Two-stage Estimators for Spatial Confounding

Public health data are often spatially dependent, but standard spatial regression methods can suffer from bias and invalid inference when the independent variable is associated with spatially-correlated residuals. This could occur if, for example, there is an unmeasured environmental contaminant associated with the independent and outcome variables in a spatial regression analysis. Geoadditive structural equation modeling (gSEM), in which an estimated spatial trend is removed from both the explanatory and response variables before estimating the parameters of interest, has previously been proposed as a solution, but there has been little investigation of gSEM's properties with point-referenced data. We link gSEM to results on double machine learning and semiparametric regression based on two-stage procedures. We propose using these semiparametric estimators for spatial regression using Gaussian processes with Mat\`ern covariance to estimate the spatial trends, and term this class of estimators Double Spatial Regression (DSR). We derive regularity conditions for root-$n$ asymptotic normality and consistency and closed-form variance estimation, and show that in simulations where standard spatial regression estimators are highly biased and have poor coverage, DSR can mitigate bias more effectively than competitors and obtain nominal coverage.