Modeling Time-Varying Dependencies Between Positive-Valued High-Frequency Time Series

Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 213)


Multiplicative error models (MEM) became a standard tool for modeling conditional durations of intraday transactions, realized volatilities, and trading volumes. The parametric estimation of the corresponding multivariate model, the so-called vector MEM (VMEM), requires a specification of the joint error term distribution, which is due to the lack of multivariate distribution functions on \(\mathbb{R}_{+}^{d}\) defined via a copula. Maximum likelihood estimation is based on the assumption of constant copula parameters and therefore leads to invalid inference if the dependence exhibits time variations or structural breaks. Hence, we suggest to test for time-varying dependence by calibrating a time-varying copula model and to re-estimate the VMEM based on identified intervals of homogenous dependence. This paper summarizes the important aspects of (V)MEM, its estimation, and a sequential test for changes in the dependence structure. The techniques are applied in an empirical example.


Change Point Tail Dependency Copula Model Likelihood Ratio Test Statistic Archimedean Copula 
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.



The financial support from the Deutsche Forschungsgemeinschaft via SFB 649 Ökonomisches Risiko, Humboldt-Universität zu Berlin is gratefully acknowledged.


  1. 1.
    Baillie, R.T., Bollerslev, T., Mikkelsen, H.O.: Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econometrics 74, 3–30 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31(3), 307–327 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bougerol, P., Picard, N.: Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52(1–2), 115–127 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chen, X., Fan, Y.: Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. J. Econometrics 135(1–2), 125–154 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cipollini, F., Gallo, G.M.: Automated variable selection in vector multiplicative error models. Comput. Stat. Data Anal. 54(11), 2470–2486 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Conrad, C., Haag, B.R.: Inequality constraints in the fractionally integrated GARCH model. J. Financ. Econometrics 4(3), 413–449 (2006)CrossRefGoogle Scholar
  7. 7.
    Engle, R.: Dynamic conditional correlation: a simple class of multivariate GARCH models. J. Bus. Econ. Stat. 20(3), 339–350 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Engle, R.F.: Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica 50(4), 987–1007 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Engle, R.F., Russell, J.R.: Autoregressive conditional duration: A new model for irregularly spaced transaction data. Econometrica 66(5), 1127–1162 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Genest, C., Rivest, L.P.: Statistical inference procedures for bivariate Archimedean copulas. J. Am. Stat. Assoc. 88, 1034–1043 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Härdle, W.K., Okhrin, O., Okhrin, Y.: Time varying hierarchical Archimedean copulae. SFB 649 Discussion Paper 2010, 018, SFB 649, Economic Risk, Berlin (2010)Google Scholar
  12. 12.
    Hautsch, N.: Econometrics of Financial High-Frequency Data. Springer, Berlin (2012)zbMATHCrossRefGoogle Scholar
  13. 13.
    Hosking, J.R.M.: Fractional differencing. Biometrika 68(1), 165–176 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jasiak, J.: Persistence in intertrade durations. Finance 19, 166–195 (1998)Google Scholar
  15. 15.
    Joe, H.: Families of m-variate distributions with given margins and \(m(m - 1)/2\) bivariate dependence parameters. In: Rüschendorf, L., Schweizer, B., Taylor, M. (eds.) Distribution with Fixed Marginals and Related Topics. IMS Lecture Notes – Monograph Series. Institute of Mathematical Statistics, Hayward (1996)Google Scholar
  16. 16.
    Lee, S., Hansen, B.: Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Economet. Theor. 10, 29–52 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    McNeil, A.J.: Sampling nested archimedean copulas. J. Stat. Comput. Simulat. 78, 567–581 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    McNeil, A.J., Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and l 1 norm symmetric distributions. Ann. Stat. 37(5b), 3059–3097 (2009)zbMATHCrossRefGoogle Scholar
  19. 19.
    Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  20. 20.
    Okhrin, O, Okhrin, Y, Schmid, W.: On the structure and estimation of hierarchical Archimedean copulas. J. Econom. 173, 189–204 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Okhrin, O., Okhrin, Y., Schmid, W.: Properties of hierarchical Archimedean copulas. Stat. Risk Model. 30(1), 21–54 (2013)zbMATHCrossRefGoogle Scholar
  22. 22.
    Patton, A.J.: Modelling asymmetric exchange rate dependence. Int. Econ. Rev. 47, 527–556 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sklar, A.: Fonctions de répartition à n dimension et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 299–231 (1959)MathSciNetGoogle Scholar
  24. 24.
    Spokoiny, V.: Multiscale local change point detection with applications to value-at-risk. Ann. Stat. 37(3), 1405–1436 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Business and EconomicsHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations