Synthetic Difference in Differences Dmitry Arkhangelsky† Susan Athey‡ David A. Hirshberg§ Guido W. Imbens¶ Stefan Wager‖ Version July 2021 Abstract We present a new estimator for causal effects with panel data that builds on insights behind the widely used difference in differences and synthetic control methods. Relative to these methods we find, both theoretically and empirically, that this \synthetic difference in differences” estimator has desirable robustness properties, and that it performs well in settings where the conventional estimators are commonly used in practice. We study the asymptotic behavior of the estimator when the systematic part of the outcome model includes latent unit factors interacted with latent time factors, and we present conditions for consistency and asymptotic normality. Keywords: causal inference, difference in differences, synthetic controls, panel data ∗We are grateful for helpful comments and feedback from a co-editor and referees, as well as from arXiv:1812.09970v4 [stat.ME] 2 Jul 2021 Alberto Abadie, Avi Feller, Paul Goldsmith-Pinkham, Liyang Sun, Yiqing Xu, Yinchu Zhu, and seminar participants at several venues. This research was generously supported by ONR grant N00014-17-1- 2131 and the Sloan Foundation. The R package for implementing the methods developed here is available at https://github.com/synth-inference/synthdid. The associated vignette is at https://synth- inference.github.io/synthdid/. †Associate Professor, CEMFI, Madrid,
[email protected]. ‡Professor of Economics, Graduate School of Business, Stanford University, SIEPR, and NBER,
[email protected]. §Postdoctoral Fellow, Graduate School of Business and Department of Statistics, Stanford Univer- sity,
[email protected]. ¶Professor of Economics, Graduate School of Business, and Department of Economics, Stanford University, SIEPR, and NBER,
[email protected].