Evaluation Estimation Performances of Liu Type and Two-Parameter Ridge Estimators Using Monte Carlo Experiments

Abstract

Multiple linear regression model is a widely used statistical technique in social and life sciences. Ordinary Least Squares (OLS) estimator is the Best Linear Unbiased Estimator (BLUE) for the unknown population parameters of this model. Unfortunately, sometimes two or more of the regressors may be moderately or highly correlated causing multicollinearity problem. Various biased estimators are proposed to refine the ill-conditioning of X'X matrix and shrink the variance under the multicollinearity. The most popular of them is Ridge estimator. But it may worsen the fit when solving the ill-conditioning problem. Two-Parameter Ridge (2PR) and Liu Type (LT) estimators are proposed to overcome the fitting degeneration of Ridge estimator by using a tuning parameter. In this study, holding the parameter refining the ill-conditioning of X'X matrix fixed, the success of the tuning parameters of these estimators is investigated. Minimizers of Predicted Sum of Squares (PRESS) and Generalized Cross Validation (GCV) statistics are used as estimates of tuning parameters. Optimum parameter estimates are compared via their Scalar Mean Squared Errors (SMSE). It is observed that the SMSEs of estimates obtained by LT and 2PR estimators decreases when estimates of parameter refining the ill-conditioning of X'X matrix increases, and in all cases estimates obtained by the 2PR estimator are much more efficient than estimates obtained by LT and OLS estimators.