This paper extends the horseshoe prior of Carvalho et al. (2010) to the Bayesian quantile regression (HS-BQR) and provides a fast sampling algorithm that speeds up computation significantly in high dimensions. The performance of the  HS-BQR is tested on large scale Monte Carlo simulations and an empirical application relevant to macroeoncomics. The Monte Carlo design considers several sparsity structures (sparse, dense, block) and error structures (i. i.d. errors and heteroskedastic errors). A number of LASSO based estimators (frequentist and Bayesian) are pitted against the HS-BQR to better gauge the performance of the method on the different designs. The HS-BQR yields just as good, or better performance than the other estimators considered when evaluated using coefficient bias and forecast error. We find that the HS-BQR is particularly potent in sparse designs and when estimating extreme quantiles. The simulations also highlight how the high dimensional quantile estimators fail to correctly identify the quantile function of the variables when both location and scale effects are present . In the empirical application, in which we evaluate forecast densities of US inflation, the HS-BQR provides well calibrated forecast densities whose individual quantiles, have the highest pseudo R squared, highlighting its potential for Value-at-Risk estimation .
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 .
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