  In this paper we study the benefit of using the adaptive LASSO for predictive
quantile regression. It is common that predictors in predictive quantile
regression have various degrees of persistence and exhibit different signal
strengths in explaining the dependent variable. We show that the adaptive LASSO
has the consistent variable selection and the oracle properties under the
simultaneous presence of stationary, unit root and cointegrated predictors.
Some encouraging simulation and out-of-sample prediction results are reported.
