A Bayesian Approach to Multiple-Output Quantile Regression

This paper presents a Bayesian approach to multiple-output quantile regression. The unconditional model is proven to be consistent and asymptotically correct frequentist confidence intervals can be obtained. The prior for the unconditional model can be elicited as the ex-ante knowledge of the distance of the tau-Tukey depth contour to the Tukey median, the first prior of its kind. A proposal for conditional regression is also presented. The model is applied to the Tennessee Project Steps to Achieving Resilience (STAR) experiment and it finds a joint increase in tau-quantile subpopulations for mathematics and reading scores given a decrease in the number of students per teacher. This result is consistent with, and much stronger than, the result one would find with multiple-output linear regression. Multiple-output linear regression finds the average mathematics and reading scores increase given a decrease in the number of students per teacher. However, there could still be subpopulations where the score declines. The multiple-output quantile regression approach confirms there are no quantile subpopulations (of the inspected subpopulations) where the score declines. This is truly a statement of `no child left behind' opposed to `no average child left behind.'