Bayesian Multivariate Two‐Stage Volatility‐Return Models (MTS‐VR)

ABSTRACT This paper develops a class of models to capture the joint dynamics of asset returns and multivariate volatility. Owing to its two‐stage construction, this class is referred to as the multivariate two‐stage volatility‐return (MTS‐VR) model. The first stage of the MTS‐VR model focuses on modelling the dynamic trend of the multivariate stochastic volatility process. This stage is constructed by calibrating the model using time‐varying covariance matrices, which are estimated from multiple financial asset return series. Within this component of the MTS‐VR framework, we propose two new classes of dynamic volatility models formulated under a Bayesian paradigm. The first class directly models the stochastic dynamics of the covariance matrix, whereas the second separately models each asset's stochastic volatility and the dynamics of the correlation structure. The second stage of the MTS‐VR model involves the modelling of the multivariate asset returns profiles, conditional on the trend model of the multivariate volatility dynamic obtained from Stage 1. This involves a class of multivariate Student‐ t stochastic models, conditionally driven by the volatility dynamic of Stage 1 and a return trend process. We evaluate the performance of the proposed models using both simulated and financial data. The models are implemented in Rtan within a Bayesian framework. The simulation study examines the effect of the degrees of freedom parameter of the Wishart distribution in the data‐generating process. In the empirical application, the second model specification is preferred based on its superior performance.