A new mixture vector autoregressive model based on Gaussian and Student's $t$ distributions is introduced. As its mixture components, our model incorporates conditionally homoskedastic linear Gaussian vector autoregressions and conditionally heteroskedastic linear Student's $t$ vector autoregressions. For a $p$th order model, the mixing weights depend on the full distribution of the preceding $p$ observations, which leads to attractive practical and theoretical properties such as ergodicity and full knowledge of the stationary distribution of $p+1$ consecutive observations. A structural version of the model with statistically identified shocks is also proposed. The empirical application studies the effects of the Euro area monetary policy shock. We fit a two-regime model to the data and find the effects, particularly on inflation, stronger in the regime that mainly prevails before the Financial crisis than in the regime that mainly dominates after it. The introduced methods are implemented in the accompanying R package gmvarkit.Comment: arXiv admin note: text overlap with arXiv:2007.04713