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

Hardy, Nicolas and Korobilis, Dimitris (2026): Generalized Bayesian Composite Quantile Regression with an Application to Equity Premium Forecasting.

Abstract

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: Composite quantile regression, Gibbs posterior, Generalized Bayes, Learning rate calibration, Equity premium forecasting, Spike-and-slab priors
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C11 - Bayesian Analysis: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C21 - Cross-Sectional Models ; Spatial Models ; Treatment Effect Models ; Quantile Regressions; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; E - Macroeconomics and Monetary Economics > E3 - Prices, Business Fluctuations, and Cycles > E37 - Forecasting and Simulation: Models and Applications; G - Financial Economics > G1 - General Financial Markets > G17 - Financial Forecasting and Simulation

References

Adrian, T., Boyarchenko, N., and Giannone, D. (2019). Vulnerable growth. American Economic Review, 109(4), 1263–1289.

Agnoletto, D., Rigon, T., and Dunson, D.B. (2025). Bayesian inference for generalized linear models via quasi-posteriors. Biometrika, 112(2), asaf022.

Alquier, P., Ridgway, J., and Chopin, N. (2016). On the properties of variational approximations of Gibbs posteriors. Journal of Machine Learning Research, 17(1), 1–41.

Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817–858. Arnroth, L. (2025). Bayesian composite Lp-quantile regression. Metrika, 88(1), 83–97.

Bissiri, P.G., Holmes, C.C., and Walker, S.G. (2016). A general framework for updating belief distributions. Journal of the Royal Statistical Society: Series B, 78(5), 1103–1130.

Campbell, J.Y. and Thompson, S.B. (2008). Predicting excess stock returns out of sample: Can anything beat the historical average? Review of Financial Studies, 21(4), 1509–1531.

Catoni, O. (2004). Statistical Learning Theory and Stochastic Optimization. Lecture Notes in Mathematics, Vol. 1851. Springer, Berlin.

Chernozhukov, V. and Hong, H. (2003). An MCMC approach to classical estimation. Journal of Econometrics, 115(2), 293–346.

Clark, T.E. and West, K.D. (2007). Approximately normal tests for equal predictive accuracy in nested models. Journal of Econometrics, 138(1), 291–311.

Frazier, D.T., Drovandi, C., and Kohn, R. (2023). Calibrated generalized Bayesian inference. arXiv preprint arXiv:2311.15485.

Gelman, A., Gilks, W.R., and Roberts, G.O. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. The Annals of Applied Probability, 7(1), 110–120.

Glosten, L.R., Jagannathan, R., and Runkle, D.E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779–1801.

Goyal, A. and Welch, I. (2008). A comprehensive look at the empirical performance of equity premium prediction. Review of Financial Studies, 21(4), 1455–1508.

Goyal, A., Welch, I., and Zafirov, A. (2024). A comprehensive 2022 look at the empirical performance of equity premium prediction. Review of Financial Studies, 37(11), 3490–3557.

Grünwald, P. and van Ommen, T. (2017). Inconsistency of Bayesian inference for misspecified linear models, and a proposal for repairing it. Bayesian Analysis, 12(4), 1069–1103.

Han, H., Linton, O., Oka, T., and Whang, Y.-J. (2016). The cross-quantilogram: Measuring quantile dependence and testing directional predictability between time series. Journal of Econometrics, 193(1), 251–270.

Huang, H. and Chen, Z. (2015). Bayesian composite quantile regression. Journal of Statistical Computation and Simulation, 85(18), 3744–3754.

Huber, P.J. (1967). The behavior of maximum likelihood estimates under non-standard conditions. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 221–233. University of California Press, Berkeley.

Jiang, W. and Tanner, M.A. (2008). Gibbs posterior for variable selection in high-dimensional classification and data mining. Annals of Statistics, 36(5), 2207–2231.

Kai, B., Li, R., and Zou, H. (2010). Local composite quantile regression smoothing: An efficient and safe alternative to local polynomial regression. Journal of the Royal Statistical Society Series B: Statistical Methodology, 72(1), 49–69.

Koenker, R. (2005). Quantile Regression. Econometric Society Monographs, No. 38. Cambridge University Press, Cambridge.

Koenker, R. and Bassett, G. (1978). Regression quantiles. Econometrica, 46(1), 33–50.

Kozumi, H. and Kobayashi, G. (2011). Gibbs sampling methods for Bayesian quantile regression. Journal of Statistical Computation and Simulation, 81(11), 1565–1578.

Ledoit, O. and Wolf, M. (2008). Robust performance hypothesis testing with the Sharpe ratio. Journal of Empirical Finance, 15(5), 850–859.

Linton, O. and Whang, Y.-J. (2007). The quantilogram: With an application to evaluating directional predictability. Journal of Econometrics, 141(1), 250–282.

Lyddon, S.P., Holmes, C.C., and Walker, S.G. (2019). General Bayesian updating and the loss-likelihood bootstrap. Biometrika, 106(2), 465–478.

Murphy, S.A. and Van der Vaart, A.W. (2000). On profile likelihood. Journal of the American Statistical Association, 95(450), 449–465.

Neely, C.J., Rapach, D.E., Tu, J., and Zhou, G. (2014). Forecasting the equity risk premium: The role of technical indicators. Management Science, 60(7), 1772–1791.

Politis, D.N. and White, H. (2004). Automatic block-length selection for the dependent bootstrap. Econometric Reviews, 23(1), 53–70.

Powell, J.L. (1991). Estimation of monotonic regression models under quantile restrictions. In Barnett, W.A., Powell, J., and Tauchen, G. (Eds.), Nonparametric and Semiparametric Methods in Econometrics, 357–384. Cambridge University Press, Cambridge.

Ročková, V. and George, E.I. (2018). The spike-and-slab LASSO. Journal of the American Statistical Association, 113(521), 431–444.

Sriram, K., Ramamoorthi, R.V., and Ghosh, P. (2013). Posterior consistency of Bayesian quantile regression based on the misspecified asymmetric Laplace density. Bayesian Analysis, 8(4), 879–898.

Syring, N. and Martin, R. (2019). Calibrating general posterior credible regions. Biometrika, 106(2), 479–495.

Syring, N. and Martin, R. (2022). Direct Gibbs posterior inference on risk minimizers: Construction, concentration, and calibration. In Handbook of Statistics, Vol. 47, 1–41. Elsevier.

White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1), 1–25.

Winter, S., Melikechi, O., and Dunson, D.B. (2026). Sequential Gibbs posteriors with applications to principal component analysis. Biometrika. Forthcoming.

Wu, P.-S. and Martin, R. (2023). A comparison of learning rate selection methods in generalized Bayesian inference. Bayesian Analysis, 18(1), 105–132.

Yang, Y. and He, X. (2012). Bayesian empirical likelihood for quantile regression. Annals of Statistics, 40(2), 1102–1131.

Yang, Y., Wang, H.J., and He, X. (2016). Posterior inference in Bayesian quantile regression with asymmetric Laplace likelihood. International Statistical Review, 84(3), 327–344.

Yu, K. and Moyeed, R.A. (2001). Bayesian quantile regression. Statistics & Probability Letters, 54(4), 437–447.

Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418–1429.

Zou, H. and Yuan, M. (2008). Composite quantile regression and the oracle model selection theory. Annals of Statistics, 36(3), 1108–1126.