We compare predictive performance of a multitude of alternative Bayesian vector autoregression (VAR) models allowing for cointegration and time‐varying conditional covariances, described by different multivariate stochastic volatility (MSV) models, including their hybrids with multivariate GARCH processes (MSV‐MGARCH), as well as t‐GARCH and Markov‐switching structures. The forecast accuracy is evaluated mainly through predictive Bayes factors, but energy scores and the probability integral transform are also used. Two empirical studies, for the US and Polish economies, are based on a small model of monetary policy comprising inflation, unemployment and interest rate. The results indicate that capturing conditional heteroskedasticity by some MSV‐MGARCH specifications contributes the most to the forecasting power of the VAR/VEC model.