  We show that the activation knot of a potentially non-stationary regressor on
the adaptive Lasso solution path in autoregressions can be leveraged for
selection-free inference about a unit root. The resulting test has asymptotic
power against local alternatives in $1/T$ neighbourhoods, unlike post-selection
inference methods based on consistent model selection. Exploiting the
information enrichment principle devised by Reinschl\"ussel and Arnold
arXiv:2402.16580 [stat.ME] to improve the Lasso-based selection of ADF models,
we propose a composite statistic and analyse its asymptotic distribution and
local power function. Monte Carlo evidence shows that the combined test
dominates the comparable post-selection inference methods of Tibshirani et al.
[JASA, 2016, 514, 600-620] and may surpass the power of established unit root
tests against local alternatives. We apply the new tests to groundwater level
time series for Germany and find evidence rejecting stochastic trends to
explain observed long-term declines in mean water levels.
