Synthetic Difference In Differences
We present a new perspective on the Synthetic Control (SC) method as a weighted least squares regression estimator with time fixed effects and unit weights. This perspective suggests a generalization with two way (both unit and time) fixed effects, and both unit and time weights, which can be interpreted as a unit and time weighted version of the standard Difference In Differences (DID) estimator. We find that this new Synthetic Difference In Differences (SDID) estimator has attractive properties compared to the SC and DID estimators. Formally we show that our approach has double robustness properties: the SDID estimator is consistent under a wide variety of weighting schemes given a well-specified fixed effects model, and SDID is consistent with appropriately penalized SC weights when the basic fixed effects model is misspecified and instead the true data generating process involves a more general low-rank structure (e.g., a latent factor model). We also present results that justify standard inference based on weighted DID regression. Further generalizations include unit and time weighted factor models.
This research was generously supported by ONR grant N00014-17-1-2131 and the Sloan Foundation. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
Susan Athey serves on the boards of directors of Expedia (EXPE), Lending Club (LC), Rover, Ripple, Turo, Innovations for Poverty Action, and CoinCenter. She previously had a long term consulting relationship with Microsoft. She also advises venture capital firms X/Seed Capital and NYCA Partners.Guido W. Imbens
I have consulted for Microsoft Corporation, Facebook, Amazon, and Lilly Corporation.