Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-15T05:30:51.858Z Has data issue: false hasContentIssue false
  • English
  • Français

HIDDEN MARKOV STRUCTURES FOR DYNAMIC COPULAE

Published online by Cambridge University Press:  22 December 2014

Wolfgang Karl Härdle ,
Ostap Okhrin  and
Weining Wang
Wolfgang Karl Härdle
Affiliation:
Humboldt-Universität zu Berlin
Ostap Okhrin
Affiliation:
Humboldt-Universität zu Berlin
Weining Wang*
Affiliation:
Humboldt-Universität zu Berlin
*
*Address correspondence to Weining Wang, Hermann-Otto-Hirschfeld Junior Professor in Nonparametric Statistics and Dynamic Risk Management at the Ladislaus von Bortkiewicz Chair of Statistics of Humboldt-Universität zu Berlin, Spandauer Straße 1, 10178 Berlin, Germany; e-mail: wangwein@cms.hu-berlin.de.
Get access Rights & Permissions [Opens in a new window]

Abstract

Understanding the time series dynamics of a multi-dimensional dependency structure is a challenging task. Multivariate covariance driven Gaussian or mixed normal time varying models have only a limited ability to capture important features of the data such as heavy tails, asymmetry, and nonlinear dependencies. The present paper tackles this problem by proposing and analyzing a hidden Markov model (HMM) for hierarchical Archimedean copulae (HAC). The HAC constitute a wide class of models for multi-dimensional dependencies, and HMM is a statistical technique for describing regime switching dynamics. HMM applied to HAC flexibly models multivariate dimensional non-Gaussian time series.

We apply the expectation maximization (EM) algorithm for parameter estimation. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application. This example is motivated by a local adaptive analysis that yields a time varying HAC model. We compare its forecasting performance with that of other classical dynamic models. In another, second, application, we model a rainfall process. This task is of particular theoretical and practical interest because of the specific structure and required untypical treatment of precipitation data.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 5 , October 2015 , pp. 981 - 1015
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adrian, T. & Brunnermeier, M.K. (2011) CoVaR, Staff Reports 348, Federal Reserve Bank of New York.Google Scholar
Ailliot, P., Thompson, C., & Thomson, P. (2009) Space-time modeling of precipitation by using a hidden Markov model and censored Gaussian distributions. Journal of the Royal Statistical Society 58, 405–426.Google Scholar
Bickel, P.J., Ritov, Y., & Rydén, T. (1998) Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Annals of Statistics 26(4), 1614–1635.CrossRefGoogle Scholar
Bickel, P.J. & Rosenblatt, M. (1973) On some global measures of the deviations of density function estimates. The Annals of Statistics 1, 1071–1095.CrossRefGoogle Scholar
Bradley, R. (1986) Basic properties of strong mixing conditions. In Eberlein, E. & Taqqu, M.S. (eds.), Dependence in Probability and Statistics, pp. 165–192. Birkhauser.CrossRefGoogle Scholar
Caia, Z., Chen, X., Fan, Y., & Wang, X. (2006) Selection of Copulas with Applications in Finance. Working paper. Available athttp://www.economics.smu.edu.sg/femes/2008/papers/219.pdf.Google Scholar
Cappé, O., Moulines, E., & Rydén, T. (2005) Inference in Hidden Markov Models. Springer-Verlag.CrossRefGoogle Scholar
Chen, X. & Fan, Y. (2005) Estimation of copula-based semiparametric time series models. Journal of Econometrics 130(2), 307–335.CrossRefGoogle Scholar
Chen, X. & Fan, Y. (2006) Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. Journal of Econometrics 135, 125–154.CrossRefGoogle Scholar
Dempster, A., Laird, N., & Rubin, D. (1977) Maximum likelihood from incomplete data via the em algorithm (with discussion). Journal of the Royal Statistical Society B 39, 1–38.Google Scholar
Engle, R. (2002) Dynamic conditional correlation. Journal of Business and Economic Statistics 20(3), 339–350.CrossRefGoogle Scholar
Fuh, C.-D. (2003) SPRT and CUSUM in hidden Markov models. Annals of Statistics 31(3), 942–977.CrossRefGoogle Scholar
Gao, X. & Song, P.X.-K. (2011) Composite likelihood EM algorithm with applications to multivariate hidden Markov model. Statistica Sinica 21, 165–185.Google Scholar
Giacomini, E., Härdle, W.K., & Spokoiny, V. (2009) Inhomogeneous dependence modeling with time-varying copulae. Journal of Business and Economic Statistics 27(2), 224–234.CrossRefGoogle Scholar
Hamilton, J. (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57(2), 357–384.CrossRefGoogle Scholar
Härdle, W., Herwartz, H., & Spokoiny, V. (2003) Time inhomogeneous multiple volatility modeling. Journal of Financial econometrics 1(1), 55–95.CrossRefGoogle Scholar
Härdle, W.K., Okhrin, O., & Okhrin, Y. (2013) Dynamic structured copula models. Statistics & Risk Modeling 30(4), 361–388.CrossRefGoogle Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman & Hall.Google Scholar
Leroux, B.G. (1992) Maximum-likelihood estimation for hidden Markov models. Stochastic Processes and their Applications 40, 127–143.CrossRefGoogle Scholar
Liu, W. & Wu, W. (2010) Simultaneous nonparametric inference of time series. The Annals of Statistics 38, 2388–2421.CrossRefGoogle Scholar
McLachlan, G. & Peel, D. (2000) Finite Mixture Models. Wiley.CrossRefGoogle Scholar
McNeil, A.J. & Nešlehová, J. (2009) Multivariate Archimedean copulas, d-monotone functions and l 1 norm symmetric distributions. Annals of Statistics 37(5b), 3059–3097.CrossRefGoogle Scholar
Nelsen, R.B. (2006) An Introduction to Copulas. Springer-Verlag.Google Scholar
Okhrin, O., Okhrin, Y., & Schmid, W. (2013) On the structure and estimation of hierarchical Archimedean copulas. Journal of Econometrics 173, 189–204.CrossRefGoogle Scholar
Okimoto, T. (2008) Regime switching for dynamic correlations. Journal of Financial and Quantitative Analysis 43(3), 787–816.CrossRefGoogle Scholar
Patton, A.J. (2004) On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. Journal of Financial Econometrics 2, 130–168.CrossRefGoogle Scholar
Pelletier, D. (2006) Regime switching for dynamic correlations. Journal of Econometrics 131, 445–473.CrossRefGoogle Scholar
Rabiner, L.R. (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of IEEE 77(2), 257–286.CrossRefGoogle Scholar
Rodriguez, J.C. (2007) Measuring financial contagion: A copula approach. Journal of Empirical Finance 14, 401–423.CrossRefGoogle Scholar
Savu, C. & Trede, M. (2010) Hierarchical Archimedean copulas. Quantitative Finance 10, 295–304.CrossRefGoogle Scholar
Sklar, A. (1959) Fonctions dé repartition á n dimension et leurs marges. Publications de l’Institut de statistique de l’Universit de Paris 8, 299–231.Google Scholar
Whelan, N. (2004) Sampling from Archimedean copulas. Quantitative Finance 4, 339–352.CrossRefGoogle Scholar