  The Panel Vector Autoregressive (PVAR) model is a popular tool for
macroeconomic forecasting and structural analysis in multi-country applications
since it allows for spillovers between countries in a very flexible fashion.
However, this flexibility means that the number of parameters to be estimated
can be enormous leading to over-parameterization concerns. Bayesian
global-local shrinkage priors, such as the Horseshoe prior used in this paper,
can overcome these concerns, but they require the use of Markov Chain Monte
Carlo (MCMC) methods rendering them computationally infeasible in high
dimensions. In this paper, we develop computationally efficient Bayesian
methods for estimating PVARs using an integrated rotated Gaussian approximation
(IRGA). This exploits the fact that whereas own country information is often
important in PVARs, information on other countries is often unimportant. Using
an IRGA, we split the the posterior into two parts: one involving own country
coefficients, the other involving other country coefficients. Fast methods such
as approximate message passing or variational Bayes can be used on the latter
and, conditional on these, the former are estimated with precision using MCMC
methods. In a forecasting exercise involving PVARs with up to $18$ variables
for each of $38$ countries, we demonstrate that our methods produce good
forecasts quickly.
