  Conjugate priors allow for fast inference in large dimensional vector
autoregressive (VAR) models but, at the same time, introduce the restriction
that each equation features the same set of explanatory variables. This paper
proposes a straightforward means of post-processing posterior estimates of a
conjugate Bayesian VAR to effectively perform equation-specific covariate
selection. Compared to existing techniques using shrinkage alone, our approach
combines shrinkage and sparsity in both the VAR coefficients and the error
variance-covariance matrices, greatly reducing estimation uncertainty in large
dimensions while maintaining computational tractability. We illustrate our
approach by means of two applications. The first application uses synthetic
data to investigate the properties of the model across different
data-generating processes, the second application analyzes the predictive gains
from sparsification in a forecasting exercise for US data.
