Consistent Infill Estimability of the Regression Slope Between Gaussian Random Fields Under Spatial Confounding

TL;DR

This paper characterizes when the regression slope between Gaussian random fields can be consistently estimated under spatial confounding, providing theoretical conditions, estimators, and spectral criteria for various process families.

Contribution

It offers a novel theoretical framework for the consistent estimability of the regression slope under spatial confounding, including spectral conditions and estimators applicable to multiple Gaussian process families.

Findings

Provided sufficient conditions for slope estimability under fixed-domain asymptotics.

Derived spectral tail decay conditions for estimability verification.

Established a complete characterization for common Gaussian process families.

Abstract

The problem of estimating the slope parameter in regression between two spatial processes under confounding by an unmeasured spatial process has received widespread attention in the recent statistical literature. Yet, a fundamental question remains unsolved: when is this slope consistently estimable under spatial confounding, with existing insights being largely empirical or estimator-specific. In this manuscript, we characterize conditions for consistent estimability of the regression slope between Gaussian random fields (GRFs). Under fixed-domain (infill) asymptotics, we give sufficient conditions for consistent estimability using a novel characterization of the regression slope as the ratio of principal irregular terms of covariances, dictating the relative local behavior of the exposure and confounder processes. When estimability holds, we provide consistent estimators of the slope using local differencing (taking discrete differences or Laplacians of the processes of suitable order). Using functional analysis results on Paley-Wiener spaces, we then provide an easy-to-verify necessary condition for consistent estimability of the slope in terms of the relative spectral tail decays of the confounder and exposure. As a by-product, we establish a novel and general spectral condition on the equivalence of measures on the paths of multivariate GRFs with component fields of varying smoothnesses, a result of independent importance. We show that for the Mat\'ern, power-exponential, generalized Cauchy, and coregionalization families, the necessary and sufficient conditions become identical, thereby providing a complete characterization of consistent estimability of the slope under spatial confounding. The results are extended to accommodate measurement error using local-averaging-and-differencing based estimators. Finite sample behavior is explored via numerical experiments.