Extended Oracle Properties of Adaptive Lasso Estimators

We study the asymptotic properties of adaptive lasso estimators when some components of the parameter of interest β are strictly different than zero, while other components may be zero or may converge to zero with rate n^-δ, with δ>0, where n denotes the sample size. To achieve this objective, we analyze the convergence/divergence rates of each term in the first-order conditions of adaptive lasso estimators. First, we derive conditions that allow selecting tuning parameters in order to ensure that adaptive lasso estimates of n^-δ-components indeed collapse to zero. Second, in this case, we also derive asymptotic distributions of adaptive lasso estimators for nonzero components. When δ>1/2, we obtain the usual n^1/2-asymptotic normal distribution, while when 0<δ ≤ 1/2, we show nδ-consistency combined with (biased) n^1/2-δ-asymptotic normality for nonzero components. We call these properties, Extended Oracle Properties. These results allow practitioners to exclude in their model the asymptotically negligible variables and make inferences on the asymptotically relevant variables.