  We propose a novel approach to elicit the weight of a potentially
non-stationary regressor in the consistent and oracle-efficient estimation of
autoregressive models using the adaptive Lasso. The enhanced weight builds on a
statistic that exploits distinct orders in probability of the OLS estimator in
time series regressions when the degree of integration differs. We provide
theoretical results on the benefit of our approach for detecting stationarity
when a tuning criterion selects the $\ell_1$ penalty parameter. Monte Carlo
evidence shows that our proposal is superior to using OLS-based weights, as
suggested by Kock [Econom. Theory, 32, 2016, 243-259]. We apply the modified
estimator to model selection for German inflation rates after the introduction
of the Euro. The results indicate that energy commodity price inflation and
headline inflation are best described by stationary autoregressions.
