More on Liu-Type Estimator in Linear Regression

Abstract

Recently, Liu [Liu, K. (2003). Using Liu-type estimator to combat collinearity. Commun. Statist. Theory Methods 32:1009–1020] introduced the Liu-type estimator to combat collinearity in linear regression. The Liu-type estimator can be applied in two ways. First, when the effect of collinearity is moderate, the Liu-type estimator can be used as a means to further improve the performance of ridge regression. In this case, k selected by existing methods for ridge regression usually is random. Here, we generalize the results in Liu (2003) to the random k case and prove that when k is selected by Hoerl–Kennard formula, the Liu-type estimator is still able to further improve ridge regression. Second, when collinearity is severe, we suggest to choose k based on condition number consideration, which is different from the existing selection methods, and then use the second parameter in Liu-type estimator to make adjustment. We use simulation to provide some empirical justification for the choice of k proposed in Liu (2003).