Vector autoregressions with stochastic volatility in both the conditional mean and variance are commonly used to estimate the macroeconomic effects of uncertainty shocks. Despite their popularity, intensive computational demands when estimating such models have made out-of-sample forecasting exercises impractical, particularly when working with large data sets.
In the new CAMP working paper 04/2021, Cross, Hou, Koop and Poon, propose an efficient Markov chain Monte Carlo (MCMC) algorithm for posterior and predictive inference in such models that facilitates such exercises. The key insight underlying the algorithm is that the (log-)conditional densities of the log volatilities possess Hessian matrices that are banded. This enables the authors to build upon recent advances in band and sparse matrix algorithms for state space models. In a simulation exercise, they evaluate the new algorithm numerically and establish its computational and statistical efficiency over a conventional particle filter based algorithm. Using macroeconomic data for the US they find that such models generally deliver more accurate point and density forecasts over a conventional benchmark in which stochastic volatility only enters the variance of the model.