Treatment Effects with Correlated Spillovers: Bridging Discrete and Continuous Methods

This paper develops a continuous functional framework for treatment effects propagating through geographic space and economic networks. We derive a master equation from three independent economic foundations -- heterogeneous agent aggregation, market equilibrium, and cost minimization -- establishing that the framework rests on fundamental principles rather than ad hoc specifications. The framework nests conventional econometric models -- autoregressive specifications, spatial autoregressive models, and network treatment effect models -- as special cases, providing a bridge between discrete and continuous methods. A key theoretical result shows that the spatial-network interaction coefficient equals the mutual information between geographic and network coordinates, providing a parameter-free measure of channel complementarity. The Feynman-Kac representation characterizes treatment effects as accumulated policy exposure along stochastic paths representing economic linkages, connecting the continuous framework to event study methodology. The no-spillover case emerges as a testable restriction, creating a one-sided risk profile where correct inference is maintained regardless of whether spillovers exist. Monte Carlo simulations confirm that conventional estimators exhibit 25-38% bias when spillovers are present, while our estimator maintains correct inference across all configurations including the no-spillover case.