Sufficient conditions for asymptotic refinement of the cluster pairs bootstrap with discrete regressors

Establish sufficient conditions under which the cluster pairs bootstrap achieves asymptotic refinement for inference in linear regression with clustered errors when the regressors are discrete and cluster distributions are non-identical, rather than relying on the classical Cramer's condition that excludes discrete regressors.

Background

The paper highlights limitations of the standard proofs for asymptotic refinement of the cluster pairs bootstrap in clustered regression settings: they require the classical Cramer's condition on the regressors, which fails for discrete variables commonly used in practice (e.g., binary policy dummies).

The authors’ analytic approach avoids resampling and does not impose Cramer's condition by conditioning on regressors, but for the cluster pairs bootstrap they note an unresolved gap: finding sufficient conditions that allow discrete regressors and non-identical cluster distributions while still delivering asymptotic refinement.

They reference weaker versions of Cramer's condition (e.g., Bai and Rao, 1991) but do not extend the cluster pairs bootstrap theory under such conditions, indicating this remains unresolved within the scope of their work.

References

We could not find a sufficient condition for the cluster pairs bootstrap's asymptotic refinement that allows for discrete regressors and non-identical distributions. A weaker version of the Cram er's condition has been proposed, e.g., in , but the cluster pairs bootstrap's asymptotic refinement without the classical Cram er's condition is beyond the scope of this paper.

— Refined Cluster Robust Inference  (2603.24786 - Gafarov et al., 25 Mar 2026) in Introduction, footnote following the paragraph beginning “Third, the standard proof for the cluster pairs bootstrap’s asymptotic refinement excludes discrete regressors.”