Heterogeneous Effects of Endogenous Treatments with Interference and Spillovers in a Large Network

This paper studies the identification and estimation of heterogeneous effects of an endogenous treatment under interference and spillovers in a large single-network setting. We model endogenous treatment selection as an equilibrium outcome that explicitly accounts for spillovers and derive conditions guaranteeing the existence and uniqueness of this equilibrium. We then identify heterogeneous marginal exposure effects (MEEs), which may vary with both the treatment status of neighboring nodes and unobserved heterogeneity. We develop estimation strategies and establish their large-sample properties. Equipped with these tools, we analyze the heterogeneous effects of import competition on U.S. local labor markets in the presence of interference and spillovers. We find negative MEEs, consistent with the existing literature. However, these effects are amplified by spillovers in the presence of treated neighbors and among localities that tend to select into lower levels of import competition. These additional empirical findings are novel and would not be credibly obtainable without the econometric framework proposed in this paper.

💡 Research Summary

This paper develops a comprehensive econometric framework for identifying and estimating heterogeneous marginal exposure effects (MEEs) of an endogenous binary treatment when both treatment assignment and outcomes are subject to network interference and spillovers. The authors model treatment choice as a Bayesian Nash equilibrium in a discrete game on a large single network. Each node i observes public covariates Z_i, private information ν_i, and the network adjacency matrix A. The payoff from taking the treatment includes a linear term Z_i′β_D and a peer‑effect term λ · (average treatment probability of i’s neighbors). Assuming logistic errors for ν_i and a moderate peer‑effect bound |λ| < 4, they prove that the best‑response mapping is a contraction, guaranteeing existence and uniqueness of the equilibrium (Lemma 2.1). Consequently, the equilibrium choice probability P_i(Z,A) is a well‑defined, observable function of the data.

In the first stage, the equilibrium condition reduces to a standard logistic regression with an endogenous regressor that depends on neighbors’ choice probabilities. Under a full‑rank condition on the instrument matrix (Assumption 3.1), the structural parameters β_D and λ are point‑identified (Theorem 3.1). The authors then introduce a low‑dimensional “exposure mapping” T(i,D,A) that aggregates the high‑dimensional treatment configuration D into a scalar summary. This device allows them to define MEEs that can vary with both the treatment status of neighboring nodes and unobserved heterogeneity, without imposing restrictive functional forms on the outcome equation.

For estimation, the paper adopts the ψ‑dependence framework of Kojevnikov, Menzel, and Santos (2021), which accommodates arbitrary network dependence by controlling the decay of dependence with graph distance. Using generalized method of moments (GMM), the authors construct moment conditions based on the first‑stage equilibrium and the second‑stage outcome model. They establish a uniform law of large numbers (ULLN) for ψ‑dependent arrays and prove that the GMM estimator is consistent and converges stably to a mixed normal distribution, where the asymptotic variance‑covariance matrix is random because it reflects common shocks that generate both covariates and the network structure. This result aligns with the robust inference literature (Andrews 2005; Kuersteiner & Prucha 2013, 2020) and ensures that conventional Wald‑type tests remain valid.

The empirical application revisits the impact of Chinese import competition on U.S. local labor markets using the Autor‑et‑al. (2013) commuting‑zone dataset. Treating each commuting zone as a node, the authors first estimate the treatment‑choice game, finding evidence of strategic substitution: zones facing higher import exposure tend to reduce manufacturing employment, and this decision is influenced by neighboring zones’ exposure levels. In the second stage, they estimate MEEs and discover substantial heterogeneity: (i) zones with treated neighbors experience a larger (more negative) effect of import competition on employment than isolated zones, and (ii) zones that would have chosen lower levels of import exposure (i.e., “low‑exposure” types) exhibit amplified negative effects when they are actually exposed. These patterns are invisible to standard two‑stage least squares that ignore network interference.

Overall, the paper makes three major contributions. First, it integrates endogenous treatment selection and network interference into a single equilibrium model, providing conditions for identification that are novel in the causal‑inference literature. Second, it extends ψ‑dependence‑based asymptotics to a nonlinear GMM setting with mixed‑normal limits, offering a robust inference tool for large‑network data. Third, the empirical findings demonstrate that ignoring spillovers can severely understate both the magnitude and the heterogeneity of treatment effects in policy‑relevant contexts such as trade competition. The framework is readily extensible to dynamic settings, multiple overlapping networks, and alternative forms of spillovers, opening avenues for future research.