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 diffusion-based models of information propagation, we establish conditions under which spatial and temporal boundaries share common dynamics, derive identification results, and propose consistent estimators. Monte Carlo simulations demonstrate the performance of our methods under various data-generating processes. The framework provides tools for detecting when local treatments become systemic and identifying critical thresholds for policy intervention.