Decomposition of Spillover Effects Under Misspecification: Pseudo-true Estimands and a Local-Global Extension

Applied work under interference typically models outcomes as functions of own treatment and a low-dimensional exposure mapping of others' treatments, even when that mapping may be misspecified. We ask what policy object such exposure-based procedures target. Taking the marginal policy effect as primitive, we show that any researcher-chosen exposure mapping induces a unique pseudo-true outcome model: the best approximation to the underlying potential outcomes within the class of functions that depend only on that mapping. This yields a decomposition of the marginal policy effect into exposure-based direct and spillover effects, and each component optimally approximates its oracle counterpart, with a sign-preserving interpretation under monotonicity. We then study a structured misspecification setting in which outcomes depend on both network spillovers and a global equilibrium channel, while the analyst may model only one. In this setting, we obtain a sharper asymptotic decomposition into direct, local, and global components, implying that existing estimators recover their respective oracle channel-specific effects even when the other channel is present but omitted from the maintained model. The analysis also yields phase transitions in convergence rates and higher-order expansions for Z-estimators. A semi-synthetic experiment calibrated to a large cash-transfer study illustrates the empirical relevance of the framework.