We introduce a semi-parametric Bayesian framework for a simultaneous
analysis of linear quantile regression models. A simultaneous analysis is essential
to attain the true potential of the quantile regression framework, but is computa-
tionally challenging due to the associated monotonicity constraint on the quantile
curves. For a univariate covariate, we present a simpler equivalent characterization
of the monotonicity constraint through an interpolation of two monotone curves.
The resulting formulation leads to a tractable likelihood function and is embedded
within a Bayesian framework where the two monotone curves are modeled via lo-
gistic transformations of a smooth Gaussian process. A multivariate extension is
suggested by combining the full support univariate model with a linear projection
of the predictors. The resulting single-index model remains easy to ¯t and provides
substantial and measurable improvement over the ¯rst order linear heteroscedastic
model. Two illustrative applications of the proposed method are provided.