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The Augmented Synthetic Control Method
Open accessAbstract
The synthetic control method (SCM) is a popular approach for estimating the impact of a treatment on a single unit in panel data settings. The “synthetic control” is a weighted average of control units that balances the treated unit’s pretreatment outcomes and other covariates as closely as possible. A critical feature of the original proposal is to use SCM only when the fit on pretreatment outcomes is excellent. We propose Augmented SCM as an extension of SCM to settings where such pretreatment fit is infeasible. Analogous to bias correction for inexact matching, augmented SCM uses an outcome model to estimate the bias due to imperfect pretreatment fit and then de-biases the original SCM estimate. Our main proposal, which uses ridge regression as the outcome model, directly controls pretreatment fit while minimizing extrapolation from the convex hull. This estimator can also be expressed as a solution to a modified synthetic controls problem that allows negative weights on some donor units. We bound the estimation error of this approach under different data-generating processes, including a linear factor model, and show how regularization helps to avoid over-fitting to noise. We demonstrate gains from Augmented SCM with extensive simulation studies and apply this framework to estimate the impact of the 2012 Kansas tax cuts on economic growth. We implement the proposed method in the new augsynth R package.
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Citations
Cites
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Hastie, T., Tibshirani, R., & Friedman, J. (2009) · Springer Series in Statistics · doi:10.1007/978-0-387-84858-7 · Cited by 25,970
Abadie, A., Diamond, A., & Hainmueller, J. (2010) · Journal of the American Statistical Association · doi:10.1198/jasa.2009.ap08746 · Cited by 5,513 · cited 12×
“This underscores the importance of the recommendation in Abadie, Diamond, and Hainmueller (2010, 2015) to use SCM only in settings with excellent pretreatment fit.”
InfluentialThe economic costs of conflict: a case study of the Basque Country
Abadie, A., & Gardeazabal, J. (2003) · American Economic Review · doi:10.1257/000282803321455188 · Cited by 4,517 · cited 6×
“The synthetic control method (SCM) is a popular approach for estimating the impact of a treatment on a single unit in panel data settings with a modest number of control units and with many pre-treatment periods (Abadie and Gardeazabal, 2003; Abadie et al., 2010, 2015).”
BackgroundMethodInfluentialLogistic Regression in Rare Events Data
King, G., & Zeng, L. (2001) · Political Analysis · doi:10.1093/oxfordjournals.pan.a004868 · Cited by 4,316 · cited 2×
“McCullagh and Nelder (1989) and King and Zeng (2001) show that the MLE for logistic regression can be badly biased when the response is a rare event.”
BackgroundEstimation of Regression Coefficients When Some Regressors Are Not Always Observed
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“Equation (10), by contrast, is analogous to standard doubly robust estimation (Robins, Rotnitzky, and Zhao 1994), which begins with the outcome model but then re-weights to balance residuals.”
MethodComparative Politics and the Synthetic Control Method
Abadie, A., Diamond, A., & Hainmueller, J. (2014) · American Journal of Political Science · doi:10.1111/ajps.12116 · Cited by 2,493
Comparative Politics and the Synthetic Control MethodPreprint
Abadie, A., Diamond, A., & Hainmueller, J. (2014) · SSRN Electronic Journal · doi:10.2139/ssrn.1950298 · Cited by 2,217 · cited 14×
“This underscores the importance of the recommendation in Abadie, Diamond, and Hainmueller (2010, 2015) to use SCM only in settings with excellent pretreatment fit.”
InfluentialOn the Application of Probability Theory to Agricultural Experiments. Essay on Principles. Section 9
Neyman, J., Dabrowska, D. M., & Speed, T. P. (1990) · Statistical Science · doi:10.1214/ss/1177012031 · Cited by 1,680
Synthetic Difference-in-Differences
Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021) · American Economic Review · doi:10.1257/aer.20190159 · Cited by 1,518
Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects
Abadie, A. (2021) · Journal of Economic Literature · doi:10.1257/jel.20191450 · Cited by 1,486
Bias-Corrected Matching Estimators for Average Treatment Effects
Abadie, A., & Imbens, G. W. (2011) · Journal of Business & Economic Statistics · doi:10.1198/jbes.2009.07333 · Cited by 1,416
The State of Applied Econometrics: Causality and Policy Evaluation
Athey, S. C., & Imbens, G. W. (2017) · The Journal of Economic Perspectives · doi:10.1257/jep.31.2.3 · Cited by 1,406
Randomization Analysis of Experimental Data: The Fisher Randomization Test Comment
Rubin, D. B. (1980) · Journal of the American Statistical Association · doi:10.2307/2287653 · Cited by 1,368
Generalized Synthetic Control Method: Causal Inference with Interactive Fixed Effects Models
Xu, Y. (2017) · Political Analysis · doi:10.1017/pan.2016.2 · Cited by 938 · cited 5×
“Specifically, we use the Generalized Synthetic Control Method (Xu, 2017) to estimate a factor model with three latent factors based on the series of log Gross State Product (GSP) per capita, N = 50, T0 = 89.”
MethodInfluentialThe Dangers of Extreme Counterfactuals
King, G., & Zeng, L. (2005) · Political Analysis · doi:10.1093/pan/mpj004 · Cited by 549 · cited 2×
“See King and Zeng (2006) for a discussion of extrapolation in constructing counterfactuals.”
BackgroundMethodMatrix Completion Methods for Causal Panel Data Models
Athey, S., Bayati, M., Doudchenko, N., Imbens, G., & Khosravi, K. (2018) · doi:10.3386/w25132 · Cited by 497
Regional policy evaluation: interactive fixed effects and synthetic controls.
Gobillon, L., & Magnac, T. (2016) · Review of Economics and Statistics · doi:10.1162/rest_a_00537 · Cited by 424 · cited 4×
“1 Bias for a general weighting estimator Following the setup in Abadie et al. (2010), we assume that there are J latent time-varying factors μt = {μjt}, j = 1, .”
BackgroundMethodInfluentialBalancing, Regression, Difference-In-Differences and Synthetic Control Methods: A Synthesis
Doudchenko, N., & Imbens, G. (2016) · doi:10.3386/w22791 · Cited by 410
Entropy Balancing is Doubly Robust
Zhao, Q., & Percival, D. (2016) · Journal of Causal Inference · doi:10.1515/jci-2016-0010 · Cited by 402
Approximate Residual Balancing: Debiased Inference of Average Treatment Effects in High Dimensions
Athey, S. C., Imbens, G. W., & Wager, S. (2018) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/rssb.12268 · Cited by 373 · cited 1×
“See Athey, Imbens, and Wager (2018) for analogous results on balancing weights in high dimensional cross-sectional settings.”
BackgroundStable weights that balance covariates for estimation with incomplete outcome data
Zubizarreta, J. R. (2015) · Journal of the American Statistical Association · doi:10.1080/01621459.2015.1023805 · Cited by 369 · cited 1×
“een treated and control units, rather than balancing them indirectly by rst estimating the propensity score coecients. 3 Examples includeHainmueller(2011),Graham et al.(2012),Imai and Ratkovic(2013),Zubizarreta (2015),Tan(2017), andWang and Zubizarreta(2018). Several also combine weighting and outcome modeling, includ…”
ResultA Penalized Synthetic Control Estimator for Disaggregated Data
Abadie, A., & L’Hour, J. (2021) · Journal of the American Statistical Association · doi:10.1080/01621459.2021.1971535 · Cited by 245
Synthetic controls with imperfect pretreatment fit
Ferman, B., & Pinto, C. C. D. X. (2021) · Quantitative Economics · doi:10.3982/qe1596 · Cited by 236
Oaxaca-Blinder as a Reweighting Estimator
Kline, P. (2011) · American Economic Review · doi:10.1257/aer.101.3.532 · Cited by 182 · cited 2×
“This formulation covers several estimators, including penalized SCM, entropy balancing (Hainmueller, 2011), Oaxaca-Blinder weights (Kline, 2011), and minimal approximately balancing weights (Wang and Zubizarreta, 2018).”
MethodInverse probability tilting for moment condition models with missing data
Graham, B. S., De Xavier Pinto, C. C., & Egel, D. (2012) · The Review of Economic Studies · doi:10.1093/restud/rdr047 · Cited by 174
Model-Assisted Survey Estimation with Modern Prediction Techniques
Breidt, F. J., & Opsomer, J. D. (2017) · Statistical Science · doi:10.1214/16-sts589 · Cited by 151
Robbins, M. W., Saunders, J. M., & Kilmer, B. (2016) · Journal of the American Statistical Association · doi:10.1080/01621459.2016.1213634 · Cited by 143
Covariate balancing propensity score by tailored loss functions
Zhao, Q. (2019) · The Annals of Statistics · doi:10.1214/18-aos1698 · Cited by 129
On the role of covariates in the synthetic control method
Botosaru, I., & Ferman, B. (2019) · Econometrics Journal · doi:10.1093/ectj/utz001 · Cited by 128 · cited 10×
“Botosaru and Ferman (2017) and Kaul et al. (2018) explore the role of auxiliary covariates in SCM and note several pathologies. Ferman and Pinto (2018) consider the behavior of SCM when the weights fail to exactly balance the lagged outcomes, showing that the resulting SCM weights do not converge to oracle weights whe…”
BackgroundInfluentialAugmented minimax linear estimation
Hirshberg, D. A., & Wager, S. (2021) · The Annals of Statistics · doi:10.1214/21-aos2080 · Cited by 111
Statistical Inference for Average Treatment Effects Estimated by Synthetic Control Methods
Li, K. T. (2019) · Journal of the American Statistical Association · doi:10.1080/01621459.2019.1686986 · Cited by 110 · cited 1×
“…inference for the SCM and variants, going beyond the original proposal in Abadie and Gardeazabal (2003) and Abadie, Diamond, and Hainmueller (2010, 2015); see, for example, Kathleen (2020), Toulis and Shaikh (2018), Cattaneo, Feng, and Titiunik (2019), and Cher-nozhukov, Wuthrich, and Zhu (2018).”
BackgroundSynthetic Difference In Differences
Arkhangelsky, D., Athey, S. C., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2019) · National Bureau of Economic Research · doi:10.3386/w25532 · Cited by 94
Tan, Z. (2019) · Biometrika · doi:10.1093/biomet/asz059 · Cited by 88
Robust Synthetic ControlPreprint
Amjad, M. J., Devavrat Shah, B., & Shen, D. (2017) · RePEc: Research Papers in Economics · doi:10.48550/arxiv.1711.06940 · Cited by 79
Prediction Intervals for Synthetic Control Methods
Cattaneo, M. D., Feng, Y., & Titiunik, R. (2021) · Journal of the American Statistical Association · doi:10.1080/01621459.2021.1979561 · Cited by 78
An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic ControlsPreprint
Chernozhukov, V., Wüthrich, K., & Zhu, Y. (2021) · RePEc: Research Papers in Economics · doi:10.6084/m9.figshare.14473870 · Cited by 55
Panel parametric, semiparametric, and nonparametric construction of counterfactuals
Hsiao, C., & Zhou, Q. (2019) · Journal of Applied Econometrics · doi:10.1002/jae.2702 · Cited by 55 · cited 1×
“These include the matrix completion method in Athey et al. (2017), the generalized synthetic control method in Xu (2017), and the combined approaches in Hsiao et al. (2018).”
Samartsidis, P., Seaman, S. R., Presanis, A. M., Hickman, M., & Angelis, D. D. (2019) · Statistical Science · doi:10.1214/19-sts713 · Cited by 35
Imperfect Synthetic Controls: Did the Massachusetts Health Care Reform Save Lives?
Powell, D. (2018) · RAND Corporation eBooks · doi:10.7249/wr1246 · Cited by 29
Two tales of two U.S. states: Regional fiscal austerity and economic performance
Rickman, D. S., & Wang, H. (2017) · Regional Science and Urban Economics · doi:10.1016/j.regsciurbeco.2017.10.008 · Cited by 24 · cited 1×
“In 2010, Sam Brownback was elected governor of Kansas, having run on a platform emphasizing tax cuts and deficit reduction (see Rickman and Wang 2018, for further discussion and analysis).”
BackgroundInterpretable sensitivity analysis for balancing weights
Soriano, D., Ben‐Michael, E., Bickel, P. J., Feller, A., & Pimentel, S. D. (2023) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1093/jrsssa/qnad032 · Cited by 20
Self-citationCovariate Balancing Propensity Score by Tailored Loss FunctionsPreprint
Zhao, Q. (2016) · arXiv (Cornell University) · doi:10.48550/arxiv.1601.05890 · Cited by 18
Tan, Z. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1710.08074 · Cited by 17
On the Properties of the Synthetic Control Estimator with Many Periods and Many ControlsPreprint
Ferman, B. (2021) · RePEc: Research Papers in Economics · doi:10.6084/m9.figshare.15135426 · Cited by 15
Imperfect Synthetic Controls: Did the Massachusetts Health Care Reform Save Lives?Preprint
Powell, D. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3192710 · Cited by 14 · cited 1×
“Powell (2018) instead allowed for extrapolation via a Frisch-Waugh-Lovell-style projection, which similarly generalizes the typical SCM setting.”
MethodCombining Matching and Synthetic Control to Trade off Biases from Extrapolation and Interpolation
Kellogg, M., Mogstad, M., Pouliot, G., & Torgovitsky, A. (2020) · National Bureau of Economic Research · doi:10.3386/w26624 · Cited by 11
Inference on average treatment effects in aggregate panel data settings
Chernozhukov, V., Wüthrich, K., & Zhu, Y. (2019) · doi:10.1920/wp.cem.2019.3219 · Cited by 9
Minimal Approximately Balancing Weights: Asymptotic Properties and Practical ConsiderationsPreprint
Wang, Z., & Zubizarreta, J. R. (2017) · arXiv (Cornell University) · Cited by 9
Randomization Tests in Observational Studies with Staggered Adoption of TreatmentPreprint
Shaikh, A. M., & Toulis, P. (2019) · SSRN Electronic Journal · doi:10.2139/ssrn.3509581 · Cited by 9
Hsiao, C. H., & Zhou, Q. (2018) · RePEc: Research Papers in Economics · Cited by 4
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- Ben-Michael, E., Feller, A., & Rothstein, J. (2019), “Synthetic Controls With Staggered Adoption,” Journal of the Royal Statistical Society, Series B, Forthcoming, arXiv: 1912.03290. · arXiv
- 10.1080/01621459.2021.1920957 · DOI · Paper page
- Vovk, V., Gammerman, A., & Shafer, G. (2005), Algorithmic Learning in a Random World, Boston, MA: Springer.
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What's trending in difference-in-differences? A synthesis of the recent econometrics literature
Roth, J., Sant’Anna, P. H. C., Bilinski, A., & Poe, J. (2023) · Journal of Econometrics · doi:10.1016/j.jeconom.2023.03.008 · Cited by 2,027
Synthetic Difference-in-Differences
Arkhangelsky, D., Athey, S., Hirshberg, D. A., Imbens, G. W., & Wager, S. (2021) · American Economic Review · doi:10.1257/aer.20190159 · Cited by 1,518
Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects
Abadie, A. (2021) · Journal of Economic Literature · doi:10.1257/jel.20191450 · Cited by 1,486
Matrix Completion Methods for Causal Panel Data Models
Athey, S., Bayati, M., Doudchenko, N., Imbens, G., & Khosravi, K. (2018) · doi:10.3386/w25132 · Cited by 497
Liu, L., Wang, Y., & Xu, Y. (2024) · American Journal of Political Science · doi:10.1111/ajps.12723 · Cited by 390
Metadata
- Authors
- Ben-Michael, E., Feller, A., & Rothstein, J.
- Affiliations
- Department of Statistics, Institute for Quantitative Social Sciences, Harvard University
- University of California System (US)
- Corresponding author
- Feller, Avi
- Venue
- Journal of the American Statistical Association
- Publisher
- Informa UK Limited
- Published
- 2021-07-29
- Volume
- 116
- Issue
- 536
- Pages
- 1789-1803
- ISSN
- 0162-1459 (ISSN-L)
- 1537-274X
- OpenAlex
- W3123564515
- Cited by
- 462
- References
- 42
- Citations by year
- 2026: 93 · 2025: 110 · 2024: 79 · 2023: 69 · 2022: 58 · 2021: 33
- FWCI
- 44.66
- Type
- journal article
- Language
- en
- Open access
- green
- doi.org
- Locations
- Journal of the American Statistical Association (publishedVersion)
- National Bureau of Economic Research (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- arXiv (Cornell University) (submittedVersion)
- National Bureau of Economic Research (submittedVersion)
- eScholarship (California Digital Library) (submittedVersion)
- Figshare (submittedVersion, cc-by)
- arXiv (Cornell University)
- Figshare (cc-by)
- Funders
- Institute of Education Sciences (R305D200010)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
- Social Sciences › Public Administration › Evaluation and Performance Assessment
- Keywords
- synthetic control method, treatment effect estimation, bias correction, ridge regression, factor models, regularization, overfitting, economic growth
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