Tests of serial independence for continuous multivariate time series based on a Möbius decomposition of the independence empirical copula process

Article

Abstract

Genest and Rémillard have recently studied tests of randomness based on a decomposition of the serial independence empirical copula process into a finite number of asymptotically independent sub-processes. A generalization of this decomposition that can be used to test serial independence in the continuous multivariate time series framework is investigated. The weak limits of the Cramér–von Mises statistics derived from the various processes under consideration are determined. As these statistics are not distribution-free, the consistency of the bootstrap methodology is investigated. Extensive simulations are used to study the finite-sample behavior of the tests for continuous time series of dimension one to three, and comparisons with the portmanteau test are provided, as well as, in the one-dimensional case, with the ranked-based version of the Brock, Dechert, and Scheinkman test. Finally, the studied tests are applied to a real trivariate financial time series.

Keywords

Serial copula Test of serial independence Empirical process Möbius decomposition Cramér–von Mises statistic Bootstrap Permutation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beran R., Bilodeau M., de Micheaux P.L. (2007) Nonparametric tests of independence between random vectors. Journal of Multivariate Analysis, 98(9): 1805–1824MathSciNetMATHCrossRefGoogle Scholar
  2. Blum J., Kiefer J., Rosenblatt M. (1961) Distribution free tests of independence based on the sample distribution function. Annals of Mathematical Statistics, 32: 485–498MathSciNetMATHCrossRefGoogle Scholar
  3. Chatterjee S., Yilmaz M. (1992) Chaos, fractals and statistics. Statistical Science, 7: 49–68MathSciNetMATHCrossRefGoogle Scholar
  4. Deheuvels P. (1981) A non parametric test for independence. Publications de l’Institut de Statistique de l’Université de Paris, 26: 29–50MathSciNetMATHGoogle Scholar
  5. Delgado M. (1996) Testing serial independence using the sample distribution function. Journal of Time Series Analysis, 17(3): 271–285MathSciNetMATHCrossRefGoogle Scholar
  6. Fermanian J.D., Radulovic D., Wegkamp M. (2004) Weak convergence of empirical copula processes. Bernoulli, 10(5): 847–860MathSciNetMATHCrossRefGoogle Scholar
  7. Fisher R. (1932) Statistical methods for research workers. Olivier and Boyd, LondonMATHGoogle Scholar
  8. Genest C., Rémillard B. (2004) Tests of independence and randomness based on the empirical copula process. Testm, 13(2): 335–369MATHCrossRefGoogle Scholar
  9. Genest C., Ghoudi K., Rémillard B. (2007a) Rank-based extensions of the Brock, Dechert, and Scheinkman test. Journal of the American Statistical Association, 102(480): 1363–1376MathSciNetMATHCrossRefGoogle Scholar
  10. Genest C., Quessy J.F., Rémillard B. (2007b) Asymptotic local efficiency of Cramér–von Mises tests for multivariate independence. The Annals of Statistics, 35: 166–191MathSciNetMATHCrossRefGoogle Scholar
  11. Ghoudi K., Kulperger R., Rémillard B. (2001) A nonparametric test of serial independence for times series and residuals. Journal of Multivariate Analysis, 79: 191–218MathSciNetMATHCrossRefGoogle Scholar
  12. Hallin M., Puri M. (1995) A multivariate Wald-Wolfowitz rank test against serial dependence. The Canadian Journal of Statistics, 23(1): 55–65MathSciNetMATHCrossRefGoogle Scholar
  13. Hosking J. (1980) The multivariate portmanteau statistic. Journal of the American Statistical Association, 75(371): 602–608MathSciNetMATHCrossRefGoogle Scholar
  14. Johansen S. (1995) Likelihood-based inference in cointegrated vector autoregressive models. Oxford University Press, New YorkMATHCrossRefGoogle Scholar
  15. Kosorok M. (2008) Introduction to empirical processes and semiparametric inference. Springer, New YorkMATHCrossRefGoogle Scholar
  16. Ljung G., Box G. (1978) On a measure of lack of fit in time series models. Biometrika, 65: 297–303MATHCrossRefGoogle Scholar
  17. McNeil A., Frey R., Embrechts P. (2005) Quantitative risk management. Princeton University Press, Princeton Series in Finance. New JerseyMATHGoogle Scholar
  18. Rota G.C. (1964) On the foundations of combinatorial theory. I. Theory of Möbius functions. Z Wahrscheinlichkeitstheorie und Verw Gebiete, 2: 340–368MathSciNetMATHCrossRefGoogle Scholar
  19. Schweizer B., Wolff E. (1981) On nonparametric measures of dependence for random variables. The Annals of Statistics, 9(4): 879–885MathSciNetMATHCrossRefGoogle Scholar
  20. Skag H., Tjøstheim D. (1993) A nonparametric test of serial independence based on the empirical distribution function. Biometrika, 80: 591–602MathSciNetCrossRefGoogle Scholar
  21. Sklar A. (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris, 8: 229–231MathSciNetGoogle Scholar
  22. Tippett L. (1931) The method of statistics. Williams and Norgate, LondonGoogle Scholar
  23. van der Vaart A., Wellner J. (1996) Weak convergence and empirical processes. Springer, New YorkMATHGoogle Scholar
  24. Yan, J., Kojadinovic, I. (2008). Copula: Multivariate dependence with copulas. R package version 0.8-3Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2009

Authors and Affiliations

  1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand
  2. 2.Department of StatisticsUniversity of ConnecticutStorrsUSA

Personalised recommendations