Modelling Volatility and Correlations with a Hidden Markov Decision Tree

Mardi | 2011-02-01

Philippe CHARLOT – libre

The goal of the present paper is to present a new multivariate GARCH model with time varying conditional correlation. Since the seminal work of Bollerslev (1990), conditional correlation models have become a attractive field in economics. Different specifications have been developed to study both empirical findings and practical use like asymmetry, change in regime but also estimation of large correlation matrix (see, e.g. Silvennoinen and Teräsvirta (2009) for a survey of recent advances). Among this field of research, our work focus on change in regime specification based on tree structure. Indeed, tree-structured dynamic correlation models has been developed to analyse volatility and covolatility asymmetries (see Dellaportas and Vrontos (2007)) or linking the dynamics of the individual volatilities with the dynamics of the correlations (see Audrino and Trojani (2006)). The common approach of these models is to partitioning the space of time series recursively using binary decisions. This can be interpreted as a deterministic decision tree. At the opposite, the approach that we adopt for this paper is developed around the idea of hierarchical architecture with a Markov temporal structure. Our model is based on an extension of Hidden Markov Model (HMM) introduced by Jordan, Ghahramani, and Saul (1997). It is a factorial and coupled HMM. Hence, our model is based is a stochastic decision tree liking the dynamics of univariate volatility with the dynamics of the correlations. It can be view as a HMM which is both factorial and dependent coupled. The factorial decomposition provides a factorized state space. This state space decomposition is done using state dependent and time-varying transition probabilities given an input variable. The top level of the tree can be seen as a master process and the following levels as slave processes. The constraint of a level on the following is done via a coupling transition matrix which produce the ordered hierarchy of the structure. As the links between decision states are driven with Markovian dynamics andthe switch from one level to the following is done via a coupling transition matrix, this architecture gives a fully probabilistic decision tree. Estimation is done in one step using maximum likelihood. We also perform an empirical analysis of our model using real financial time series. Results show that our hidden tree-structured model can be an interesting alternative to deterministic decision tree.