Econometrics Journal (2007), volume 10, pp. 503–520. doi: 10.1111/j.1368-423X.2007.00219.x Modelling volatility asymmetries: a Bayesian analysis of a class of tree structured multivariate GARCH models P. DELLAPORTAS AND I. D. VRONTOS Department of Statistics, Athens University of Economics and Business, Patission 76, 10434 Athens, Greece Email: petros@aueb.gr, vrontos@aueb.gr First version received: September 2005; final version accepted: April 2007 Summary A new class of multivariate threshold GARCH models is proposed for the analysis and modelling of volatility asymmetries in financial time series. The approach is based on the idea of a binary tree where every terminal node parametrizes a (local) multivariate GARCH model for a specific partition of the data. A Bayesian stochastic method is developed and presented for the analysis of the proposed model consisting of parameter estimation, model selection and volatility prediction. A computationally feasible algorithm that explores the posterior distribution of the tree structure is designed using Markov chain Monte Carlo stochastic search methods. Simulation experiments are conducted to assess the performance of the proposed method, and an empirical application of the proposed model is illustrated using real financial time series. Key words: Autoregressive conditional heteroscedasticity, Bayesian inference, Markov chain Monte Carlo, Stochastic search, Tree structured models. 1. INTRODUCTION Modelling and forecasting time-varying conditional variances and covariances between asset returns is crucial for many issues in financial econometrics, such as asset pricing, portfolio selection and risk management. Since the seminal work of Engle (1982) and Bollerslev (1986) on Autoregressive Conditional Heteroscedasticity (ARCH) and Generalized ARCH (GARCH), several univariate and multivariate GARCH-type models have been proposed and widely applied in financial time series; see, for example, Bollerslev, Chou and Kroner (1992), Bollerslev, Engle and Nelson (1994), Li, Ling and McAleer (2002), Bauwens, Laurent and Rombouts (2006) and Gourieroux (1997), for comprehensive surveys of theoretical and empirical developments and applications. While univariate GARCH models are able to capture several features often met in financial data, such as thick tails and volatility clustering, they cannot capture an important stylized fact, the asymmetric volatility effect. This is observed when a negative return shock leads to a higher volatility than a positive return shock of the same magnitude. Several univariate models have been proposed to capture this property; see, for example, the Exponential GARCH model of Nelson (1991), the asymmetric models of Glosten, Jagannathan and Runkle (1993) and Engle and Ng (1993) and the Threshold ARCH model of Zakoian (1994). C  Royal Economic Society 2007. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA, 02148, USA.