A Necessary and Sufficient Condition for Size Controllability of Heteroskedasticity Robust Test Statistics

TL;DR

The paper addresses finite-sample size control for heteroskedasticity-robust test statistics in linear regression, with a focus on testing a single linear restriction. It shows that, when testing a single restriction (q=1), a necessary and sufficient condition governs the existence of finite-size critical values, replacing the previous sufficiency-only condition from PP21. The weaker, uncorr form of the condition is shown to be necessary and sufficient for the existence of a smallest finite critical value, ensuring that tests like HC0-HC4 have correct finite-sample size under this criterion; if violated, the size collapses to 1. These results, built on invariance arguments and extended to broader heteroskedasticity models, provide practical guidance for constructing size-controlled tests and confidence intervals via test inversion, with the existing algorithms from PP21 readily applicable. Overall, the work resolves ambiguity around size controllability for a fundamental case and sharpens the theoretical foundation for robust inference under heteroskedasticity.

Abstract

We revisit size controllability results in Pötscher and Preinerstorfer (2025) concerning heteroskedasticity robust test statistics in regression models. For the special, but important, case of testing a single restriction (e.g., a zero restriction on a single coefficient), we povide a necessary and sufficient condition for size controllability, whereas the condition in Pötscher and Preinerstorfer (2025) is, in general, only sufficient (even in the case of testing a single restriction).

Theorems & Definitions (8)

• Theorem 2.1: Theorem 5.1(b,c) and Propositions 5.5(b) and 5.7(b) in PP21 for the case $q=1$
• Theorem 2.2
• Lemma A.1
• Lemma A.2
• Lemma A.3
• Lemma A.4
• Lemma B.1
• Theorem B.2