  We propose a novel variational Bayes approach to estimate high-dimensional
vector autoregression (VAR) models with hierarchical shrinkage priors. Our
approach does not rely on a conventional structural VAR representation of the
parameter space for posterior inference. Instead, we elicit hierarchical
shrinkage priors directly on the matrix of regression coefficients so that (1)
the prior structure directly maps into posterior inference on the reduced-form
transition matrix, and (2) posterior estimates are more robust to variables
permutation. An extensive simulation study provides evidence that our approach
compares favourably against existing linear and non-linear Markov Chain Monte
Carlo and variational Bayes methods. We investigate both the statistical and
economic value of the forecasts from our variational inference approach within
the context of a mean-variance investor allocating her wealth in a large set of
different industry portfolios. The results show that more accurate estimates
translate into substantial statistical and economic out-of-sample gains. The
results hold across different hierarchical shrinkage priors and model
dimensions.
