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.

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