Bayesian Smoothed Quantile Regression

TL;DR

BSQR introduces a kernel-smoothed, differentiable likelihood for Bayesian quantile regression to overcome non-smoothness and inferential misalignment in standard ALD-based BQR. It establishes posterior consistency, a Bernstein–von Mises theorem under misspecification, and a generalized Wilks phenomenon with calibrated uncertainty, while enabling efficient gradient-based MCMC (HMC/NUTS). The framework yields substantial out-of-sample improvements in prediction and sampler efficiency across simulations and a real-world systemic risk application, with robust finite-sample performance under sparsity and high dimensions. Its modular, plug-and-play structure supports extensions to hierarchical, latent-variable, and time-varying models, offering a practical, rigorous toolkit for distributional econometrics.

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.

Figures (4)

• Figure 1: Smoothing the quantile objective. Comparison between the non-differentiable check loss (dashed gray) and the kernel-smoothed loss $L_h(\cdot; \tau)$ (solid blue). The smoothing removes the singularity at the origin (red circle), enabling gradient-based sampling via HMC.
• Figure 2: Geometric intuition of Bayesian calibration. Under model misspecification, the uncalibrated posterior (red) is typically overly concentrated (overconfident). The scale parameter $\theta$ rescales the posterior covariance (blue) to match the frequentist sampling variance (dashed gray), ensuring valid asymptotic coverage.
• Figure 3: Evolution of dynamic downside and upside systemic risk betas for JPM.
• Figure 4: Sensitivity analysis of the downside beta ($\beta(0.05)$) to bandwidth choice. Posterior distributions for JPM are estimated on the 252-day window ending July 7, 2020. The figure compares the cross-validated bandwidth ($h_{\text{CV}}$, black dashed) with undersmoothed ($0.5h_{\text{CV}}$, orange long-dashed) and oversmoothed ($2.0h_{\text{CV}}$, blue solid) alternatives. While increasing bandwidth systematically shifts the posterior location to the right, the qualitative conclusion of high systemic risk ($\beta > 1$) remains robust.

Theorems & Definitions (37)

• Theorem 1: Posterior consistency of BSQR
• Proposition 1: Finite-sample global existence
• Proposition 2: Rate of convergence
• Remark 1: Justification for undersmoothing
• Theorem 2: Bernstein-von Mises theorem under misspecification
• Remark 2: Choice of kernel and asymptotic universality
• Remark 3: Regularity and efficiency gains
• Corollary 1: Asymptotic coverage and generalized Wilks' phenomenon
• Remark 4: Bayesian implementation strategy
• Theorem 3: Propriety under improper uniform prior for $\boldsymbol{\beta}$