  This paper extends the horseshoe prior of Carvalho et al. (2010) to Bayesian
quantile regression (HS-BQR) and provides a fast sampling algorithm for
computation in high dimensions. The performance of the proposed HS-BQR is
evaluated on Monte Carlo simulations and a high dimensional Growth-at-Risk
(GaR) forecasting application for the U.S. The Monte Carlo design considers
several sparsity and error structures. Compared to alternative shrinkage
priors, the proposed HS-BQR yields better (or at worst similar) performance in
coefficient bias and forecast error. The HS-BQR is particularly potent in
sparse designs and in estimating extreme quantiles. As expected, the
simulations also highlight that identifying quantile specific location and
scale effects for individual regressors in dense DGPs requires substantial
data. In the GaR application, we forecast tail risks as well as complete
forecast densities using the McCracken and Ng (2020) database. Quantile
specific and density calibration score functions show that the HS-BQR provides
the best performance, especially at short and medium run horizons. The ability
to produce well calibrated density forecasts and accurate downside risk
measures in large data contexts makes the HS-BQR a promising tool for
nowcasting applications and recession modelling.
