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

TL;DR

This paper addresses how to interpret exposure-based spillover effects when the exposure mapping is misspecified. It introduces a design-based primitive, the marginal policy effect (MPE), and shows that for any fixed exposure mapping there exists a unique pseudo-true outcome model that best approximates the true potential outcomes; the MPE then decomposes into direct and spillover components within this pseudo-true framework. In a structured local–global environment, the MPE further splits into direct, local, and global channels, with Li–Wager-type estimators remaining robust for the local channel and augmented designs enabling identification of the global channel. The authors provide estimators and asymptotic results for each component, plus simulations and a semi-synthetic cash-transfer experiment illustrating recoverability of the three channels. The framework offers a unified lens to interpret interference analyses under misspecification and guides practical design choices to isolate and estimate channel-specific effects in empirical settings.

Abstract

Applied work with 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. This raises a basic question: what policy object are exposure-based estimands implicitly targeting, and how should we interpret their direct and spillover components relative to the underlying policy question? We take as primitive the marginal policy effect, defined as the effect of a small change in the treatment probability under the actual experimental design, and show that any researcher-chosen exposure mapping induces a unique pseudo-true outcome model. This model is the best approximation to the underlying potential outcomes that depends only on the user-chosen exposure. Utilizing that representation, the marginal policy effect admits a canonical decomposition into exposure-based direct and spillover effects, and each component provides its optimal approximation to the corresponding oracle objects that would be available if interference were fully known. We then focus on a setting that nests important empirical and theoretical applications in which both local network spillovers and global spillovers, such as market equilibrium, operate. There, the marginal policy effect further decomposes asymptotically into direct, local, and global channels. An important implication is that many existing methods are more robust than previously understood once we reinterpret their targets as channel-specific components of this pseudo-true policy estimand. Simulations and a semi-synthetic experiment calibrated to a large cash-transfer experiment show that these components can be recovered in realistic experimental designs.

Figures (4)

• Figure 1: Index outcome model with $\mathtt{linear}$ link function and fixed sample size $n=1000$. Dashed curves indicate oracle ADE / local AIE / global AIE, while solid curves are Monte Carlo averages.
• Figure 2: Index outcome model with non-linear link functions and fixed sample size $n=1000$. ADE, local AIE, and global AIE are plotted as functions of the mixing parameter $u$. Dashed curves indicate oracle values, while solid curves are Monte Carlo averages.
• Figure 3: MSE of the ADE and the local and global AIE estimators against $n$ in log-log scale, across two $(\kappa,\alpha)$ regimes for linear and nonlinear (cosine) outcome DGP.
• Figure 4: Monte Carlo sampling distributions for the ADE, local AIE, and global AIE estimators in the Filmer-calibrated design.

Theorems & Definitions (31)

• Theorem 3.2
• Proposition 3.3
• Corollary 3.4
• Theorem 5.4
• Theorem 5.5
• Remark 5.6
• Theorem 5.7
• Theorem 5.8
• Corollary 5.9
• proof : Proof of Theorem \ref{['thm:mpe equal direct plus indirect']}