t(G−1) distribution for CV2/CV3 cluster-robust t-statistics

Establish whether t-statistics constructed using CV2 and CV3 cluster-robust variance estimators follow, under fixed-G asymptotics analogous to Bester–Conley–Hansen (2011), an asymptotic t(G−1) distribution, and derive the precise conditions under which this approximation is theoretically justified.

Background

Bester–Conley–Hansen (2011) provide a fixed-G asymptotic result implying that the CV1-based t-statistic is asymptotically t(G−1). In practice, researchers frequently use the same t(G−1) reference distribution for t-statistics formed with CV2 and CV3, even though no analogous proof has been established. Clarifying this would strengthen the theoretical basis for widely used small-sample inference procedures with CV2 and CV3.

References

Thus, although there are to my knowledge no theoretical results like those of \citet*{BCH_2011} to justify the use of the $t(G-1)$ distribution with CV${2}$ and CV${3}$ standard errors, it is conventional to use this distribution with $t$-statistics based on any of the three CRVEs.

— When Can We Trust Cluster-Robust Inference?  (2604.02000 - MacKinnon, 2 Apr 2026) in Subsection 2.2 (Inference Using CRVEs)