Advertisement

AStA Advances in Statistical Analysis

, Volume 94, Issue 1, pp 1–31 | Cite as

De copulis non est disputandum

Copulae: an overview
  • Wolfgang Karl Härdle
  • Ostap OkhrinEmail author
Original Paper

Abstract

Normal distribution of residuals is a traditional assumption in multivariate models. It is, however, not very often consistent with real data. Copulae allow for an extension of dependency models to nonellipticity and for separation of margins from the dependency. This paper provides a survey of copulae where different copula classes, estimation and simulation techniques and goodness-of-fit tests are considered. In the empirical section we apply different copulae to the static and dynamic Value-at-Risk of portfolio returns and Profit-and-Loss function.

Copulae Multivariate dependence Value-at-Risk 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barbe, P., Genest, C., Ghoudi, K., Rémillard, B.: On Kendalls’s process. J. Multiv. Anal. 58, 197–229 (1996) zbMATHCrossRefGoogle Scholar
  2. Breymann, W., Dias, A., Embrechts, P.: Dependence structures for multivariate high-frequency data in finance. Quant. Finance 1, 1–14 (2003) CrossRefMathSciNetGoogle Scholar
  3. Chen, X., Fan, Y., Patton, A.: Simple tests for models of dependence between multiple financial time series, with applications to US equity returns and exchange rates. Discussion paper 483, Financial Markets Group, London School of Economics (2004) Google Scholar
  4. Clayton, D.G.: A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65, 141–151 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  5. Deutsch, H., Eller, R.: Derivatives and Internal Models. Macmillan Press, New York (1999) Google Scholar
  6. Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986) zbMATHGoogle Scholar
  7. Dobrić, J., Schmid, F.: A goodness of fit test for copulas based on Rosenblatt’s transformation. Comput. Stat. Data Anal. 51, 4633–4642 (2007) zbMATHCrossRefGoogle Scholar
  8. Embrechts, P., McNeil, A.J., Straumann, D.: Correlation and dependence in risk management: properties and pitfalls. In: RISK, pp. 69–71 (1999) Google Scholar
  9. Fermanian, J.-D.: Goodness-of-fit tests for copulas. J. Multiv. Anal. 95(1), 119–152 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. Frank, M.J.: On the simultaneous associativity of f(x,y) and x+yf(x,y). Aequ. Math. 19, 194–226 (1979) zbMATHCrossRefGoogle Scholar
  11. Frees, E., Valdez, E.: Understanding relationships using copulas. N. Am. Actuar. J. 2, 1–125 (1998) zbMATHMathSciNetGoogle Scholar
  12. Frey, R., McNeil, A.J.: Dependent defaults in models of portfolio credit risk. J. Risk 6(1), 59–92 (2003) Google Scholar
  13. Genest, C., Rémillard, B.: Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann. Inst. Henri Poincaré. Probab. Stat. 44(6), 1096–1127 (2008) zbMATHCrossRefGoogle Scholar
  14. Genest, C., Rivest, L.-P.: A characterization of Gumbel family of extreme value distributions. Stat. Probab. Lett. 8, 207–211 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  15. Genest, C., Rivest, L.-P.: Statistical inference procedures for bivariate Archimedean copulas. J. Am. Stat. Assoc. 88, 1034–1043 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  16. Giacomini, E., Härdle, W.K., Spokoiny, V.: Inhomogeneous dependence modeling with time-varying copulae. J. Bus. Econ. Stat. 27(2), 224–234 (2009) CrossRefGoogle Scholar
  17. Gumbel, E.J.: Distributions des valeurs extrêmes en plusieurs dimensions. Publ. Inst. Stat. Univ. Paris 9, 171–173 (1960) zbMATHMathSciNetGoogle Scholar
  18. Hoeffding, W.: Masstabinvariante Korrelationstheorie. Schriften Math. Inst. Inst. Angewandte Math. Univ. Berlin 5(3), 179–233 (1940) Google Scholar
  19. Hoeffding, W.: Masstabinvariante Korrelationsmasse für diskontinuierliche Verteilungen. Arch. Math. Wirtschafts Sozialforschung 7, 49–70 (1941) Google Scholar
  20. Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997) zbMATHGoogle Scholar
  21. Joe, H., Xu, J.J.: The estimation method of inference functions for margins for multivariate models. Technical Report 166, Department of Statistics, University of British Columbia (1996) Google Scholar
  22. Marshall, A.W., Olkin, J.: Families of multivariate distributions. J. Am. Stat. Assoc. 83, 834–841 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  23. McNeil, A.J.: Sampling nested Archimedean copulas. J. Stat. Comput. Simul. 78(6), 567–581 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  24. Nelsen, R.B.: An Introduction to Copulas. Springer, New York (2006) zbMATHGoogle Scholar
  25. Okhrin, O., Okhrin, Y., Schmid, W.: On the structure and estimation of hierarchical Archimedean copulas. J. Econom. (2009a) Google Scholar
  26. Okhrin, O., Okhrin, Y., Schmid, W.: Properties of hierarchical Archimedean copulas. SFB 649 Discussion Paper 2009-014. Sonderforschungsbereich 649, Humboldt-Universität zu Berlin, Germany. Available at http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2009-014.pdf (2009b)
  27. Patton, A.J.: On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. J. Financ. Econom. 2, 130–168 (2004) Google Scholar
  28. Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23, 470–472 (1952) zbMATHCrossRefMathSciNetGoogle Scholar
  29. Savu, C., Trede, M.: Hierarchical Archimedean copulas. Discussion Paper, University of Muenster (2006) Google Scholar
  30. Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959) MathSciNetGoogle Scholar
  31. Wang, W., Wells, M.: Model selection and semiparametric inference for bivariate failure-time data. J. Am. Stat. Assoc. 95, 62–76 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  32. Whelan, N.: Sampling from Archimedean copulas. Quant. Finance 4, 339–352 (2004) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.C.A.S.E.–Center for Applied Statistics and EconomicsHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations