Consistency of common spatial estimators under spatial confounding

Abstract: This paper addresses the asymptotic performance of popular spatial regression estimators of the linear effect of an exposure on an outcome under `spatial confounding" -- the presence of an unmeasured spatially-structured variable influencing both the exposure and the outcome. We first show that the estimators from ordinary least squares (OLS) and restricted spatial regression are asymptotically biased under spatial confounding. We then prove a novel main result on the consistency of the generalized least squares (GLS) estimator using a Gaussian process (GP) working covariance matrix in the presence of spatial confounding under infill (fixed domain) asymptotics. The result holds under very general conditions -- for any exposure with some non-spatial variation (noise), for any spatially continuous fixed confounder function, using any Mat\ern or square exponential kernel used to construct the GLS estimator, and without requiring Gaussianity of errors. Finally, we prove that spatial estimators from GLS, GP regression, and spline models that are consistent under confounding by a fixed function will also be consistent under endogeneity or confounding by a random function, i.e., a stochastic process. We conclude that, contrary to claims in some literature on spatial confounding, traditional spatial estimators are capable of estimating linear exposure effects under spatial confounding as long as there is some noise in the exposure. We support our theoretical arguments with simulation studies.