Beyond the Oracle Property: Adaptive LASSO in Cointegrating Regressions

Abstract: This paper establishes new asymptotic results for the adaptive LASSO estimator in cointegrating regression models. We study model selection probabilities, estimator consistency, and limiting distributions under both standard and moving-parameter asymptotics. We also derive uniform convergence rates and the fastest local-to-zero rates that can still be detected by the estimator, complementing and extending the results of Lee, Shi, and Gao (2022, Journal of Econometrics, 229, 322--349). Our main findings include that under conservative tuning, the adaptive LASSO estimator is uniformly $T$-consistent and the cut-off rate for local-to-zero coefficients that can be detected by the procedure is $1/T$. Under consistent tuning, however, both rates are slower and depend on the tuning parameter. The theoretical results are complemented by a detailed simulation study showing that the finite-sample distribution of the adaptive LASSO estimator deviates substantially from what is suggested by the oracle property, whereas the limiting distributions derived under moving-parameter asymptotics provide much more accurate approximations. Finally, we show that our results also extend to models with local-to-unit-root regressors and to predictive regressions with unit-root predictors.