Generalized Bayesian Composite Quantile Regression with an Application to Equity Premium Forecasting

Composite quantile regression (CQR) is a robust and efficient estimator under heavy-tailed and contaminated errors. Existing Bayesian extensions rely on working likelihoods that require latent-variable augmentation and can deliver poorly calibrated credible intervals. We develop generalized Bayesian CQR, which exponentiates the composite quantile loss directly, targeting the same objective as frequentist CQR. Because generalized Bayes replaces point optimization with posterior averaging over the loss surface, it is especially relevant under heavy-tailed errors where the composite quantile loss flattens near its minimum. In generalized Bayes posterior dispersion depends on a learning rate that we calibrate by matching marginal variances to their frequentist sandwich counterparts. The resulting credible intervals achieve near-nominal coverage in cross-sectional settings and substantially reduce the undercoverage of i.i.d.\ intervals under serial dependence, with a residual shortfall under high persistence that mirrors the finite-sample bias of frequentist HAC inference. The calibration has a closed-form solution under flat priors and extends to normal and spike-and-slab LASSO priors for shrinkage and variable selection. Sampling uses standard Metropolis-Hastings with no latent variables, achieving roughly 100-fold computational gains over likelihood-based Bayesian CQR at a common quantile grid. Monte Carlo experiments show competitive or improved point estimation relative to frequentist CQR, reliable coverage, and robust variable selection across Gaussian, heavy-tailed, and contaminated error distributions. An equity premium forecasting application demonstrates that the efficiency and robustness gains translate into economically meaningful improvements in out-of-sample portfolio performance.