Continuous Treatment Effects with Spatial and Network Spillovers

Abstract: This paper develops a continuous functional framework for treatment effects that propagate through geographic space and economic networks. We derive a master equation governing propagation from three economic foundations -- heterogeneous agent aggregation, market equilibrium, and cost minimization -- establishing that the framework rests on fundamental principles rather than ad hoc specifications. A key result shows that the spatial-network interaction coefficient equals the mutual information between geographic and market coordinates. The Feynman-Kac representation decomposes effects into inherited and accumulated components along stochastic paths representing economic linkages. The framework nests the no-spillover case as a testable restriction. Monte Carlo simulations demonstrate that conventional estimators -- two-way fixed effects, difference-in-differences, and generalized propensity score -- exhibit 25-38% bias and severe undercoverage when spillovers exist, while our estimator maintains correct inference regardless of whether spillovers are present. Applying the framework to U.S. minimum wage policy, we reject the no-spillover null and find total effects at state borders four times larger than direct effects -- conventional methods capture only one-quarter of policy impact. Structural estimates reveal spatial diffusion consistent with commuting-distance labor mobility, network diffusion consistent with quarterly supply chain adjustment, and significant spatial-network interaction reflecting geographic clustering of industries. Entropy-based fragility diagnostics outperform standard centrality measures by 56-76% in predicting labor market disruptions, identifying all high-risk state-industry pairs during 2020-2021 with six-month advance warning.