  In this paper we propose the adaptive lasso for predictive quantile
regression (ALQR). Reflecting empirical findings, we allow predictors to have
various degrees of persistence and exhibit different signal strengths. The
number of predictors is allowed to grow with the sample size. We study
regularity conditions under which stationary, local unit root, and cointegrated
predictors are present simultaneously. We next show the convergence rates,
model selection consistency, and asymptotic distributions of ALQR. We apply the
proposed method to the out-of-sample quantile prediction problem of stock
returns and find that it outperforms the existing alternatives. We also provide
numerical evidence from additional Monte Carlo experiments, supporting the
theoretical results.
