Synthetic Interventions: Extending Synthetic Controls to Multiple Treatments

The synthetic controls (SC) methodology is a prominent tool for policy evaluation in panel data applications, typically grounded in a low-rank matrix factor model where potential outcomes are governed by low-dimensional latent factors over units and time. In this article, we present the synthetic interventions (SI) framework, an extension of the SC framework to accommodate multiple interventions. Fundamental to SI is a low-rank tensor factor model that builds on the matrix structure by embedding a latent factorization over interventions within the temporal factors, and assuming that unit-level latent factors are shared across both time and interventions. To operationalize this framework, we generalize standard SC-based estimators by borrowing strength across interventions, all while preserving their core principles. We establish consistency for one estimator variant and show that a bias-corrected version achieves asymptotic normality, thereby enabling valid inference. Our statistical results are supported by simulation studies and an empirical revisit of a canonical SC tobacco control case study, where we explore related questions not previously examined.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2025.1590.