Skip to main content


  • Guannan Liu (a1), Wei Long (a2), Xinyu Zhang (a3) and Qi Li (a4)

A mixture copula is a linear combination of several individual copulas that can be used to generate dependence structures not belonging to existing copula families. Because different pairs of markets may exhibit quite different dependence structures in empirical studies, mixture copulas are useful in modeling the dependence in financial data. Therefore, rather than selecting a single copula based on certain criteria, we propose using a model averaging approach to estimate financial data dependence structures in a mixture copula framework. We select weights (for averaging) by a J-fold Cross-Validation procedure. We prove that the model averaging estimator is asymptotically optimal in the sense that it minimizes the squared estimation loss. Our simulation results show that the model averaging approach outperforms some competing methods when the working mixture model is misspecified. Using 12 years of data on daily returns from four developed economies’ stock indexes, we show that the model averaging approach more accurately estimates their dependence structures than some competing methods.

Corresponding author
*Address correspondence to Xinyu Zhang, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China; and Qingdao University, Qingdao, China; email:
Hide All

We would like to thank the Editor, Peter C.B. Phillips, Co-Editor, Dennis Kristensen, and three anonymous referees for their insightful comments that greatly improved our article. We would also like to thank Yanqin Fan and Xiaohong Chen for their help during the revision process of this article. Liu’s research is supported by the National Natural Science Foundation of China (Grant No. 71803160) and the Fundamental Research Funds for the Central Universities (project number 20720171061). Long’s research is partially supported by the Carol Lavin Bernick Faculty Grants at Tulane University. Zhang and Li’s research is partially supported by National Natural Science Foundation of China (projects 71522004, 11471324, and 71631008 for Zhang; 71722011 and 71601130 for Li).

Hide All
Aloui, R., Aïssa, M., & Nguyen, D. (2011) Global financial crisis, extreme interdependences, and contagion effects: The role of economic structure? Journal of Banking & Finance 35, 130141.
Ando, T. & Li, K. (2014) A model-averaging approach for high-dimensional regression. Journal of the American Statistical Association 109, 254265.
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817858.
Cai, Z. & Wang, X.. (2014) Selection of mixed copula model via penalized likelihood. Journal of the American Statistical Association 109, 788801.
Chen, X. & Fan, Y. (2006a) Estimation of copula-based semiparametric time series models. Journal of Econometrics 130, 307335.
Chen, X. & Fan, Y. (2006b) Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics 135, 125154.
Cheng, X. & Hansen, B. (2015) Forecasting with factor-augmented regression: A frequentist model averaging approach. Journal of Econometrics 186, 280293.
Chollete, L., Peña, V., & Lu, C.. (2005) Comovement of international financial markets. Unpublished manuscript.
Chollete, L., Heinen, A., & Valdesogo, A. (2009) Modeling international financial returns with a multivariate regime-switching copula. Journal of Financial Econometrics 7, 437480.
Fan, J. & Li, R. (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 13481360.
Fan, Y. & Patton, A. (2014) Copulas in econometrics. Annual Review of Economics 6, 179200.
Gao, Y., Zhang, X., Wang, S., Chong, T., & Zou, G. (2018). Frequentist model averaging for threshold models. Annals of the Institute of Statistical Mathematics, first published online 12 January 2018. doi:10.1007/s10463-017-0642-9.
Genest, C. & Rivest, L. (1993) Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association 88, 10341043.
Hansen, B. & Racine, J. (2012) Jackknife model averaging. Journal of Econometrics 167, 3846.
Hansen, B. (2007) Least squares model averaging. Econometrica 75, 11751189.
Hu, L. (2006) Dependence patterns across financial markets: A mixed copula approach. Applied Financial Economics 16, 717729.
Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall.
Li, D. (2000) On default correlation: A copula function approach. Journal of Fixed Income 9, 4354.
Longin, F. & Solnik, B. (2001) Extreme correlation of international equity markets. The Journal of Finance 56, 649676.
Ma, Y. & Zhu, L. (2012) A semiparametric approach to dimension reduction. Journal of the American Statistical Association 497, 168179.
Manner, H. & Reznikova, O. (2012) A survey on time-varying copulas: Specification, simulations and application. Econometric Reviews 31, 654687.
Nelsen, R. (2006) An Introduction to Copulas, 2nd ed. Springer.
Patton, A. (2006) Modelling asymmetric exchange rate dependence. International Economic Review 47, 527556.
Patton, A. (2012) A review of copula models for economic time series. Journal of Multivariate Analysis 110, 418.
Rodriguez, J. (2007) Measuring financial contagion: A copula approach. Journal of Empirical Finance 14, 401423.
Shao, J. (1997) An asymptotic theory for linear model selection. Statistica Sinica 7, 221264.
Sklar, A. (1959) Fonctions de répartition à n dimensions et leurs marges. Publication de l’Institut de Statistique de l’Universite de Paris 8, 229231.
Zhang, X. (2010) Model Averaging and its Applications. Ph.D. thesis, Academy of Mathematics and Systems Science, Chinese Academy of Sciences.
Zhang, X., Wan, A.T.K., & Zou, G. (2013) Model averaging by Jackknife criterion in models with dependent data. Journal of Econometrics 174, 8294.
Zhang, X., Yu, D., Zou, G., & Liang, H. (2016) Optimal model averaging estimation for generalized linear models and generalized linear mixed-effects models. Journal of the American Statistical Association 111, 17751790.
Zimmer, D.M. (2012) The role of copulas in the housing crisis. Review of Economics and Statistics 94, 607620.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *