Achieving parsimony in Bayesian vector autoregressions with the horseshoe prior

In the context of a vector autoregression (VAR) model, or any multivariate regression model, the number of relevant predictors may be small relative to the information set that is available. It is well known that forecasts based on (un-penalized) least squares estimates can overfit the data and lead to poor predictions. Since the Minnesota prior was proposed, there have been many methods developed aiming at improving prediction performance. The horseshoe prior is proposed in the context of a Bayesian VAR. The horseshoe prior is a unique shrinkage prior scheme in that it shrinks irrelevant signals rigorously to 0 while allowing large signals to remain large and practically unshrunk. In an empirical study, it is shown that the horseshoe prior competes favorably with shrinkage schemes commonly used in Bayesian VAR models as well as with a prior that imposes true sparsity in the coefficient vector. Additionally, the use of particle Gibbs with backwards simulation is proposed for the estimation of the time-varying volatility parameters. A detailed description of relevant MCMC methods is provided in the supplementary material.