Robust VIF regression with application to variable selection in large data sets

Abstract

The sophisticated and automated means of data collection used by an increasing number of institutions and companies leads to extremely large data sets. Subset selection in regression is essential when a huge number of covariates can potentially explain a response variable of interest. The recent statistical literature has seen an emergence of new selection methods that provide some type of compromise between implementation (computational speed) and statistical optimality (e.g., prediction error minimization). Global methods such as Mallows’ $C_{p}$ have been supplanted by sequential methods such as stepwise regression. More recently, streamwise regression, faster than the former, has emerged. A recently proposed streamwise regression approach based on the variance inflation factor (VIF) is promising, but its least-squares based implementation makes it susceptible to the outliers inevitable in such large data sets. This lack of robustness can lead to poor and suboptimal feature selection. In our case, we seek to predict an individual’s educational attainment using economic and demographic variables. We show how classical VIF performs this task poorly and a robust procedure is necessary for policy makers. This article proposes a robust VIF regression, based on fast robust estimators, that inherits all the good properties of classical VIF in the absence of outliers, but also continues to perform well in their presence where the classical approach fails.