Before revision:  With the presence of stationary, unit root and cointegrated predictors, we show that the adaptive LASSO maintains the consistent variable selection and the oracle property due to its penalty scheme that accommodates the system of regressors .
After revision:  With the presence of stationary, unit root and cointegrated predictors, we show that the adaptive LASSO asymptotically breaks cointegrated groups although it cannot wipe out all inactive cointegrating variables. This new finding motivates a simple but novel post-selection adaptive LASSO, which we call the twin adaptive LASSO (TAlasso), to fix variable selection inconsistency. TAlasso's penalty scheme accommodates the system of regressors .