Bayesian Smoothed Quantile Regression

TL;DR

The paper introduces Bayesian smoothed quantile regression (BSQR), a novel approach that improves inference accuracy, computational efficiency, and practical applicability over traditional methods by using a differentiable likelihood and kernel smoothing.

Contribution

BSQR provides a fully differentiable likelihood for Bayesian quantile regression, reducing bias, enabling gradient-based inference, and establishing theoretical guarantees including posterior consistency.

Findings

Reduces out-of-sample prediction error by up to 50%.

Improves sampling efficiency by up to 80%.

Uncovers regime shifts in financial systemic risk analysis.

Abstract

The standard asymmetric Laplace framework for Bayesian quantile regression (BQR) suffers from a fundamental decision-theoretic misalignment, yielding biased finite-sample estimates, and precludes gradient-based computation due to non-smoothness. We propose Bayesian smoothed quantile regression (BSQR), a principled framework built on a kernel-smoothed, fully differentiable likelihood. Methodologically, the symmetrizing property of our objective reduces inferential bias and aligns the posterior mean with the true conditional quantile. Theoretically, we establish posterior consistency and a Bernstein--von Mises theorem under misspecification, delivering asymptotic normality and valid frequentist coverage via a generalized Wilks phenomenon, while guaranteeing global posterior existence unlike empirical likelihood approaches. Computationally, BSQR enables Hamiltonian Monte Carlo for BQR, alleviating high-dimensional mixing bottlenecks. In simulations, BSQR reduces out-of-sample prediction error by up to 50% and improves sampling efficiency by up to 80% relative to asymmetric Laplace benchmarks, with uniform and triangular kernels performing particularly well. In a financial application to asymmetric systemic risk, BSQR uncovers distinct regime shifts around the COVID-19 period and yields sharper yet well-calibrated predictive quantiles, underscoring its practical relevance.