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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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Causal Inference About the Effects of Interventions From Observational Studies in Medical Journals
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Synthetic controls with imperfect pretreatment fit
Ferman, B., & Pinto, C. C. D. X. (2021) · Quantitative Economics · doi:10.3982/qe1596 · Cited by 236
“Ben-Michael et al. (2018) derive finite-sample bounds on the bias of the SC estimator, and show that the bounds they derive do not converge to zero when J is fixed and T0 ! 1.”
BackgroundInfluentialAn Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls
Chernozhukov, V., Wüthrich, K., & Zhu, Y. (2021) · Journal of the American Statistical Association · doi:10.1080/01621459.2021.1920957 · Cited by 176
“Our inference method also works in conjunction with more recently proposed modified versions of SC, such as the augmented SC estimator of Ben-Michael, Feller, and Rothstein (2018).”
MethodDo pilot zones for green finance reform and innovation promote energy savings? Evidence from China
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Heterogeneous emission trading schemes and green innovation
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“18 A detailed description of the method can be found in Ben-Michael et al. (2021).”
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MethodDid the ACA Medicaid expansion save lives?
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On synthetic difference-in-differences and related estimation methods in Stata
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“A key element of estimation, as in the case of synthetic control estimation (Ben-Michael et al. 2021; Abadie et al. 2010), is the adequate modeling of pre-treatment trends.”
InfluentialSynthetic Difference-in-Differences EstimationPreprint
Clarke, D., Pailañir, D., Athey, S. C., & Imbens, G. W. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4346540 · Cited by 98
“This has attracted considerable attention in both empirical applications and theoretical extensions, with recent advances including debiasing procedures (Ben-Michael et al. 2021) which can additionally house multiple treatment units (Abadie and L’Hour 2021), more flexible weighting schemes, or constant fixed differenc…”
BackgroundInfluentialConditional cash lotteries increase COVID-19 vaccination rates
Barber, A., & West, J. (2021) · Journal of Health Economics · doi:10.1016/j.jhealeco.2021.102578 · Cited by 97
“For nontrivial cases, Ben-Michael et al. (2021) demonstrate how Eq.”
Synthetic 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
“Methodologically, our work draws most directly from the literature on SC meth-ods, including Abadie and Gardeazabal ( 2003 ) ; Abadie, Diamond, and Hainmueller ( 2010, 2015 ; Abadie and L’Hour ( 2016 ) ; Doudchenko and Imbens ( 2016 ) ; and Ben-Michael, Feller, and Rothstein ( 2018 ) .”
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
“3 in Chernozhukov et al. (2018) that P( √ T0‖γ̂ − γ‖∞ > c(1− α)) ≤ α+ CT−c 0 .”
BackgroundInfant Deaths After Texas’ 2021 Ban on Abortion in Early Pregnancy
Gemmill, A., Margerison, C. E., Stuart, E. A., & Bell, S. O. (2024) · Archives of Pediatrics and Adolescent Medicine · doi:10.1001/jamapediatrics.2024.0885 · Cited by 78
Synthetic Control Estimator: A Tool for Comparative Case Studies in Economic History
Gilchrist, D. J., Emery, T., Garoupa, N. M., & Spruk, R. (2020) · Journal of Economic Surveys · doi:10.1111/joes.12493 · Cited by 81
Acharya, B., & Dhakal, C. (2021) · JAMA Network Open · doi:10.1001/jamanetworkopen.2021.38238 · Cited by 65
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“Based on the potential outcomes framework for causal inference (12, 13, 36, 37), we used the augmented synthetic control method to analyze the pandemic's progression in comparable municipalities that underwent different lockdown interventions (30, 31).”
MethodCausal analysis of central bank holdings of corporate bonds under interference
Mäkinen, T., Li, F., Mercatanti, A., & Silvestrini, A. (2022) · Economic Modelling · doi:10.1016/j.econmod.2022.105873 · Cited by 54
Low-intensity fires mitigate the risk of high-intensity wildfires in California’s forests
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“Statistical Analysis We used an augmented synthetic control approach (53).”
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Doucette, M. L., Ward, J. A., McCourt, A. D., Webster, D. W., & Crifasi, C. K. (2022) · Journal of Urban Health · doi:10.1007/s11524-022-00627-5 · Cited by 44
Texas’ 2021 Ban on Abortion in Early Pregnancy and Changes in Live Births
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Self-citationMesserle, R., & Schreyögg, J. (2023) · The European Journal of Health Economics · doi:10.1007/s10198-023-01645-z · Cited by 44
“Fixest, Augsynth, and SynthDiD were primarily used as packages for estimation [40, 53, 54].”
MethodKaplan, S., White, J. S., Madsen, K. A., Basu, S., Villas-Boas, S. B., & Schillinger, D. (2024) · JAMA Health Forum · doi:10.1001/jamahealthforum.2023.4737 · Cited by 35
Zhang, Z., Deng, H., Yan, M., Feng, C. Q., & Sun, H. (2025) · International Review of Economics & Finance · doi:10.1016/j.iref.2025.104265 · Cited by 37
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The impact of e-procurement on institutional quality
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Treatment effects in interactive fixed effects models with a small number of time periods
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“Let ATTDID denote the expression on the right hand side of Equation (10) but allowing for the possibility that the parallel trends assumption does not hold (so that, in this case, ATTDID may not be equal to the ATT ).”
BackgroundConsidering multiple governance levels in epidemiologic analysis of public policies
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Causality in Econometrics: Choice vs Chance
Imbens, G. W. (2022) · Econometrica · doi:10.3982/ecta21204 · Cited by 36
“The basic synthetic control method as well as various extensions (Arkhangelsky, Athey, Hirshberg, Imbens, and Wager (2021), Ben-Michael, Feller, and Rothstein (2021)), have in a short time found many applications in a wide range of fields, including studies of the effects of Brexit and the effects…”
MethodOn the implied weights of linear regression for causal inference
Chattopadhyay, A., & Zubizarreta, J. R. (2022) · Biometrika · doi:10.1093/biomet/asac058 · Cited by 43
Samartsidis, P., Seaman, S. R., Presanis, A. M., Hickman, M., & Angelis, D. D. (2019) · Statistical Science · doi:10.1214/19-sts713 · Cited by 35
Rico, J., Wang, Z., Panlasigui, S., Loucks, C., Swenson, J. J., & Pfaff, A. S. P. (2022) · Journal of Environmental Economics and Management · doi:10.1016/j.jeem.2022.102780 · Cited by 30
Ham, D. W., & Miratrix, L. W. (2024) · The Annals of Applied Statistics · doi:10.1214/24-aoas1872 · Cited by 39
The PCDID Approach: Difference-in-Differences When Trends Are Potentially Unparallel and Stochastic
Chan, M. K., & Kwok, S. S. M. (2021) · Journal of Business and Economic Statistics · doi:10.1080/07350015.2021.1914636 · Cited by 31
“The PCDID approach is related to the large literature on factor-augmented regression models (e.g., Stock and Watson 2002; Bernanke, Boivin, and Eliasz 2005; Bai and Ng 2006), which forecast one or several time series using a large number of predictor series.”
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Counterfactual Analysis With Artificial Controls: Inference, High Dimensions, and Nonstationarity
Masini, R., & Medeiros, M. C. (2021) · Journal of the American Statistical Association · doi:10.1080/01621459.2021.1964978 · Cited by 27
“See also Amjad, Shah, and Shen (2018), Arkhangelsky et al. (2019), and Ben-Michael, Feller, and Rothstein (2019).”
BackgroundSheridan, P. E., McElroy, S., Casey, J. A., & Benmarhnia, T. (2022) · Epidemiology · doi:10.1097/ede.0000000000001539 · Cited by 26
Multilevel Calibration Weighting for Survey Data
Ben‐Michael, E., Feller, A., & Hartman, E. (2023) · Political Analysis · doi:10.1017/pan.2023.9 · Cited by 26
“7 If we allow unbounded extrapolation and set L = −∞ and U = ∞, the resulting estimator will be equivalent to linear regression weights from a linear regression of the outcome on first-order indicatorsD (1) (Ben-Michael et al. 2021).”
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The how and why of Bayesian nonparametric causal inference
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“…are two very popular topics Wiesen-farth et al., 2014; Adhikari et al., 2020), to principal stratification (Schwartz et al., 2011), to high-dimensional panel data settings (Antonelli et al., 2020), and to replace synthetic control methods with flexible model-based methods (Ben-Michael et al., 2021).”
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Do price reductions attract customers in urban public transport? A synthetic control approach
Wallimann, H., Blättler, K., & von Arx, W. (2023) · Transportation Research Part A Policy and Practice · doi:10.1016/j.tra.2023.103700 · Cited by 24
Synthetic Controls with Staggered Adoption
Ben‐Michael, E., Feller, A., & Rothstein, J. (2021) · National Bureau of Economic Research · doi:10.3386/w28886 · Cited by 24
Self-citationLiu, B., Bryson, J. R., Sevinç, D., Cole, M. A., Elliott, R., Bartington, S., Bloss, W. J., & Shi, Z. (2023) · Environmental and Resource Economics · doi:10.1007/s10640-023-00794-2 · Cited by 26
“Both hourly observed and weather normalised pollutant concentrations were converted into weekly averages to run the second-stage analysis, i.e., the Ridge Augmented Synthetic Control Method (ASCM).”
BackgroundMethodInfluentialCommunity notes reduce engagement with and diffusion of false information online
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Counterfactual Analysis and Inference With Nonstationary Data
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Menchetti, F., Cipollini, F., & Mealli, F. (2022) · Econometrics Journal · doi:10.1093/ectj/utac024 · Cited by 38
Government debt, European Institutions and fiscal rules: a synthetic control approach
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“In applying the SCM, the approach of this paper follows the seminal work of Abadie and Gardeazabal (2003) and Abadie et al. (2010) on the SCM as well as the bias-corrected SCM in line with (Abadie & L’hour, J., 2021; Ben-Michael et al., 2021; Abadie, 2021) as a large-scale robustness analysis.”
Synthetic Control as Online Linear Regression
Chen, J. (2023) · Econometrica · doi:10.3982/ecta20720 · Cited by 22
“…typically derived under a lowrank linear factor model or a vector autoregressive model of the outcomes (see, among others, Abadie et al., 2010, Ben-Michael et al., 2019, 2021, Ferman and Pinto, 2021, Viviano and Bradic, 2019).2 While these statistical guarantees formally hold under these…”
BackgroundLiang, Y., & He, X. (2024) · Economic Analysis and Policy · doi:10.1016/j.eap.2024.10.012 · Cited by 20
Effects of U.S. State Medical Cannabis Laws on Treatment of Chronic Noncancer Pain
McGinty, E. E., Tormohlen, K. N., Seewald, N. J., Bicket, M. C., McCourt, A. D., Rutkow, L., White, S., & Stuart, E. A. (2023) · Annals of Internal Medicine · doi:10.7326/m23-0053 · Cited by 20
“Statistical Analysis We used an augmented synthetic control approach (31), which weights comparison states so that the outcomes during the prelaw period are identical in the intervention and comparison states.”
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Did Ohio’s Vaccine Lottery Increase Vaccination Rates? A Pre-Registered, Synthetic Control Study
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“2021), and additionally including daily estimates of individual mobility provided by Google trends (Barber and West 2021): Lastly, we varied the modeling technique used to compare the traditional synthetic control approach to an augmented synthetic control model (Ben-Michael et al. 2021a), exploring a technique that r…”
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Samartsidis, P., Seaman, S. R., Montagna, S., Charlett, A., Hickman, M., & Angelis, D. D. (2020) · Journal of the Royal Statistical Society Series A (Statistics in Society) · doi:10.1111/rssa.12569 · Cited by 20
“…time series observational data, including frequentist approaches based on the FA model (Gobillon and Magnac, 2016; Chan and Kwok, 2016; Xu, 2017; Li, 2018) and synthetic control-type approaches (Abadie et al. , 2010; Hsiao et al. , 2012; Doudchenko and Imbens, 2016; Ben-Michael et al. , 2018).”
An Alternative to Synthetic Control for Models with Many Covariates Under Sparsity
Bléhaut, M., D’Haultfœuille, X., L’Hour, J., & Tsybakov, A. B. (2023) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-031-30114-8_12 · Cited by 18
Samartsidis, P., Seaman, S. R., Presanis, A. M., Hickman, M., & Angelis, D. D. (2019) · Apollo (University of Cambridge) · doi:10.17863/cam.40086 · Cited by 18
Synthetic Learner: Model-free inference on treatments over time
Viviano, D., & Bradić, J. (2022) · Journal of Econometrics · doi:10.1016/j.jeconom.2022.07.006 · Cited by 18
“Alternatives for SC include ridge-regression (Ben-Michael et al., 2018), kernel balancing (Hazlett and Xu, 2018), structural models with auto-regressive elements (Blackwell and Glynn, 2018), generalized SC (Xu, 2017; Doudchenko and Imbens, 2016).”
MethodGottlieb, A., Harper, T., & Jung, H. (2024) · Law & Policy · doi:10.1111/lapo.12252 · Cited by 17
Integrating Artificial Intelligence into Causal Research in Epidemiology
Matthay, E. C., Neill, D. B., Titus, A. R., Desai, S. M., Troxel, A. B., Cerdá, M., Diaz, I. M., Santacatterina, M., & Thorpe, L. E. (2025) · Current Epidemiology Reports · doi:10.1007/s40471-025-00359-5 · Cited by 18
“For example, synthetic control methods create artificial control groups by taking weighted averages of the outcomes in untreated units, with the weights selected algorithmically by minimizing differences in confounder values between the treated and synthetic control units [113, 114].”
Learning lessons from over-crediting to ensure additionality in forest carbon credits
Swinfield, T., Williams, A. M., Coomes, D. A., Dales, M., Ferris, P., Guizar‐Coutiño, A., Hartup, J., Holland, J., Jaffer, S., Jones, J. P. G., et al. (2026) · Nature Communications · doi:10.1038/s41467-026-71552-3 · Cited by 17
“Quasi-experimental methods are advancing all the time and methods exist which combine features of difference-in-differences and synthetic control, such as interactive fi xed-effects, augmented synthetic controls and synthetic difference-in-differences 49 – 51 .”
Connealy, N. T., Piza, E. L., Arietti, R. A., Mohler, G., & Carter, J. G. (2024) · Journal of Experimental Criminology · doi:10.1007/s11292-024-09617-w · Cited by 16
“…staggered adoption, including synthetic difference-in-difference (Arkhangelsky et al., 2021), partially pooled staggered synthetic control methods (Ben-Michael et al., 2021, 2022), and group time averages for two-way fixed effects for difference-in-difference with multiple time periods (Callaway…”
The sooner the better: lives saved by the lockdown during the COVID-19 outbreak. The case of Italy
Cerqueti, R., Coppier, R., Girardi, A., & Ventura, M. (2021) · Econometrics Journal · doi:10.1093/ectj/utab027 · Cited by 16
“…correction (Powel, 2018; Arkhangelesky et al., 2019; Chernozhukov et al., 2019b). consequently, the effectiveness of non-pharmaceutical interventions (see, e.g., Bonacini et al., 2020; Cameletti, 2020; Eckardt et al., 2020; Lolli et al., 2020; Palladino et al., 2020, Peracchi and Terlizzese, 2020).”
BackgroundThe effect of doubling the minimum wage on employment and earnings in Mexico
Campos-Vázquez, R. M., & Esquivel, G. (2021) · Economics Letters · doi:10.1016/j.econlet.2021.110124 · Cited by 16
Estimating the effects of a California gun control program with multitask Gaussian processes
Ben‐Michael, E., Arbour, D. T., Feller, A., Franks, A. M., & Raphael, S. P. (2023) · The Annals of Applied Statistics · doi:10.1214/22-aoas1654 · Cited by 20
“There is a large literature on extensions and modifications to SCM; see Doudchenko and Imbens (2016) and Abadie (2019).”
MethodResultInfluentialSelf-citationThe effect of international sanctions on the size of the middle class in Iran
Farzanegan, M. R., & Habibi, N. (2025) · European Journal of Political Economy · doi:10.1016/j.ejpoleco.2025.102749 · Cited by 23
The Impact of the German Fuel Discount on Prices at the Petrol Pump
Seiler, V., & Stoeckmann, N. (2023) · German Economic Review · doi:10.1515/ger-2022-0108 · Cited by 15
Deregulation of public civilian gun carrying and violent crimes: A longitudinal analysis 1981–2019
Doucette, M. L., Crifasi, C. K., McCourt, A. D., Ward, J. A., Fix, R. L., & Webster, D. W. (2023) · Criminology & Public Policy · doi:10.1111/1745-9133.12638 · Cited by 15
Target Trial Emulation for Evaluating Health Policy
Seewald, N. J., McGinty, E. E., & Stuart, E. A. (2024) · Annals of Internal Medicine · doi:10.7326/m23-2440 · Cited by 15
Casey, J. A., Tartof, S. Y., Davis, M. F., Nachman, K. E., Price, L. B., Liu, C. H., Yu, K. C., Gupta, V., Innes, G. K., Tseng, H. F., et al. (2023) · Environmental Health Perspectives · doi:10.1289/ehp11221 · Cited by 16
“…a regularized outcome regression model (using ridge regression with the penalty parameter chosen via cross-validation) to improve pre-policy fi t. 26 In this analysis, control states are weighted — forming the synthetic control — such that their weighted average outcomes and covariates in the…”
InfluentialDoes the Swiss Debt Brake Induce Sound Federal Finances? A Synthetic Control Analysis
Pfeil, C. F., & Feld, L. P. (2023) · Public Finance Review · doi:10.1177/10911421231191566 · Cited by 14
“For further analyses, see Firpo and Possebom (2018) as well as Ben-Michael, Feller, and Rothstein (2021).”
The short‐term macroeconomic impact of populism: A case study of Poland
Brzeziński, M., & Sałach, K. (2024) · Economics and Politics · doi:10.1111/ecpo.12273 · Cited by 14
Nurturing nature: The role of green finance in reviving urban biodiversity
Gao, Z., Zhao, Y., Li, L., & Hao, Y. (2025) · The Quarterly Review of Economics and Finance · doi:10.1016/j.qref.2025.102005 · Cited by 14
Hospital quality risk standardization via approximate balancing weights
Keele, L. J., Ben‐Michael, E., Feller, A., Kelz, R. R., & Miratrix, L. W. (2023) · The Annals of Applied Statistics · doi:10.1214/22-aoas1629 · Cited by 18
“Here, we choose to penalize the sum of the squared weights,∑ Zi=j γ̂2i , though other penalties are possible; see Ben-Michael et al. (2021).”
BackgroundMethodInfluentialSelf-citationThe Cross of Gold: Brazilian Treasure and the Decline of Portugal
Kedrosky, D., & Palma, N. P. G. (2025) · The Journal of Economic History · doi:10.1017/s0022050725100764 · Cited by 13
Kesari, A. (2022) · Journal of Empirical Legal Studies · doi:10.1111/jels.12331 · Cited by 13
How Counterfactual Fairness Modelling in Algorithms Can Promote Ethical Decision-Making
Schutter, L. D., & Cremer, D. D. (2023) · International Journal of Human-Computer Interaction · doi:10.1080/10447318.2023.2247624 · Cited by 15
“Specifically, ML models can create synthetic con-trol conditions (i.e., a counterfactual scenario absent of a particular cause) that can be used to estimate causal effects (Ben-Michael et al., 2021).”
MethodTakahata, K., Suetsugu, H., Fukaya, K., & Shirota, S. (2024) · Environmental Data Science · doi:10.1017/eds.2024.5 · Cited by 13
Do Foreign Buyer Taxes Affect House Prices?
Hartley, J., Ma, L., Wächter, S. M., & Zevelev, A. A. (2024) · Journal of Real Estate Research · doi:10.1080/08965803.2024.2376955 · Cited by 13
The Long-Run Effects of R&D Place-Based Policies: Evidence from Russian Science Cities
Schweiger, H., Stepanov, A., & Zacchia, P. (2022) · American Economic Journal Economic Policy · doi:10.1257/pol.20200289 · Cited by 13
Introduction to the Special Section on Synthetic Control Methods
Abadie, A., & Cattaneo, M. D. (2021) · Journal of the American Statistical Association · doi:10.1080/01621459.2021.2002600 · Cited by 12
“Uncertainty Quantification/Inference: Cattaneo et al. (2021), Chernozhukov et al. (2021), Shaikh and Toulis (2021) Inference with synthetic control estimators is often carried out by permutation techniques after adopting a benchmark distribution for the assignment mechanism (see, e.g., Abadie et al., 2010; Firpo and P…”
BackgroundMethod
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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