Tuning parameter selectors for the smoothly clipped absolute deviation method

TLDR

The smoothly clipped absolute deviation (SCAD) penalty is a powerful regression shrinkage and selection method that automatically selects important variables and achieves oracle‑efficient estimators, but its performance hinges on choosing an appropriate tuning parameter. This study proposes a Bayesian information criterion (BIC)–based selector for the SCAD tuning parameter to consistently recover the true model. The BIC selector evaluates candidate tuning parameters by minimizing the BIC criterion applied to the SCAD penalized least squares fit. Compared with generalized cross‑validation, which induces noticeable overfitting, the BIC selector consistently identifies the true model, as confirmed by simulation studies and an empirical application to Female Labor Supply data.

Abstract

The penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method. It not only automatically and consistently selects the important variables, but also produces estimators which are as efficient as the oracle estimator. However, these attractive features depend on appropriate choice of the tuning parameter. We show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily, with a nonignorable overfitting effect in the resulting model. In addition, we propose a BIC tuning parameter selector, which is shown to be able to identify the true model consistently. Simulation studies are presented to support theoretical findings, and an empirical example is given to illustrate its use in the Female Labor Supply data.