  Explanatory variables in a predictive regression typically exhibit low signal
strength and various degrees of persistence. Variable selection in such a
context is of great importance. In this paper, we explore the pitfalls and
possibilities of the LASSO methods in this predictive regression framework. In
the presence of stationary, local unit root, and cointegrated predictors, we
show that the adaptive LASSO cannot asymptotically eliminate all cointegrating
variables with zero regression coefficients. This new finding motivates a novel
post-selection adaptive LASSO, which we call the twin adaptive LASSO (TAlasso),
to restore variable selection consistency. Accommodating the system of
heterogeneous regressors, TAlasso achieves the well-known oracle property. In
contrast, conventional LASSO fails to attain coefficient estimation consistency
and variable screening in all components simultaneously. We apply these LASSO
methods to evaluate the short- and long-horizon predictability of S\&P 500
excess returns.
