Double/Debiased Machine Learning for Functional-Form-Robust Spatial Autoregression

Spatial autoregressive inference is typically conditional on the spatial weights matrix, W, even though the underlying interaction structure is often unknown and empirical conclusions can be sensitive to its specification. This paper develops double/debiased machine learning inference for low-dimensional SAR parameters when the spatial interaction operator is learned flexibly from potentially endogenous characteristics. Within a maintained admissible support, interaction strength is generated by an unknown function of geographic and socioeconomic characteristics, making inference robust to functional form specification of the weights within that support. Endogeneity in the characteristics generating W is addressed through a nonlinear control function based on locally relevant first-stage residual information. Because the learned operator enters both the spatial lag and spatially transformed instruments, treating the estimated W as known generally leaves a first-order generated-W effect. I construct an operator-orthogonal SAR-IV/GMM score that removes this leading sensitivity and combine it with buffered spatial cross-fitting that separates evaluation-score footprints from nuisance-training observations. Under near epoch dependence on a spatially mixing innovation field and target-relevant nuisance rate and regularity conditions, the estimator is asymptotically linear and root-n normal. Monte Carlo simulations show improved finite-sample inference relative to nonorthogonal alternatives when the interaction function is misspecified, weight generating characteristics are endogenous, and observations are spatially dependent. In a U.S. application, diabetes estimates vary with the choice of W, showing the sensitivity of SAR inference to the interaction structure. Even for the same learned W, results differ across inferential methods, highlighting the importance of inference when W is learned.