On the Oracle Property of the Adaptive Lasso in Stationary and Nonstationary Autoregressions

We show that the Adaptive LASSO is oracle efficient in stationary and non-stationary autoregressions. This means that it estimates parameters consistently, selects the correct sparsity pattern, and estimates the coefficients belonging to the relevant variables at the same asymptotic efficiency as if only these had been included in the model from the outset. In particular this implies that it is able to discriminate between stationary and non-stationary autoregressions and it thereby constitutes an addition to the set of unit root tests.
However, it is also shown that the Adaptive LASSO has no power against shrinking alternatives of the form c/T where c is a constant and T the sample size if it is tuned to perform consistent model selection. We show that if the Adaptive LASSO is tuned to performed conservative model selection it has power even against shrinking alternatives of this form.
Monte Carlo experiments reveal that the Adaptive LASSO performs particularly well in the presence of a unit root while being at par with its competitors in the stationary setting.