Probability Surveys

Copulas and long memory

Rustam Ibragimov and George Lentzas

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This paper focuses on the analysis of persistence properties of copula-based time series. We obtain theoretical results that demonstrate that Gaussian and Eyraud-Farlie-Gumbel-Morgenstern copulas always produce short memory stationary Markov processes. We further show via simulations that, in finite samples, stationary Markov processes, such as those generated by Clayton copulas, may exhibit a spurious long memory-like behavior on the level of copulas, as indicated by standard methods of inference and estimation for long memory time series. We also discuss applications of copula-based Markov processes to volatility modeling and the analysis of nonlinear dependence properties of returns in real financial markets that provide attractive generalizations of GARCH models. Among other conclusions, the results in the paper indicate non-robustness of the copula-level analogues of standard procedures for detecting long memory on the level of copulas and emphasize the necessity of developing alternative inference methods.

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Probab. Surveys, Volume 14 (2017), 289-327.

Received: March 2014
First available in Project Euclid: 12 December 2017

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Long memory processes short memory processes copulas measures of dependence autocorrelations persistence volatility GARCH

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Ibragimov, Rustam; Lentzas, George. Copulas and long memory. Probab. Surveys 14 (2017), 289--327. doi:10.1214/14-PS233.

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  • Baillie, R. T. (1996), ‘Long memory processes and fractional integration in econometrics’, Journal of Econometrics 73, 5–59.
  • Beare, B. (2010), ‘Copulas and temporal dependence’, Econometrica 78, 395–410.
  • Bollerslev, T. and Mikkelsen, H. O. (1996), ‘Modeling and pricing long memory in stock market volatility’, Journal of Econometrics 73, 151–184.
  • Cambanis, S. (1991), On Eyraud-Farlie-Gumbel-Morgenstern random processes, in ‘Advances in probability distributions with given marginals (Rome, 1990)’, Vol. 67 of Mathematics and its Applications, Kluwer Academic Publishing, Dordrecht, pp. 207–222.
  • Campbell, J. Y., Lo, A. W. and MacKinlay, A. (1997), The econometrics of financial markets, Princeton University Press, Princeton.
  • Chen, X. and Fan, Y. (2004), ‘Evaluating density forecasts via copula approach’, Finance Research Letters 1, 74–84.
  • Chen, X. and Fan, Y. (2006), ‘Estimation of copula-based semiparametric time series models’, Journal of Econometrics 130, 307–335.
  • Chen, X., Wu, W. B. and Yi, Y. (2009), ‘Efficient estimation of copula-based semiparametric Markov models’, Annals of Statistics 37, 4214–4253.
  • Cherubini, U., Luciano, E. and Vecchiato, W., eds (2004), Copula methods in finance, Wiley, Chichester.
  • Comte, F. and Renault, E. (1998), ‘Long memory in continuous-time stochastic volatility models’, Mathematical Finance 8, 291–323.
  • Cont, R. (2001), ‘Empirical properties of asset returns: Stylized facts and statistical issues’, Quantitative Finance 1, 223–236.
  • Cuadras, C. (2009), ‘Constructing copula functions with weighted geometric means’, Journal of Statistical Planning and Inference 139, 3766–3772.
  • Cuadras, C. and Diaz, W. (2012), ‘Another generalization of the bivariate fgm distribution with two-dimensional extensions’, Acta et Commentationes Universitatis Tartuensis de Mathematica 16, 3–12.
  • Darsow, W. F., Nguyen, B. and Olsen, E. T. (1992), ‘Copulas and Markov processes’, Illinois Journal of Mathematics 36, 600–642.
  • Davis, R. A. and Mikosch, T. (1998), ‘The sample autocorrelations of heavy-tailed processes with applications to ARCH’, Annals of Statistics 26, 2049–2080.
  • de la Peña, V. H., Ibragimov, R. and Sharakhmetov, S. (2006), Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series, in J. Rojo, ed., ‘2nd Erich L. Lehmann Symposium – Optimality, IMS Lecture Notes – Monograph Series’, Vol. 49, Institute of Mathematical Statistics, Beachwood, Ohio, USA, pp. 183–209. Available at
  • Doukhan, P., Fermanian, J.-D. and Lang, G. (2004), ‘Copulas of a vector-valued stationary weakly dependent process’, Working paper, CREST.
  • Doukhan, P., Oppenheim, G. and Taqqu, M. S., eds (2003), Theory and applications of long-range dependence, Birkhäuser, Boston.
  • Drost, F. C. and Nijman, T. E. (1993), ‘Temporal aggregation of garch processes’, Econometrica 61, 909–927.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997), Modelling extremal events for insurance and finance, Springer, New York.
  • Embrechts, P., McNeil, A. J. and Straumann, D. (2002), Correlation and dependence in risk management: properties and pitfalls, in ‘Risk management: value at risk and beyond (Cambridge, 1998)’, Cambridge Univeristy Press, Cambridge, pp. 176–223.
  • Fermanian, J.-D., Radulović, D. and Wegkamp, M. (2004), ‘Weak convergence of empirical copula processes’, Bernoulli 10, 847–860.
  • Fruhwirth-Schnatter, S. (2006), Finite mixture models and Markov switching models, Wiley Series in Probability and Statistics, Springer, New York.
  • Granger, C. W. J. (2003), ‘Time series concepts for conditional distributions’, Oxford Bulletin of Economics and Statistics 65, 689–701 Suppl. S.
  • Granger, C. W. J., Teräsvirta, T. and Patton, A. J. (2006), ‘Common factors in conditional distributions for bivariate time series’, Journal of Econometrics 132, 43–57.
  • Granger, C. W., Maasoumi, E. and Racine, J. (2004), ‘A dependence metric for possibly nonlinear processes’, Journal of Time Series Analysis 25, 649–669.
  • Hosking, J. R. M. (1996), ‘Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series’, Journal of Econometrics 73, 261–284.
  • Hu, L. (2006), ‘Dependence patterns across financial markets: methods and evidence’, Applied Financial Economics 16, 717–729.
  • Ibragimov, M., Ibragimov, R. and Walden, J. (2015), Heavy-Tailed Distributions and Robustness in Economics and Finance, Vol. 214 of Lecture Notes in Statistics, Springer, New York.
  • Ibragimov, R. (2009a), ‘Copula-based characterizations for higher-order Markov processes’, Econometric Theory 25, 819–846.
  • Ibragimov, R. (2009b), Heavy tailed densities, in S. N. Durlauf and L. E. Blume, eds, ‘The New Palgrave Dictionary of Economics Online’, Palgrave Macmillan, New York.
  • Ibragimov, R., Jaffee, D. and Walden, J. (2009), ‘Nondiversification traps in catastrophe insurance markets’, Review of Financial Studies 22, 959–993.
  • Ibragimov, R. and Lentzas, G. (2008), ‘Copulas and long memory’, Harvard University Research Discussion Paper 2160.
  • Ibragimov, R. and Prokhorov, A. (2016), ‘Heavy tails and copulas: Limits of diversification revisited’, Economics Letters 149, 102–107.
  • Ibragimov, R. and Prokhorov, A. (2017), Heavy Tails and Copulas: Topics in Dependence Modelling in Economics and Finance, World Scientific, New York.
  • Joe, H. (1989), ‘Relative entropy measures of multivariate dependence’, Journal of the American Statistical Association 84(405), 157–164.
  • Joe, H. (1997), Multivariate models and dependence concepts, Vol. 73 of Monographs on Statistics and Applied Probability, Chapman & Hall, London.
  • Kendall, M. G. and Stuart, A. (1973), The advanced theory of statistics, Vol. 2, third edn, Hafner Publishing Co., New York.
  • Kimeldorf, G. and Sampson, A. (1975), ‘Uniform representations of bivariate distributions’, Communications in Statistics 4, 617–627.
  • Lancaster, H. O. (1957), ‘Some properties of the bivariate normal distribution considered in the form of contingency table’, Biometrika 44, 289–292.
  • Lin, G. (1987), ‘Relationships between two extensions of farlie–gumbel– morgenstern distribution’, Annals of the Institute of Statistical Mathematics 39, 129–140.
  • Lo, A. W. (1991), ‘Long-term memory in stock market prices’, Econometrica 59, 1279–1313.
  • Loretan, M. and Phillips, P. C. B. (1994), ‘Testing the covariance stationarity of heavy-tailed time series’, Journal of Empirical Finance 1, 211–248.
  • Lowin, J. (2007), The Fourier copula: Theory and applications, Senior thesis, Harvard University.
  • Marshall, A. W. and Olkin, I. (1979), Inequalities: theory of majorization and its applications, Vol. 143 of Mathematics in Science and Engineering, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York.
  • McLachlan, G. J. and Peel, D. (2000), Finite mixture models, Wiley Series in Probability and Statistics, John Wiley & Sons, New York.
  • McNeil, A. J., Frey, R. and Embrechts, P. (2015), Quantitative risk management: Concepts, Techniques and Tools, Princeton Series in Finance, Princeton University Press, Princeton, NJ.
  • Meddahi, N. and Renault, E. (2004), ‘Temporal aggregation of volatility models’, Journal of Econometrics 119, 355–379.
  • Mikosch, T. and Stărică, C. (2000), ‘Limit theory for the sample autocorrelations and extremes of a GARCH $(1,1)$ process’, Annals of Statistics 28, 1427–1451.
  • Nelsen, R. B. (1999), An introduction to copulas, Vol. 139 of Lecture Notes in Statistics, Springer-Verlag, New York.
  • Nelsen, R. B., Quesada-Molina, J. and Rodriguez-Lallena, J. (1997), ‘Bivariate copulas with cubic sections’, Nonparametric Statistics 7, 205–220.
  • Nze, P. A. and Doukhan, P. (2004), ‘Weak dependence: models and applications to econometrics’, Econometric Theory 20, 995–1045.
  • Patton, A. (2006), ‘Modelling asymmetric exchange rate dependence’, International Economic Review 47, 527–556.
  • Robinson, P. M. and Zaffaroni, P. (1998), ‘Nonlinear time series with long memory: a model for stochastic volatility’, Journal of Statistical Planning and Inference 68, 359–371.
  • Schweizer, B. and Wolff, E. F. (1981), ‘On nonparametric measures of dependence for random variables’, Annals of Statistics 9, 879–885.
  • Sharakhmetov, S. and Ibragimov, R. (2002), ‘A characterization of joint distribution of two-valued random variables and its applications’, Journal of Multivariate Analysis 83, 389–408.