Exact inference for heterogeneous clustered designs

Construct a finite-sample exact reduction for Gaussian clustered regression with heterogeneous within-cluster designs and unequal cluster sizes under unknown within-cluster covariance, without restricting inference to the asymptotic cluster-robust e-process treatment.

Background

The exact Gaussian results apply either when all outcomes share common regressors or, for shared-coefficient clustered regression, when clusters have a common within-cluster design that permits compression to identically distributed Gaussian summaries.

With heterogeneous within-cluster designs or unequal cluster sizes, the paper states that no i.i.d. Gaussian summary reduction is available. The resulting cluster-robust procedures are treated only asymptotically, leaving open whether an exact finite-sample reduction and corresponding anytime-valid inference can be obtained under unknown covariance.

References

The second concerns the scope of the exact results. Theorem~\ref{thm:main} covers regressors common to the outcome coordinates, cluster-level covariates included, and Proposition~\ref{prop:common} a common within-cluster design with shared coefficients. Heterogeneous within-cluster designs and unequal cluster sizes, for which no i.i.d.\ Gaussian summary reduction is available, are covered only asymptotically; a general exact reduction for them under unknown covariance remains outside the results proved here.

— Anytime-Valid Inference in Linear Models with Clustered Outcomes: Exact Multivariate Tests and Cluster-Robust Confidence Sequences  (2610.25613 - Lindon, 19 Oct 2026) in Section 6, Discussion; Appendix E, Section E.5