Spectral adjustment for spatial confounding

Adjusting for an unmeasured confounder is generally intractable problem, but in the spatial setting it may be possible under certain conditions. In this paper, we derive necessary conditions on coherence between treatment variable of interest and that ensure causal effect estimable. We specify our model assumptions spectral domain to allow different degrees confounding at resolutions. The key assumption ensures identifiability present global scales dissipates local scales. show equivalent adjusting global-scale by adding a spatially smoothed version mean response variable. Within general framework, propose sequence adjustment methods range from parametric adjustments based Matern function more robust semi-parametric use smoothing splines. These ideas are applied areal geostatistical data both simulated real datasets