Three's a crowd: Identification challenges in the triple difference model with spillover effects

TL;DR

This paper tackles identification challenges in triple-difference designs when spillover effects contaminate control groups. It demonstrates that the conventional TD estimator generally cannot identify the true treatment effect $ATT$ or spillover $ASU$ in the presence of interference, and introduces the Double-Triple Difference (DTD) framework with two parallel-trend-in-trends conditions to separately identify $ATT$ and $ASU$, via unconditional and conditional variants including doubly robust implementations. The authors provide formal identification results, discuss estimation strategies, and validate them through Monte Carlo simulations and an empirical application to a Campania Special Economic Zone (SEZ), where spillovers plausibly operate through linked sectors and geography. Substantial findings show that TD underestimates the treatment effect when spillovers are positive, while DTD recovers both the direct and spillover effects; evidence from the Campania case confirms meaningful positive effects and spillovers, underscoring the method’s practical relevance for policy evaluation under interference.

Abstract

The paper studies identification in triple-difference designs when spillover effects contaminate one or more control groups. We show that, under conventional identifying assumptions, the triple-difference model fails to identify both the treatment effect and the spillover effect under such interference. To overcome this limitation, we propose an alternative specification, the double-triple-difference model, and explicitly formalize identifying assumptions and spillover structures required for consistent identification of both effects. We derive formal identification results and assess the performance of the proposed model through Monte Carlo simulations. An empirical application evaluating a Special Economic Zone in Italy is provided.

Figures (10)

• Figure 1: Two strata, each comprising two groups. Treated units belong to group $\mathcal{T}_1$, and units affected by spillovers belong to group $\mathcal{I}_1$. Units in stratum $\mathcal{S}_0$ are completely unaffected, both directly and indirectly, by the policy.
• Figure 2: Double-triple difference framework with two strata and three groups per stratum.
• Figure 3: Bias of TD and DTD estimators for scenarios SUTVA, 1.0, 1.1, 1.2 at varying interference sets sizes (10% and 50% of the original control sets).
• Figure 4: Bias of TD and DTD estimators for scenarios 2.0 and 2.1 (with multiple-spillover) at varying interference sets sizes (10% and 50% of the original control sets).
• Figure 5: Bias of the doubly-robust TD and DTD estimators under SUTVA and spillovers at varying sample sizes.

Theorems & Definitions (26)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof