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

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 Matérn, 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.

Figures (8)

• Figure 1: Region of consistent estimability of the slope $\beta$ in regression of $Y=X\beta + W$ on $X$ under spatial confounding in $\mathbb R^d$ for Matérn processes, as concluded from Corollary \ref{['cor:matern']}. Here $(X,W)$ is jointly a bivariate Matérn process with smoothnesses $\nu_X$ and $\nu_W$ and cross-smoothness $\nu_{XW} > \nu_X$. The region where $\beta$ is consistently estimable is color coded by the minimum order of differencing/Laplacian needed for a consistent estimator.
• Figure 2: Local-averaging-and-differencing based estimation of $\beta$ when we observe $(\tilde{X},Z)$, a measurement error contaminated version of $(X,Y)$. The left figure corresponds to the averaging part. For each point in the coarser grid, the bivariate $(\tilde{X},Z)$ process over the finer sub-grid around it is averaged to create an averaged process for that point. The right figure corresponds to taking differences/discrete Laplacians. This is done by summing over the differences in the averaged process at the blue point and each of its neighbors (red points).
• Figure 3: Estimates of $\beta$ for regression between Gaussian random fields $Y=X\beta+W$ and $X$ when both the exposure $X$ and the unmeasured confounder $W$ have Matérn covariances with smoothnesses $\nu_X$ and $\nu_W$ respectively.
• Figure 4: Comparison of empirical standard deviations and theoretical bound on the standard deviations for the numerator and denominator of the Laplacian based consistent estimators of $\beta$.
• Figure S1: Summary metrics for difference-based and GLS estimators.

Theorems & Definitions (60)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 1: Matérn covariance
• Corollary 2
• Corollary 3: Power exponential covariance
• Corollary 4: Generalized Cauchy
• Corollary 5