Consistency of common spatial estimators under spatial confounding
研究了空间混杂下几种空间回归估计量（如普通最小二乘、限制性空间回归、广义最小二乘等）的渐近性质，证明只要暴露变量存在非空间变异，传统空间估计量仍能一致估计线性暴露效应。
This article 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 and restricted spatial regression are asymptotically biased under spatial confounding. We then prove a novel result on the infill consistency of the generalized least squares estimator using a working covariance matrix from a Matérn or squared exponential kernel, in the presence of spatial confounding. The result holds under very mild assumptions, accommodating any exposure with some nonspatial variation, any spatially continuous fixed confounder function, and non-Gaussian errors in both the exposure and the outcome. Finally, we prove that spatial estimators from generalized least squares, Gaussian process 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 some claims in the 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.