Inferences Based on Robust Regression Estimators When There Is Multicolinearity

Abstract

The paper deals with the goal of testing hypotheses about the slope parameters of a linear regression model when there is multicolinearity. A heteroscedastic method was recently derived based on a ridge estimator, but it does not guard against the deleterious impact of outliers. Several robust analogs of the ridge estimator have been proposed that might deal with this concern. The goal here is to find a robust method that performs reasonably well in simulations.

Robust regression: Testing global hypotheses about the slopes when there is multicollinearity or heteroscedasticity

“…Following Wilcox (), four types of distributions were used. More precisely, values for the error term, ϵ, were generated from one of four g ‐and‐ h distributions (Hoaglin, ) that contain the standard normal distribution as a special case.…”

“…Again following Wilcox (), three choices for λ( X ) were used: λ( X ) = 1 (homoscedasticity), λ( X ) = | X 1 | + 1 and λ( X ) = 1/(| X 1 | + 1). These three variance patterns are labeled VP 1, VP 2 and VP 3, respectively.…”

“…Following closely the description and notation used by Wilcox (), it is assumed that …”

“…The motivation for S ( k ) stems from a simple extension of the results in Long and Ervin () who derived the so‐called HC3 estimate of the standard error of the least squares estimator that allows heteroscedasticity. Wilcox () proposed testing with …”

“…When is false, the ridge estimator is biased even when is true, which results in poor control over the Type I error probability. Wilcox () noted that no robust method was found for testing that competes well with the OLS estimator, in terms of power, under normality and homoscedasticity.…”