Multiple Hypothesis Testing Based on the Ordinary Least Squares Regression Estimator when there is Heteroscedasticity

Abstract

Consider the regression model Y _i = β₀ + β₁ X _{i 1} +...+ β _p X _iP + λ(X _{i 1},..., X _ip )ε _i , where the function λ is used to model heteroscedasticity. Assume that the ordinary least squares estimator is used. This note addresses the problem of testing H ₀:β _j = 0 for each j (j =1,..., p), with the goal of controlling the probability of at least one Type I error. Based on published results dealing with the special case p = 1, there is an obvious guess regarding a general strategy aimed at accomplishing this goal: use in part a modification of the percentile bootstrap method. This note reports simulation results indicating that a simpler approach performs well for p ≤ 2. In particular, use a basic (unmodified) percentile boot-strap method in conjunction with the Bonferroni inequality.