Bayesian Vector Autoregressions with Stochastic Volatility

This paper proposes a Bayesian approach to a vector autoregression with stochastic
volatility, where the multiplicative evolution of the precision matrix is driven by a
multivariate beta variate. Exact updating formulas are given to the nonlinear filtering of
the precision matrix. Estimation of the autoregressive parameters requires numerical
methods: an importance-sampling based approach is explained here.
KEYWORDS:Stochastic volatility, Bayesian vector autoregression, conjugacy, multivariate
beta distribution, vector autoregression.
1. INTRODUCTION
THISPAPER INTRODUCES Bayesian vector autoregressions with stochastic volatility.
In contrast to multivariate autoregressive conditional heteroskedasticity
(ARCH), the stochastic volatility setup here models the error precision matrix as
an unobserved component with shocks drawn from a multivariate beta distribution.
This allows the interpretation of a sudden large movement in the data as
the result of a draw from a distribution with a randomly increased but unobserved
variance. Exploiting a conjugacy between Wishart distributions and
multivariate singular beta distributions, the integration over the unobserved
shock to the precision matrix can be performed in closed form, leading to a
generalization of the standard Kalman-Filter formulas to the nonlinear filtering
problem at hand. Estimating the autoregressive parameters requires numerical
methods, however. The paper focusses on an importance-sampling based approach.