Spatial self-confounding: smoothness-related estimation bias in spatial regression models

Abstract The estimation of regression parameters in spatially referenced data plays a crucial role across various scientific domains. A common approach involves employing an additive regression model to capture the relationship between observations and covariates, accounting for spatial variability not explained by the covariates through a Gaussian random field. We study the effect of misspecified covariates, in particular when the misspecification changes the smoothness. We analyse the theoretical properties of the generalized least-squares estimator under infill asymptotics, and show that the estimator can have counter-intuitive properties. In particular, the estimated regression coefficients can converge to zero as the number of observations increases if the covariates are too rough, despite high correlations between observations and covariates. This has important implications for practical applications as the importance of rough covariates can be severely underestimated, leading to incorrect scientific conclusions. We also show that the estimates can diverge to infinity under certain conditions, which can also lead to incorrect conclusions in practical applications. Through an application to temperature and precipitation data, we show that both behaviours can be observed for real data. Finally, we propose adding a smoothing step in the regression and show both theoretically and practically that this can solve the problem.