Justyna Mokrzycka , Anna Pajor

(Polish) PDF


Copulas have become one of most popular tools used in modelling the dependencies among financial time series. The main aim of the paper is to formally assess the relative explanatory power of competing bivariate Copula-AR-GARCH models, which differ in assumptions on the conditional dependence structure represented by particular copulas. For the sake of comparison the Copula-AR-GARCH models are estimated using the maximum likelihood method, and next they are informally compared and ranked according to the values of the Akaike (AIC) and of the Schwarz (BIC) information criteria. We apply these tools to the daily growth rates of four sub-indices of the stock index WIG published by the Warsaw Stock Exchange. Our results indicate that the informal use of the information criteria (AIC or BIC) leads to very similar ranks of models as compared to those obtained by the use of the formal Bayesian model comparison.


Copula, Copula-AR-GARCH model, Bayesian model comparison


Abramowitz M., Stegun N., (1968), Handbook of Mathematical Functions, Dover Publications, New York.

Akaike H., (1973), Information Theory and an Extension of the Maximum Likelihood Principle, 2nd International Symposium on Information Theory, w: Petrov B. N., Csáki F., (red.), Akadémia Kiado, Budapest, 267–281.

Chib S., (1995), Marginal Likelihood from the Gibbs Output, Journal of the American Statistical Association, 90, 1313–1321.

Chib S., Jeliazkov I., (2001), Marginal Likelihood from the Metropolis-Hastings Output, Journal of the Ameri can Statistical Association, 96, 270–281.

Choroś B., Ibragimov R., Permiakova E., (2010), Copula Estimation, w: Jaworski P., Hardle W., Durante F., Rychlik T., (red.), Copula Theory and Its Applications, Springer Verlag, Berlin Heidelberg.

Craiu V. R., Sabeti A., (2012), In Mixed Company: Bayesian Inference for Bivariate Conditional Copula Models with Discrete and Continuous Outcomes, Journal of Multivariate Analysis, 110, 106–120.

de Bruijn N. G., (1970), Asymptotic Methods in Analysis, North-Holland, Amsterdam.

Doman R., (2011), Zastosowania kopuli w modelowaniu dynamiki zależności na rynkach finansowych, Wydawnictwo Uniwersytetu Ekonomicznego w Poznaniu, Poznań.

Doman M., Doman R., (2014), Dynamika zależności na globalnym rynku finansowym, Difin SA, Warszawa.

Durante F., Sempi C., (2016), Principles of Copula Theory, CRS Press, Taylor and Francis Group LLC.

Fang Y., Madsen L., Liu L., (2014), Comparison of Two Methods to Check Copula Fitting, IAENG International Journal of Applied Mathematics, 44 (1), IJAM_44_1_07.

Genest C., Rémillard B., Beaudoin D., (2009), Goodness-of-Fit Tests for Copulas: A Review and a Power Study, Insurance: Mathematics and Economics, 44, 199–213.

Jaworski P., (2012), Wybrane zagadnienia modelowania zmienności na rynkach finansowych z wykorzystaniem kopuli i procesów GARCH, http://docplayer.pl/1206688-Wybrane-zagadnienia-modelowaniazmiennosci-na-rynkach-finansowych-z-wykorzystaniem-kopuli-i-procesow-garch.html (dostęp: 5.05.2016)

Jeffreys H., (1961), Theory of Probability, Oxford University Press, London.

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

Joe H., (2005), Asymptotic Efficiency of the Two-Stage Estimation Method for Copula-Based Models, Journal of Multivariate Analysis, 94, 401–419.

Joe H., Xu J. J., (1996), The Estimation Method of Inference Function for Margins for Multivariate Models, Technical Report no. 166, Department of Statistics, University of British Columbia, Vancouver. https://open.library.ubc.ca/cIRcle/collections/facultyresearchandpublications/52383/items/1.0225985 (dostęp: 6.05.2016)

Jondeau E., Rockinger M., (2006), The Copula-GARCH Model of Conditional Dependencies: An International Stock Market Application, Journal of International Money and Finance, 25, 827–853.

Kass R. E., Raftery A. E., (1995), Bayes Factors, Journal of the American Statistical Association, 90, 773–795.

Kojadinovic I., Yan J., Holmes M., (2011), Fast Large-Sample Goodness-of-Fit Tests for Copulas, Statistica Sinica, 21, 841–871.

Lenk P., (2009), Simulation Pseudo-Bias Correction to the Harmonic Mean Estimator of Integrated Likelihoods, Journal of Computational and Graphical Statistics, 18, 941–960.

Nelsen R. B., (1999), An Introduction to Copulas, Springer-Verlag, New York.

Newey W. K., West, K. D., (1987), A Simple Positive Semi-Definite Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica, 55, 703–708.

Newton M. A., Raftery A. E., (1994), Approximate Bayesian Inference by the Weighted Likelihood Bootstrap, Journal of the Royal Statistical Society B, 56 (1), 3–48.

Osiewalski J., (2001), Ekonometria bayesowska w zastosowaniach, Wydawnictwo Akademii Ekonomicznej w Krakowie, Kraków.

Osiewalski J., Pipień M., (2004), Bayesian Comparison of Bivariate ARCH-Type models for the Main Exchange Rates in Poland, Journal of Econometrics, 123, 371–391.

Osiewalski J., Steel M. F. J., (1993), A Bayesian Perspective on Model Selection, maszynopis; opublikowano w języku hiszpańskim: Una perspectiva bayesiana en sección de modelos, Cuadernos Economicos ICE, 55, 327–351.

Pajor A., (2003), Procesy zmienności stochastycznej SV w bayesowskiej analizie finansowych szeregów czasowych, Monografie: Prace Doktorskie, Nr 2, Wydawnictwo AE w Krakowie, Kraków.

Pajor A., Osiewalski J., (2013), A Note on Lenk’s Correction of the Harmonic Mean Estimator, Central European Journal of Economic Modelling and Econometrics, 5 (4), 271–275.

Patton A. J., (2001), Modelling Time Varying Exchange Rate Dependence Using the Conditional Copula, Discussion Paper 2001–09 University of California, San Diego.

Patton A. J., (2006a), Estimation of Multivariate Models for Time Series of Possibly Different Lengths, Journal of Applied Econometrics, 21 (2), 147–173.

Patton A. J., (2006b), Modelling Asymmetric Exchange Rate Dependence, International Economic Review, 47 (2), 527–556.

Patton A. J., (2012), A Review of Copula Models for Economic Time Series, Journal of Multivariate Analysis, 100, 4–18.

Patton A. J., (2013), Copula Methods for Forecasting Multivariate Time Series, w: Elliott G., Timmermann A., (red.), Handbook of Economic Forecasting, 2, Springer Verlag.

Raftery A. E., (1996), Hypothesis Testing and Model Selection, w: Gilks W. R., Spiegelhalter D. J., Richardson S., (red.), Markov Chain Monte Carlo in Practice, Chapman and Hall, London, 163–188.

Regis L., (2011), A Bayesian Copula Model for Stochastic Claims Reserving, working paper no. 227, Collegio Carlo Alberto, http://www.carloalberto.it/assets/working-papers/no.227.pdf (dostęp: 6.05.2016).

Rossi J. L., Ehlers R. S., Andrade M. G., (2012), Copula-GARCH Model Selection: A Bayesian Approach, Technical Report 88, University of Sao Paulo.

Schwarz G., (1978), Estimating the dimension of a model, Annals of Statistics, 6, 461–464.

Silva R. S., Lopes H. F., (2008), Copula, Marginal Distributions and Model Selection: a Bayesian Note, Statistics and Computing, 18 (3), 313–320.

Sklar A., (1959), Fonctions de répartition a n dimensions et leurs marges’, Publications de l’Institut de Statistique de L’Université de Paris, 8, 229–231.

Smith M. D., (2003), Modelling Sample Selection Using Archimedean Copulas, Econometrics Journal, 6, 99–123.

Smith M. S., (2013), Bayesian Approaches to Copula Modelling, w: Damien P., Dellaportas P., Polson N., Stephens D., (red.), Bayesian Theory and Applications, Oxford University Press, Oxford, 336–358. http://works.bepress.com/michael_smith/30/ (dostęp: 6.05.2016).

Tierney L., (1994), Markov Chains for Exploring Posterior Distributions (with discussion), The Annals of Statistics, 22, 1701–1762.

Zellner A., (1971), An Introduction to Bayesian Inference in Econometrics, J. Wiley, New York.

Back to top
Copyright © 2019 Statistics Poland