Improved Causal Inference on Spatial Observations: A Smoothing Spline Approach

With geographical observations, nearby places often have very similar treatments, controls, and outcomes. In such cases, even with perfect identification, difference in differences and synthetic controls return imprecise coefficients, while regression discontinuities and instrumental variables are prone to severe bias and spurious significance. This paper shows how this may be remedied by adding a spatial smoothing spline to the regression, something easily implemented in practice. The spline allows spatial structure to be separated out as a nuisance variable while simultaneously improving the bias-variance trade-off for the parameters of interest. For simulations and real examples, including a spline causes a marked shrinkage of coefficients, while standard errors change little for most types of cross-section but fall for panels.