With uncertain changes of the economic environment, macroeconomic downturns during recessions and crises can hardly be explained by a Gaussian structural shock. There is evidence that the distribution of macroeconomic variables is skewed and heavy tailed. In this paper, we contribute to the literature by extending a vector autoregression (VAR) model to account for more realistic assumptions on the multivariate distribution of macroeconomic variables. We propose a general class of generalized hyperbolic skew Student's t distribution with stochastic volatility for the innovations in the VAR model that allows us to take into account both skewness and heavy tails. Tools for Bayesian inference and model selection using a Gibbs sampler are provided. In an empirical study, we present evidence of skewness and heavy tails for monthly macroeconomic variables. The analysis also gives a clear message that skewness is a value-added feature to VAR models with heavy tails. (C) 2022 The Author(s). Published by Elsevier B.V.