Estimable group effects for strongly correlated variables in linear models

Abstract

In ordinary least-squares regression, strongly correlated predictor variables generate multicollinearity known to cause poor estimation of individual parameters of such variables and consequently difficulties in inference and prediction with the estimated model. We construct a theoretical model to study the impact of such multicollinearity on estimation of linear combinations of these parameters and uncover linear combinations that can be remarkably accurately estimated. Our results show that this type of multicollinearity represents a redistribution of information that allows some linear combinations to be extremely accurately estimated at the expense of other linear combinations becoming inestimable. Based on insights gained from studying this theoretical model, for all linear models with strongly correlated predictor variables, we develop a simple method for finding linear combinations of parameters of these variables that may be accurately estimated. Such linear combinations can be used to help resolve the aforementioned difficulties, and they provide another tool for handling multicollinearity in ordinary least-squares regression.