  We consider parameter inference for linear quantile regression with
non-stationary predictors and errors, where the regression parameters are
subject to inequality constraints. We show that the constrained quantile
coefficient estimators are asymptotically equivalent to the metric projections
of the unconstrained estimator onto the constrained parameter space. Utilizing
a geometry-invariant property of this projection operation, we propose
inference procedures - the Wald, likelihood ratio, and rank-based methods -
that are consistent regardless of whether the true parameters lie on the
boundary of the constrained parameter space. We also illustrate the advantages
of considering the inequality constraints in analyses through simulations and
an application to an electricity demand dataset.
