Synthetic Difference In Differences

This paper introduces a new perspective on the Synthetic Control (SC) method, viewing it as a weighted least squares regression estimator with time fixed effects and unit weights. This perspective allows for a generalization incorporating two-way fixed effects (unit and time) and both unit and time weights, resulting in a unit and time-weighted version of the Difference-in-Differences (DID) estimator. The authors term this new estimator Synthetic Difference-in-Differences (SDID). They demonstrate that SDID possesses attractive properties compared to SC and DID, including double robustness: SDID is consistent under various weighting schemes given a well-specified fixed effects model and with appropriately penalized SC weights when the fixed effects model is misspecified, but the true data-generating process involves a more general low-rank structure (e.g., a latent factor model). The paper also justifies standard inference based on weighted DID regression and discusses generalizations to unit and time-weighted factor models.

Dmitry Arkhangelsky, Susan Athey, David A. Hirshberg, Guido W. Imbens, Stefan Wager