  Large Bayesian VARs are now widely used in empirical macroeconomics. One
popular shrinkage prior in this setting is the natural conjugate prior as it
facilitates posterior simulation and leads to a range of useful analytical
results. This is, however, at the expense of modeling flexibility, as it rules
out cross-variable shrinkage -- i.e., shrinking coefficients on lags of other
variables more aggressively than those on own lags. We develop a prior that has
the best of both worlds: it can accommodate cross-variable shrinkage, while
maintaining many useful analytical results, such as a closed-form expression of
the marginal likelihood. This new prior also leads to fast posterior simulation
-- for a BVAR with 100 variables and 4 lags, obtaining 10,000 posterior draws
takes less than half a minute on a standard desktop. We demonstrate the
usefulness of the new prior via a structural analysis using a 15-variable VAR
with sign restrictions to identify 5 structural shocks.
