A Unified Framework for Spatial and Temporal Treatment Effect Boundaries: Theory and Identification

This paper develops a unified theoretical framework for detecting and estimating boundaries in treatment effects across both spatial and temporal dimensions. We formalize the concept of treatment effect boundaries as structural parameters characterizing regime transitions where causal effects cease to operate. Building on reaction-diffusion models of information propagation, we establish conditions under which spatial and temporal boundaries share common dynamics governed by diffusion parameters $(\delta, \lambda)$, yielding the testable prediction $d^*/\tau^* = 3.32\lambda\sqrt{\delta}$ for standard detection thresholds. We derive formal identification results under staggered treatment adoption and develop a three-stage estimation procedure implementable with standard panel data. Monte Carlo simulations demonstrate excellent finite-sample performance, with boundary estimates achieving RMSE below 10\% in realistic configurations. We apply the framework to two empirical settings: EU broadband diffusion (2006-2021) and US wildfire economic impacts (2017-2022). The broadband application reveals a scope limitation --- our framework assumes depreciation dynamics and fails when effects exhibit increasing returns through network externalities. The wildfire application provides strong validation: estimated boundaries satisfy $d^* = 198$ km and $\tau^* = 2.7$ years, with the empirical ratio (72.5) exactly matching the theoretical prediction $3.32\lambda\sqrt{\delta} = 72.5$. The framework provides practical tools for detecting when localized treatments become systemic and identifying critical thresholds for policy intervention.