Composite Likelihood Methods for Large Bayesian VARs with Stochastic Volatility

Adding multivariate stochastic volatility of a flexible form to large Vector Autoregressions
(VARs) involving over a hundred variables has proved challenging due to computational
considerations and over-parameterization concerns. The existing literature either works
with homoskedastic models or smaller models with restrictive forms for the stochastic
volatility. In this paper, we develop composite likelihood methods for large VARs with
multivariate stochastic volatility. These involve estimating large numbers of parsimonious
models and then taking a weighted average across these models. We discuss various
schemes for choosing the weights. In our empirical work involving VARs of up to 196
variables, we show that composite likelihood methods have similar properties to existing
alternatives used with small data sets in that they estimate the multivariate stochastic
volatility in a flexible and realistic manner and they forecast comparably. In very high
dimensional VARs, they are computationally feasible where other approaches involving
stochastic volatility are not and produce superior forecasts than natural conjugate prior
homoscedastic VARs.