Develop recovery-variance-corrected inference for recoverU+

Develop recovery-variance-corrected inference for the recoverU+ estimator that accounts for sample-to-sample variability from estimating the recovered spatial confounder, particularly under weak spatial confounding.

Background

The recoverU+ procedure estimates a spatial confounder from the observed data and then treats the estimated confounder as if it were an ordinary covariate. The implemented influence-function, spatial HAC, and influence-resampling bootstrap standard errors condition on the single recovered-confounder realization and therefore omit uncertainty from the recovery stage. Simulations show substantial undercoverage when spatial confounding is weak. The paper identifies a parametric field bootstrap that reruns confounder recovery and causal estimation as a practical possible correction, but states that a formally recovery-variance-corrected inferential procedure remains unresolved.

References

Recovery-variance-corrected inference is, to our knowledge, an open problem; the practical cost of the gap is contained, because the regime in which it bites, weak spatial confounding, is precisely the regime in which recoverU$+$ is least needed.

— A unified framework for estimating direct causal effect under spatial confounding and interference, with the R package spaci  (2609.28799 - Ogunsola et al., 23 Sep 2026) in Section 6, subsection “Limitations”