Difference-in-Differences under Local Dependence on Networks

Estimating causal effects under interference, where the stable unit treatment value assumption is violated, is critical in fields such as regional and public economics. Much of the existing research on causal inference under interference relies on a pre-specified “exposure mapping”. This paper focuses on difference-in-difference and proposes a nonparametric identification strategy for direct and indirect average treatment effects under local interference on an observed network. In particular, we proposed a new concept of an indirect effect measuring the total outward influence of the intervension. Based on parallel trends assumption conditional on the neighborhood treatment vector, we develop inverse probability weighted and doubly robust estimators. We establish their asymptotic properties, including consistency under misspecification of nuisance models under some regularity conditions. Simulation studies and an empirical application demonstrate the effectiveness of the proposed method.

💡 Research Summary

This paper tackles the problem of causal inference when treatment effects spill over across units connected by a network, a setting that violates the Stable Unit Treatment Value Assumption (SUTVA) underlying most conventional Difference‑in‑Differences (DID) analyses. Existing network‑interference literature typically relies on a pre‑specified exposure mapping that collapses the high‑dimensional vector of neighbors’ treatments into a low‑dimensional “exposure level.” Misspecification of that mapping can induce severe bias. The authors propose a fundamentally different, non‑parametric identification strategy that does not require any exposure mapping.

The key idea is to condition on the exact treatment vector of a fixed‑size neighborhood. For each unit i, the L nearest neighbors (according to shortest‑path distance on the observed adjacency matrix) are identified and their binary treatment indicators are stacked into a vector D_Ni of length L. By keeping L fixed, the dimensionality of the conditioning set remains manageable while still capturing the local interference that the authors assume to be confined within a theoretical range K (with L‑neighborhoods nested inside K‑neighborhoods).

Two causal parameters are defined. The Average Direct Treatment Effect on the Treated (ADTT) measures, for treated units, the expected difference in outcomes when the unit’s own treatment switches from 0 to 1 while holding the observed neighbor‑treatment vector fixed. The Average Indirect Treatment Effect on the Treated (AITT) is a novel “outward spillover” quantity: it averages, over all treated units i, the change in each neighbor j’s outcome that would occur if i were treated versus not treated, again conditioning on the rest of the neighbor configuration. This directly quantifies policy externalities rather than the more common “inward” spillover effect.

Identification rests on three assumptions: (1) No anticipation – pre‑treatment outcomes do not depend on future treatment; (2) Conditional parallel trends under interference – for any fixed neighbor‑treatment vector, the expected change in the untreated potential outcome between periods is the same for treated and control units, and an analogous condition holds for each neighbor j when conditioning on the treatment status of i’s other neighbors; (3) Sufficiency and locality of the L‑neighborhood – the chosen L is large enough to capture all relevant interference (i.e., L‑neighbors lie within the true interference radius K). Under these assumptions, the authors prove two identification theorems. Both ADTT and AITT can be expressed as inverse‑probability‑weighted (IPW) estimands. The weights involve the conditional propensity score e_i(D_Ni, z_i)=P(D_i=1|D_Ni, z_i) and the marginal propensity score π_i(z_i)=P(D_i=1|z_i). For AITT, a joint propensity e′ij(D_j, D{-i}N_j, z_i, z_j) = P(D_i=1|D_j, D_{-i}N_j, z_i, z_j) appears, reflecting the need to control for the treatment status of the focal unit’s neighbors when evaluating its effect on a specific neighbor.

Estimation proceeds via (a) a straightforward IPW estimator that plugs in estimated propensity scores, and (b) a doubly‑robust (DR) estimator that augments the IPW with outcome regression models. The DR estimator retains consistency if either the propensity model or the outcome model is correctly specified, providing protection against misspecification of nuisance functions.

Because observations are not independent but linked through a single large network, standard i.i.d. asymptotics do not apply. The authors adopt the ψ‑dependence framework of Kojevnikov et al. (2021) to handle network dependence. Under appropriate mixing conditions derived from the bounded interference range K, they establish √n‑consistency and asymptotic normality for both the IPW and DR estimators, allowing conventional Wald‑type inference.

Monte‑Carlo simulations explore performance across varying network densities, choices of L, and degrees of misspecification in the propensity and outcome models. Results show that when the exposure mapping is misspecified, traditional methods exhibit large bias, whereas the proposed approach remains approximately unbiased, albeit with larger variance when L is large (the “curse of dimensionality”).

An empirical illustration uses a two‑period panel of municipalities subject to a place‑based fiscal incentive. The network is defined by geographic adjacency. The authors estimate ADTT (the direct impact of the incentive on treated municipalities) and AITT (the average effect of a treated municipality’s incentive on the outcomes of its neighboring municipalities). Findings reveal a positive direct effect and a statistically significant outward spillover, indicating that the policy’s total welfare impact exceeds the naïve estimate that ignores interference.

The paper acknowledges limitations: (i) the need to pre‑select L, which may affect bias‑variance trade‑offs; (ii) the requirement that the full neighbor‑treatment vector be observed, which may be demanding in some applications; (iii) the strength of the conditional parallel trends assumption, which must be justified empirically. Future work could develop data‑driven methods for choosing L, incorporate dimension‑reduction techniques for high‑dimensional neighbor vectors, and extend the framework to dynamic networks or multi‑period panels.

In sum, this work provides a robust, non‑parametric DID framework for settings with local network interference, eliminating reliance on potentially misspecified exposure mappings and delivering estimators that are both theoretically sound under network dependence and practically useful for evaluating policies with spillover effects.