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

TL;DR

This paper establishes a necessary and sufficient condition for size controllability of heteroskedasticity robust test statistics in regression models, improving upon previous sufficient-only conditions.

Contribution

It provides a precise criterion for size controllability in the case of testing a single restriction, refining prior results.

Findings

Derived a necessary and sufficient condition for size controllability.

Clarified the special case of testing a single restriction.

Improved theoretical understanding of heteroskedasticity robust tests.

Abstract

We revisit size controllability results in P\"otscher 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 P\"otscher and Preinerstorfer (2025) is, in general, only sufficient (even in the case of testing a single restriction).