Jakob Stöber, Claudia Czado

Regular vine (R-vine) copulas, which are entirely constructed from bivariate copulas as building blocks, constitute a flexible class of high dimensional dependency models.Since Aas et al. (2009) introduced the R-vine copula structure in an inferential context, it has become popular for the accurate description of fixed dependency structures.However, it lacks the possibility to model dependency structures varying over time as displayed in international financial data sets.In this talk, we attempt to fill in this gap by introducing the so called Markov switching R-vine copula, generalizing work of Cholette et al. (2009).It allows for the structure and parameters of an R-vine copula to discretely change over time, conditional on a hidden underlying state variable, modeled by a Markov chain.Since an n dimensional R-vine copula has n(n-1)/2 parameters and the likelihood surface is possibly multimodal, direct numerical maximization of the likelihood function is difficult. In particular estimating the covariance of parameter estimates using the numerical Hessian matrix often yields covariance matrices which are not positive definite. To deal with these problems, we introduce a new estimation procedure based on the EM algorithm and a novel Bayesian Gibbs sampling method and asses their properties using extensive simulations.While direct application of the EM algorithm is computationally intensive, we overcome this challenge by replacing the maximization step with stepwise maximization of the copula parameters. This results in much faster computations and the stepwise EM algorithm can in particular be applied to obtain starting values for some steps of joint maximization or a Gibbs sampler.For the Bayesian estimation procedure, we extend the work of Min and Czado (2010) to general R-vines and arbitrary bivariate copulas. We incorporate further Gibbs steps to deal with the underlying regime switching model as it has been described in Kim and Nelson (1998) and demonstrate that our procedure is able to identify the true model in simulated data.We conclude with an application of Markov switching vine copulas to exchange rate data from 2005 to 2009 and demonstrate their ability to describe dependency changes during times of crisis.

**References:**

Aas, K., Czado, C., Frigessi, A., and Bakken, H. (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics 44 (2), 182-198.

Cholette, L., Heinen, A., Valdesogo, A. (2009). Modeling International Financial Returns with a Multivariate Regime-Switching Copula. Journal of Financial Econometrics 7 (4), 437-480.

Kim, C.-J., and Nelson, C. R. (1998). Business cycle turning points, a new coincident index, and tests of duration dependence based on a dynamic factor model with regime switching. The Review of Economics and Statistics 80 (2), 188-201.

Min, A., and Czado, C. (2010). Bayesian inference for multivariate copulas using pair-copula constructions. Journal of Financial Econometrics 8 (4), 511-546.

**Keywords:** Pair copula constructions; Markov switching; Regular Vines

**Biography:** Mr. Jakob Stöber is a PhD student at the Department of Mathematics at Technische Universität München. He has been working with Claudia Czado since 2010. His main research interest lies in multivariate copula modeling.