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Difference-in-Differences under Local Dependence on Networks

Akihiro Sato, Shonosuke Sugasawa

TL;DR

This paper tackles causal inference under interference by extending Difference-in-Differences to networks without relying on pre-specified exposure mappings. It introduces two estimands, the direct effect on the treated $\tau_{\mathrm{dir}}$ and the average indirect effect on neighbors $\tau_{\mathrm{ind}}$ (AITT), and derives nonparametric identification under a conditional parallel trends assumption that conditions on the neighborhood treatment vector $\boldsymbol{D}_{N_i}$. Estimation is implemented via inverse probability weighting and doubly robust methods, with variance estimated by a network-aware HAC procedure anchored in $\psi$-dependence; asymptotics hold under local dependence and alpha-mixing conditions. The approach is validated through simulations demonstrating robustness to misspecified exposure mappings and through a China SEZ policy application that reveals substantive direct effects and outward spillovers, highlighting policy externalities often missed by standard DID. Overall, the method provides a practical, robust alternative when the interference mechanism is complex or unknown and offers meaningful guidance for policy evaluation in connected settings.

Abstract

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.

Difference-in-Differences under Local Dependence on Networks

TL;DR

This paper tackles causal inference under interference by extending Difference-in-Differences to networks without relying on pre-specified exposure mappings. It introduces two estimands, the direct effect on the treated and the average indirect effect on neighbors (AITT), and derives nonparametric identification under a conditional parallel trends assumption that conditions on the neighborhood treatment vector . Estimation is implemented via inverse probability weighting and doubly robust methods, with variance estimated by a network-aware HAC procedure anchored in -dependence; asymptotics hold under local dependence and alpha-mixing conditions. The approach is validated through simulations demonstrating robustness to misspecified exposure mappings and through a China SEZ policy application that reveals substantive direct effects and outward spillovers, highlighting policy externalities often missed by standard DID. Overall, the method provides a practical, robust alternative when the interference mechanism is complex or unknown and offers meaningful guidance for policy evaluation in connected settings.

Abstract

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.
Paper Structure (31 sections, 8 theorems, 54 equations, 5 figures, 3 tables)

This paper contains 31 sections, 8 theorems, 54 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Identification
  3. Setup and potential outcomes
  4. Causal estimands
  5. Identification results
  6. Estimation and Inference
  7. Inverse probability weighted estimator
  8. Doubly robust estimator
  9. Variance estimation
  10. Asymptotic Properties
  11. Local dependence on networks
  12. Assumptions
  13. Main theorems
  14. Simulation Study
  15. Data generating process
  16. ...and 16 more sections

Key Result

Theorem 1

Under Assumptions as:NoAnticipation, as:trend, and as:neighborhood_sufficiency, ADTT is identified as where $\pi_i(z_i) \equiv P(D_i=1|z_i)$ is the marginal probability of $D_i=1$ conditional on $z_i$, and $e_i({\boldsymbol{D}}_{N_i}, z_i) = P(D_i = 1 | {\boldsymbol{D}}_{N_i}, z_i)$ is the propensity score conditional on the neighborhood treatment vector ${\boldsymbol{D}}_{N_i}$ and $z_i$. Note t

Figures (5)

  • Figure 1: Distribution of the number of treated neighbors ($S_i$) across units in a representative simulation run. This histogram shows the prevalence of different exposure levels, which is crucial for understanding the distribution of spillover effects in our data generating process.
  • Figure 2: Distribution of estimated treatment effects across Monte Carlo replications. Left: ADTT estimates. Right: AITT estimates. True values indicated by dashed lines (ADTT = 0.8, AITT = 0.4961). Methods are labeled as Proposed IPW/DR (our methods), Xu (Oracle), Xu (MO), and Xu (FM) corresponding to the exposure mapping specifications described in Section 5.2.
  • Figure 3: Bias of the proposed IPW and DR ADTT estimators and benchmark methods as a function of sample size $N \in \{300, 500, 700\}$. The proposed IPW and DR methods show decreasing bias with increasing sample size, providing empirical evidence for consistency. Benchmark methods that ignore interference or rely on misspecified exposure mappings retain substantial bias even with larger sample sizes.
  • Figure 4: Bias as a function of spatial correlation strength ($\rho_0 \in \{0.2, 0.5, 0.8\}$).
  • Figure 5: Bias as a function of the number of neighborhood features used in propensity score estimation.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Definition 3: $\psi$-dependence
  • Lemma 1: $\psi$-dependence of ADTT and AITT summand
  • Theorem 3: Consistency
  • Theorem 4: Asymptotic normality
  • Theorem 5: Consistency of variance estimator
  • Theorem 6: Double robustness of DR estimators
  • ...and 1 more