Abstract: Quantile regression estimators at a fixed quantile level rely mainly on a small subset of the observed data. As a result, efforts have been made to construct simultaneous estimations at multiple quantile levels in order to take full advantage of all observations and to improve the estimation efficiency. We propose a novel approach that links multiple linear quantile models by imposing a condition on the rank of the matrix formed by all of the regression parameters. This approach resembles a reduced-rank regression, but also shares similarities with the dimension-reduction modeling. We develop estimation and inference tools for such models and examine their optimality in terms of the asymptotic estimation variance. We use simulation experiments to examine the numerical performance of the proposed procedure, and a data example to further illustrate the method.

Key words and phrases: Check function, composite quantile regression, generalized method of moment, linear quantile regression, optimal estimating equations, quantile regression, reduced-rank regression.