Pursuing homogeneity and sparsity in simultaneous quantile regression

Quantile regression analyzes the impact of predictors on the conditional distribution of the response by focusing on a collection of conditional quantiles rather than a single conditional mean. The main goal of this work is to Pursuing Homogeneity in Simultaneous Quantile Regression (PHISQ) to get more interpretable and efficient quantile regression estimation. The new method reveals not only the predictors that are associated with the outcome, but also the true sources of <i>homogeneity</i>, the specific predictors that only have homogeneous effects on the response at the quantiles of interest, and the true sources of heterogeneity, the predictors that have heterogeneous effects across quantile levels of interest. Therefore, the new method may eventually result in a model that is considerably more parsimonious and interpretable. In addition, the new method can pool/borrow information across different quantiles to estimate the homogeneous regression parameters, and thus improve the estimation efficiency, especially at the tails. We demonstrate that the penalized PHISQ method exhibits desirable properties under mild regularity conditions. Simulation results and a real data application further validate the effectiveness of PHISQ. Supplementary materials for the article are available online.