Horseshoe Prior Bayesian Quantile Regression

TL;DR

This paper introduces a Bayesian quantile regression method using the horseshoe prior, optimized for high-dimensional data, and demonstrates its superior performance in simulations and economic risk forecasting.

Contribution

It extends the horseshoe prior to Bayesian quantile regression and develops a fast sampling algorithm suitable for high-dimensional settings.

Findings

HS-BQR outperforms alternative priors in bias and forecast error.

HS-BQR is especially effective for sparse models and extreme quantiles.

It provides accurate density forecasts and downside risk measures.

Abstract

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.