Skip to main content
SpringerLink
Log in
Book cover

Copula Theory and Its Applications pp 93–109Cite as

Pair-Copula Constructions of Multivariate Copulas

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Statistics ((LNSP,volume 198))

Abstract

In this survey we introduce and discuss the pair-copula construction method to build flexible multivariate distributions. This class includes drawable (D), canonical (C) and regular vines developed in [5] and [4]. Estimation and model selection methods are studied both in a classical as well as in a Bayesian setting. This flexible class of multivariate copulas can be applied to model complex dependencies. Literature to applications in modeling financial data as well as Bayesian belief networks are provided. It closes with a section on open problems.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aas, K.: Modelling the dependence structure of financial assets: A survey of four copulas. Tech. Rep. SAMBA/22/04, Norsk Regnesentral (2004)

    Google Scholar 

  2. Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insurance, Mathematics and Economics 44, 182–198 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baba, K., Sibuya, M.: Equivalence of partial and conditional correlation coefficients. Journal of Japanese Statistical Society 35, 1–19 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Bedford, T., Cooke, R.: Probability density decomposition for conditionally dependent random variables modeled by vines. Annals of Mathematics and Artificial Intelligence 32, 245– 268 (2001)

    Article  MathSciNet  Google Scholar 

  5. Bedford, T., Cooke, R.: Vines - a new graphical model for dependent random variables. Annals of Statistics 30(4), 1031–1068 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berg, D.: Copula goodness-of-fit testing: an overview and power comparison. The European Journal of Finance pp. 1466–4364 (2009)

    Google Scholar 

  7. Berg, D., Aas, K.: Models for construction of higher-dimensional dependence: A comparison study (2009). Forthcoming in The European Journal of Finance

    Google Scholar 

  8. Besag, J.: Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B 36, 192–236 (1974)

    MATH  MathSciNet  Google Scholar 

  9. Chib, S.: Markov chain Monte Carlo methods: Computation and inference. In: Handbook of Econometrics, pp. 3569–3649. North Holland (2001)

    Google Scholar 

  10. Chollete, L., Heinen, A., Valdesogo, A.: Modeling international financial returns with a multivariate regime switching copula (2008). Preprint

    Google Scholar 

  11. Cowell, R., Dawid, A., S., L., D., S.: Probabilistic networks and experts systems. Springer, New York (1999)

    Google Scholar 

  12. Czado, C., Gärtner, F., Min, A.: Joint Bayesian Inference of D-vines with AR(1) Margins. In: Kurowicka D. and Joe H. (eds.) Dependence Modeling- Handbook on Vine Copulas, pp. 359-330, World Scientific Publishing, Singapore. (2010)

    Google Scholar 

  13. Czado, C., Min, A., Baumann, T., Dakovic, R.: Pair-copula constructions for modeling exchange rate dependence (2008). Preprint, available under http://wwwm4. ma.tum.de/Papers/index.html

  14. Durante, F., Sempi, C.: Copula theory: an introduction. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of theWorkshop Held in Warsaw 25-26 September 2009, Springer (2010).

    Google Scholar 

  15. Embrechts, P., Lindskog, F., McNeil, A.: Modelling dependence with copulas and applications to risk management. In: S.T.Rachev (ed.) Handbook of Heavy Tailed Distributions in Finance, pp. 329–384. Elsevier, North-Holland (2003)

    Chapter  Google Scholar 

  16. Erhardt, V., Czado, C.: Sampling Count Variables with specified Pearson Correlation - a Comparison between a naive and a C-vine Sampling Approach. In: Kurowicka D. and Joe H. (eds.) 108 Claudia Czado Dependence Modeling- Handbook on Vine Copulas, pp. 359-330, World Scientific Publishing, Singapore. (2010)

    Google Scholar 

  17. Erhardt, V., Czado, C.: A Method for approximately sampling high-dimensional Count Variables with prespecified Pearson Correlation. Preprint, available under http://wwwm4. ma.tum.de/Papers/index.html, (2009)

  18. Fang, H.B., Fang, K., Kotz, S.: The meta-elliptical distributions with given marginals. Journal of Multivariate Analysis 82(1), 1–16 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fischer, M., Köck, C., Schlüter, S., Weigert, F.: An empirical analysis of multivariate copula models. Quant. Finance 9 (7), 839–854 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Frahm, G., Junker, M., Szimayer, A.: Elliptical copulas:applicability and limitations. Statist. Probab. Lett. 63(3), 275–286 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Genest, C., Rivest, L.P.: Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88(423), 1034–1043 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Genest, C., Rémillard, B., Beaudoin, D.: Omnibus goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics 44, 199–213 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Green, P.: Reversible jump markov chain monte carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Haff, I.H., Aas, K., Frigessi, A.: On the simplified pair-copula construction - simply useful or too simplistic? Tech. rep., Norwegian Computing Center, Oslo (2009)

    Google Scholar 

  25. Hanea, A., Kurowicka, D.: Mixed non-parametric continuous and discrete bayesian belief networks. In: Advances in mathematical modeling for reliability. IOS Press, Amsterdam (2008)

    Google Scholar 

  26. Hanea, A., Kurowicka, D., Cooke, R.: Hybrid method for quantifying and analyzing bayesian belief networks. Quality and Realiability Engineering 22(6), 709–729 (2006)

    Article  Google Scholar 

  27. Hanea, A., Kurowicka, D., Cooke, R., Ababei, D.: Mining and visualizing ordinal data with non-parametric continuous bbn’s. Computational Statistics and Data Analysis, in press. Doi:10.1016/j.csda.2008.09.032 (2008)

    Google Scholar 

  28. Hastings, W.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)

    Article  MATH  Google Scholar 

  29. Heinen, A., Valdesogo, A.: Asymmetric capm dependence for large dimensions: The canonical vine autoregressive copula model (2008). Preprint

    Google Scholar 

  30. Heinen, A., Valdesogo, A.: Dynamic d vines (2009). Preprint

    Google Scholar 

  31. Joe, H.: Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependence parameters. In: L. Rüschendorf and B. Schweizer and M. D. Taylor (ed.) Distributions with Fixed Marginals and Related Topics (1996)

    Google Scholar 

  32. Joe, H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London (1997)

    MATH  Google Scholar 

  33. Joe, H.: Generating random correlation matrices based on partial correlations. Journal of Multivariate Analysis 97, 2177–2189 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Joe, H., Li, H., Nikoloulopoulos, A.: Tail dependence and vine copulas. Tech. rep., Department of Mathematics, Washington State University, Pullman, WA, USA (2008)

    Google Scholar 

  35. Kolbjornsen, O., Stien, M.: The d-vine creation of non-gaussian random fields (2008). Preprint

    Google Scholar 

  36. Kurowicka, D.: Optimal truncation of vines (2009). Preprint

    Google Scholar 

  37. Kurowicka, D., Cooke, R.: A parametrization of positive definite matrices in terms of partial correlation vines. Linear Algebra and its Applications 372, 225–251 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  38. Kurowicka, D., Cooke, R.: Distribution-free continuous bayesian belief. In: Modern statistical and mathematical methods in reliabiliy. World Scientific (2005)

    Google Scholar 

  39. Kurowicka, D., Cooke, R.: Uncertainty analysis with high dimensional dependence modelling. Wiley, Chichester (2006)

    Google Scholar 

  40. Kurowicka, D., Cooke, R.: Sampling algorithms for generating joint uniform distributions using the vine-copula method. Computational Statistics & Data Analysis 51(6), 2889–2906 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  41. Lanzendörfer, J.N.: Joint estimation of parameters in multivariate normal regression with correlated errors using pair-copula constructions and an application to finance (2009). Diploma Thesis, Center for Mathematical Sciences, Technische Universität München

    Google Scholar 

  42. Lewandowski, D., Kurowicka, D., Joe, H.: Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis 100, 1989–2001 (2009) 4 Pair-copula constructions of multivariate copulas 109

    Article  MATH  MathSciNet  Google Scholar 

  43. Liebscher, E.: Modelling and estimation of multivariate copulas. Tech. rep., Working Paper, University of Applied Sciences Merseburg, Germany (2006)

    Google Scholar 

  44. McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press (2006)

    Google Scholar 

  45. de Melo Mendes, B.V., Semeraro, M.M., Leal, R.C.: Pair-copulas modeling in finance. Tech. rep., M/COPPEAD, Federal University at Rio de Janeiro, Brazil (2009)

    Google Scholar 

  46. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. Journal of Chemical Phisics 21, 1087–1092 (1953)

    Article  Google Scholar 

  47. Min, A., Czado, C.: Bayesian inference for multivariate copulas using pair-copula constructions (2009). To appear in Journal of Financial Econometrics, preprint available under http://www-m4.ma.tum.de/Papers/index.html

  48. Min, A., Czado, C.: Bayesian model selection for multivariate copulas using pair-copula constructions (2009). Preprint

    Google Scholar 

  49. Morales-Napoles, O., Cooke, R., Kurowicka, D.: About the number of vines and regular vines on n nodes (2009). Submitted to Discrete Applied Mathematics

    Google Scholar 

  50. Morillas, P.: A method to obtain new copulas from a given one. Metrika 61, 169–184 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  51. Nelsen, R.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)

    Google Scholar 

  52. Savu, C., Trede, M.: Hierarchical Archimedean copulas. In: International Conference on High Frequency Finance, Konstanz, Germany, May (2006)

    Google Scholar 

  53. Schirmacher, D., Schirmacher, E.: Multivariate dependence modeling using pair-copulas. Tech. rep., Society of Acturaries: 2008 Enterprise Risk Management Symposium, April 14-16, Chicago (2008). URL http://www.soa.org/library/monographs/other-monographs/2008/april/2008-erm-toc.aspx

  54. Sklar, A.: Fonctions dé repartition á n dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959)

    MathSciNet  Google Scholar 

  55. Smith, M., Min, A., Czado, C., Almeida, C.: Modeling longitudinal data using a pair-copula decomposition of serial dependence (2009). In revision for the Journal of the American Statistical Association

    Google Scholar 

  56. Whelan, N.: Sampling from Archimedian copulas. Quantitative Finance 4, 339–352 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Claudia Czado is supported by the Deutsche Forschungsgemeinschaft (CZ86 1-3). Special thanks to K. Aas and A. Frigessi who introduced me to this promising research area.

Author information

Authors and Affiliations

  1. Department of Mathematics, Technische Universität München, Garching, Germany

    Claudia Czado

Authors
  1. Claudia Czado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudia Czado .

Editor information

Editors and Affiliations

  1. , Faculty of Mathematics Informatics, University of Warsaw, ul. Banacha 2, Warszawa, 02097, Poland

    Piotr Jaworski

  2. , Department of Knowledge-Based, Johannes Kepler Universität, Altenberger Straße 69, Linz, 4040, Austria

    Fabrizio Durante

  3. Ladislaus von Bortkiewicz Chair of Stati, C.A.S.E. Centre for Applied Statistics a, Humboldt-Universität zu Berlin, Unter den Linden 6, Berlin, 10099, Germany

    Wolfgang Karl Härdle

  4. , Mathematical Institute, Polish Academy of Sciences, ul. Chopina 12, Toruń, 87100, Poland

    Tomasz Rychlik

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Berlin Heidelberg

About this paper

Cite this paper

Czado, C. (2010). Pair-Copula Constructions of Multivariate Copulas. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T. (eds) Copula Theory and Its Applications. Lecture Notes in Statistics(), vol 198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12465-5_4

Download citation

Publish with us

Policies and ethics

Search

Navigation