Beyond the Oracle Property: Adaptive LASSO in Cointegrating Regressions with Local-to-Unity Regressors

This paper derives new asymptotic results for the adaptive LASSO estimator in cointegrating regressions, allowing for uncertainty about whether the regressors are exact unit root processes. We study model selection probabilities, estimator consistency, and limiting distributions under standard and moving-parameter asymptotics. We further derive uniform convergence rates and the fastest local-to-zero rates detectable by the estimator under conservative and consistent tuning. For consistent tuning, we construct confidence regions that are easy to implement, uniformly valid over the parameter space, and achieve sure asymptotic coverage without requiring knowledge or estimation of local-to-unity or long-run covariance parameters. Simulation results reveal that the finite-sample distribution of the adaptive LASSO estimator can deviate substantially from the oracle property, whereas moving-parameter asymptotics provide much more accurate approximations. Consequently, in addition to being infeasible in applications due to their dependence on non-estimable nuisance parameters, oracle-based confidence regions are often too small to achieve adequate coverage in empirically relevant scenarios with small but non-zero coefficients. In contrast, the proposed confidence regions are always feasible and deliver reliable coverage across the parameter space. An empirical application to predicting the U.S. unemployment rate illustrates their practical usefulness for quantifying uncertainty around adaptive LASSO estimates.