simultaneously, every random vector is $\alpha$-stable if its marginals are $\alpha$-stable and its copula is a Gaussian copula. The result is used to build up a formula representing density functions of $\alpha$-stable random vectors with Gaussian copula. Adopting a new tool, the paper points out that pairs of GPS signals recording latitude and longitude of a fixed point have two-dimensional stable distribution, and in the most of cases, vectors of daily returns in stock market data have multivariate stable distributions with Gaussian copulas.

Keywords

stable distributions, multivariate density function, GPSdata, stock market, portfolio selection

Supporting Institution

NAFOSTED, the Vietnam National Foundation for Sciences and Technology Development

Project Number

101.03-2017.07.

Thanks

The study was partially supported by NAFOSTED, the Vietnam National Foundation for Sciences and Technology Development, under Grant number 101.03-2017.07. Thanks are due to Dr. Bui Quang Nam for the sharing the GPS data used in the application part of this study, and also to anonymous reviewers who gave valuable comments to make the article improved.

References

• [1] R. J. Adler, R. E. Feldman, M. S Taqqu. A Practical Guide to Heavy Tailed Data, Birkhäuser, Boston, 1998.
• [2] N. Bui Quang. On stable probability distributions and statistical application, Thesis, Academy of military science and technology, Ha Noi, 2016.
• [3] P. Embrechts, F. Lindskog, A. McNeil. Modelling Dependence with Copulas and Applications to Risk Management, Handbook of Heavy Tailed Distributions in Finance, 2003, ed. Rachev S., Elsevier, Chapter 8, 329–384, 2001.
• [4] E. Fama. The behavior of stock prices, Journal of Business 38, 34–105, 1965.
• [5] C. Genest, B. Rémillard. Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models, Annales de lInstitut Henri Poincaré - Probabilités et Statistiques 44 no. 6, 1096–1127, 2008.
• [6] S. M. Kogon, D. B. Williams. Characteristic function based estimation of stable parameters, In Adler R., Feldman R. and Taqqu M. (eds.) A Practical Guide to Heavy Tailed Data, Birkhäuser, Boston, MA, 311–335, 1998.
• [7] I. Kojadinovic, J. Yan. Modeling Multivariate Distributions with Continuous Margins Using the copula R Package, J. Stat. Softw. 34 no. 9, 1–20, 2010.
• [8] R. M. Kunst. Apparently stable increments in finance data: Could ARCH effects be the cause?, J. Statist. Comput. Simulation 45, 121–127, 1993.
• [9] F. Lamantia, S. Ortobelli, S. Rachev. VaR, CVaR and Time Rules with Elliptical and Asymmetric Stable Distributed Returns, Investment Management and Financial Innovations 3, Issue 4, 19–39, 2006.
• [10] J. H. McCulloch. Simple consistent estimators of stable distribution parameters, Comm. Statist. Simulation Comput. 15, 1109–1136, 1986.
• [11] J. H. McCulloch. Financial applications of stable distributions, Handbook of Statistics 14, ed. G. Maddala and C. Rao, Elsevier Science Publishers, North-Holland, 393–425, 1996.
• [12] J.P. Nolan. Multivariate elliptically contoured stable distributions: theory and estimation, Comput. Statist. 28, Issue 5, 2067–2089, 2013.
• [13] J.P. Nolan. Stable Distributions - Models for Heavy Tailed Data, Birkhauser: Boston, MA, USA, 2016.
• [14] K. J. Palmer, M. S. Ridout, B. J. T. Morgan. Modelling cell generation times using the tempered stable distribution, J. R. Stat. Soc. Ser. C. Appl. Stat. 57, 379–397, 2008.
• [15] G. Samorodnitsky, M. S. Taqqu. Stable Non-Gaussian Random Processes, New York, NY, Chapman & Hall, 1994.
• [16] P. Samuelson. Efficient portfolio selection for Pareto - Lévy investments, J. Financ. Quant. Anal. 2, 107–117, 1967.
• [17] A. Sklar. Fonctions de rèpartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universitè de Paris 8, 229–231, 1959.
• [18] M. S. Taqqu. The modeling of Ethernet data and of signals that are heavy-tailed with infinite variance. Scand. J. Stat. 829, 273–295, 2002.