Abstract: We study joint quantile regression at multiple quantile levels with high-dimensional covariates. Variable selection performed at individual quantile levels may lack stability across neighboring quantiles, making it difficult to understand and to interpret the impact of a given covariate on conditional quantile functions. We propose a Dantzig–type penalization method for sparse model selection at each quantile level which, at the same time, aims to shrink differences of the selected models across neighboring quantiles. We show model selection consistency, and investigate the stability of the selected models across quantiles. We also provide asymptotic normality of post–model–selection parameter estimation in the multiple quantile framework. We use numerical examples and data analysis to demonstrate that the proposed Dantzig–type quantile regression model selection method provides stable models for both homogeneous and heterogeneous cases.

Key words and phrases: Fused lasso, high dimensional data, model selection, quantile regression, stability.