Estimation under Multicollinearity: Application of Restricted Liu and Maximum Entropy Estimators to the Portland Cement Dataset

Abstract

A high degree of multicollinearity among the explanatory variables severely impairs estimation of regression coefficients by the Ordinary Least Squares. Several methods have been suggested to ameliorate the deleterious effects of multicollinearity. In this paper we aim at comparing the Restricted Liu estimates of regression coefficients with those obtained by applying the Maximum Entropy Leuven (MEL) family of estimators on the widely analyzed dataset on Portland cement. dataset has been obtained from an experimental investigation of the heat evolved during the setting and hardening of Portland cements of varied composition and the dependence of this heat on the percentage of four compounds in the clinkers from which the cement was produced. The relevance of the relationship between the heat evolved and the chemical processes undergone while setting takes place is best stated in the words of Woods et al.: This property is of interest in the construction of massive works as dams, in which the great thickness severely hinder the outflow of the heat. The consequent rise in temperature while the cement is hardening may result in contractions and cracking when the eventual cooling to the surrounding temperature takes place. Two alternative models have been formulated, the one with an intercept term (non-homogenous) that exhibits a very high degree of multicollinearity and the other with no intercept term (extended homogenous) that characterizes perfect multicollinearity. Our findings suggest that several members of the MEL family of estimators outperform the OLS and the Restricted Liu estimators. The MEL estimators perform well even when perfect multicollinearity is there. A few of them may outperform the Minimum Norm LS (OLS+) estimator. Since the MEL estimators do not seek extra information from the analyst, they are easy to apply. Therefore, one may rely on the MEL estimators for obtaining the coefficients of a linear regression model under the conditions of severe (including perfect) multicollinearity among the explanatory variables.