Advertisement

Flexible and Dynamic Modeling of Dependencies via Copulas

  • Irène GijbelsEmail author
  • Klaus Herrmann
  • Dominik Sznajder
Conference paper
  • 1.7k Downloads
Part of the Lecture Notes in Statistics book series (LNS, volume 217)

Abstract

In this chapter we first review recent developments in the use of copulas for studying dependence structures between variables. We discuss and illustrate the concepts of unconditional and conditional copulas and association measures, in a bivariate setting. Statistical inference for conditional and unconditional copulas is discussed, in various modeling settings. Modeling the dynamics in a dependence structure between time series is of particular interest. For this we present a semiparametric approach using local polynomial approximation for the dynamic time parameter function. Throughout the chapter we provide some illustrative examples. The use of the proposed dynamical modeling approach is demonstrated in the analysis and forecast of wind speed data.

Keywords

Dependence Structure Copula Function Association Measure Rolling Window Joint Distribution Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors thank the organizers of the “Second workshop on Industry Practices for Forecasting” (wipfor 2013) for a very simulating meeting. This research is supported by IAP Research Network P7/06 of the Belgian State (Belgian Science Policy), and the project GOA/12/014 of the KU Leuven Research Fund. The third author is Postdoctoral Fellow of the Research Foundation – Flanders, and acknowledges support from the foundation.

References

  1. 1.
    Abegaz, F., Gijbels, I., & Veraverbeke, N. (2012). Semiparametric estimation of conditional copulas. Journal of Multivariate Analysis, Special Issue on “Copula Modeling and Dependence”, 110, 43–73.Google Scholar
  2. 2.
    Acar, E. F., Craiu, R. V., & Yao, F. (2011). Dependence calibration in conditional copulas: A nonparametric approach. Biometrics, 67, 445–453.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Acar, E. F., Genest, C., & Nešlehová, J. (2012). Beyond simplified pair-copula constructions. Journal of Multivariate Analysis, Special Issue on “Copula Modeling and Dependence”, 110, 74–90.Google Scholar
  4. 4.
    Blomqvist, N. (1950). On a measure of dependence between two random variables. The Annals of Mathematical Statistics, 21, 593–600.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, S. C., & Huang, T.-M. (2007). Nonparametric estimation of copula functions for dependence modelling. The Canadian Journal of Statistics, 35, 265–282.zbMATHCrossRefGoogle Scholar
  7. 7.
    Cherubini, U., & Luciano, E. (2001). Value-at-risk trade-off and capital allocation with copulas. Economic Notes, 30, 235–256.CrossRefGoogle Scholar
  8. 8.
    Cherubini, U., Mulinacci, S., Gobbi, F., & Romagnoli S. (2011). Dynamic copula methods in finance. New York: Wiley.CrossRefGoogle Scholar
  9. 9.
    Darsow, W. F., Nguyen, B., & Olsen, E. T. (1992). Copulas and Markov processes. Illinois Journal of Mathematics, 36, 600–642.zbMATHMathSciNetGoogle Scholar
  10. 10.
    Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. Académie Royale de Belgique, Bulletin de la Classe des Sciences, 5e Série, 65, 274–292.Google Scholar
  11. 11.
    Demarta, S., & McNeil, A. J. (2005). The t copula and related copulas. International Statistical Review, 73, 111–129.zbMATHCrossRefGoogle Scholar
  12. 12.
    Fan, J., & Gijbels, I. (1996). Local polynomial modelling and its applications. London: Chapman and Hall.zbMATHGoogle Scholar
  13. 13.
    Fan, J., Farmen, M., & Gijbels, I. (1998). Local maximum likelihood estimation and inference. Journal of the Royal Statistical Society, Series B, 60, 591–608.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Fermanian, J.-D., Radulović, D., & Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli, 10, 847–860.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Genest, C., Ghoudi, K., & Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82, 543–552.zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Gini, C. (1909). Concentration and dependency ratios (in Italian). English translation in Rivista di Politica Economica, 87, 769–789 (1997).Google Scholar
  17. 17.
    Gini, C. (1912). Variability and mutability (in Italian, 156 p.). Bologna: C. Cuppini. (Reprinted in E. Pizetti & T. Salvemini (Eds.), Memorie di metodologica statistica. Rome: Libreria Eredi Virgilio Veschi (1955))Google Scholar
  18. 18.
    Gijbels, I., & Mielniczuk, J. (1990). Estimating the density of a copula function. Communications in Statistics – Theory and Methods, 19, 445–464.Google Scholar
  19. 19.
    Gijbels, I., Omelka, M., & Veraverbeke, N. (2012). Multivariate and functional covariates and conditional copulas. Electronic Journal of Statistics, 6, 1273–1306.zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Gijbels, I., Veraverbeke, N., & Omelka, M. (2011). Conditional copulas, association measures and their applications. Computational Statistics & Data Analysis, 55, 1919–1932.MathSciNetCrossRefGoogle Scholar
  21. 21.
    Gneiting, T., Larson, K., Westrick, K., Genton, M., & Aldrich, E. (2006). Calibrated probabilistic forecasting at the stateline wind energy center: The regime-switching-space-time method. Journal of the American Statistical Society, 101, 968–979.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Hafner, C., & Reznikova, O. (2010). Efficient estimation of a semiparametric dynamic copula model. Computational Statistics & Data Analysis, 54, 2609–2627.zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Hall, P., Wolff, R. C. L., & Yao, Q. (1999). Methods for estimating a conditional distribution function. Journal of the American Statistical Association, 94, 154–163.zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Hobæk Haff, I., Aas, K., & Frigessi, A. (2010). On the simplified pair-copula construction – Simply useful or too simplistic? Journal of Multivariate Analysis, 101, 1296–1310.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Hollander, M., & Wolfe, D. A. (1999). Nonparametric statistical methods (2nd ed.). New York: Wiley.zbMATHGoogle Scholar
  26. 26.
    Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis, 94, 401–419.zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Kruskal, W. H. (1958). Ordinal measures of association. Journal of the American Statistical Association, 53, 814–861.zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Nelsen, R. B. (2006). An introduction to copulas (Lecture notes in statistics, 2nd ed.). New York: Springer.Google Scholar
  29. 29.
    Omelka, M., Gijbels, I., & Veraverbeke, N. (2009). Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing. The Annals of Statistics, 37, 3023–3058.zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Patton, A. (2006). Modeling asymmetric exchange rate dependence. International Economical Review, 47, 527–556.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Patton, A. (2012). A review of copula models for economic time series. Journal of Multivariate Analysis, 110, 4–18.zbMATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Segers, J. (2012). Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions. Bernoulli, 18, 764–782.zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Segers, J., van den Akker, R., & Werker, B. J. M. (2014). Semiparametric gaussian copula models: Geometry and efificent rank-based estimation. Annals of Statistics, 42(5), 1911–1940.zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de L’Institut de Statistique de L’Université de Paris, 8, 229–231.MathSciNetGoogle Scholar
  35. 35.
    Stute, W. (1986). On almost sure convergence of conditional empirical distribution functions. The Annals of Statistics, 14, 891–901.zbMATHCrossRefGoogle Scholar
  36. 36.
    Tatsu, J., Pinson, P., Madsen, H. (2015). Space-time trajectories of wind power generation: Parametrized precision matrices under a Gaussian copula approach. Lecture Notes in Statistics 217: Modeling and Stochastic Learning for Forecasting in High Dimension, 267–296.Google Scholar
  37. 37.
    Van Keilegom, I., & Veraverbeke, N. (1997). Estimation and bootstrap with censored data in fixed design nonparametric regression. The Annals of the Institute of Statistical Mathematics, 49, 467–491.zbMATHCrossRefGoogle Scholar
  38. 38.
    Veraverbeke, N., Gijbels, I., & Omelka, M. (2014). Pre-adjusted nonparametric estimation of a conditional distribution function. Journal of the Royal Statistical Society, Series B, 76, 399–438.MathSciNetCrossRefGoogle Scholar
  39. 39.
    Veraverbeke, N., Omelka, M., & Gijbels, I. (2011). Estimation of a conditional copula and association measures. The Scandinavian Journal of Statistics, 38, 766–780.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Irène Gijbels
    • 1
    Email author
  • Klaus Herrmann
    • 1
  • Dominik Sznajder
    • 1
  1. 1.Department of Mathematics and Leuven Statistics Research Centre (LStat)KU LeuvenLeuven (Heverlee)Belgium

Personalised recommendations

Cite paper

Buy options