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Large-Sample Properties of the Synthetic Control Method under Selection on Unobservables

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  • Dmitry Arkhangelsky
  • David Hirshberg

Abstract

We analyze the synthetic control (SC) method in panel data settings with many units. We assume the treatment assignment is based on unobserved heterogeneity and pre-treatment information, allowing for both strictly and sequentially exogenous assignment processes. We show that the critical property that determines the behavior of the SC method is the ability of input features to approximate the unobserved heterogeneity. Our results imply that the SC method delivers asymptotically normal estimators for a large class of linear panel data models as long as the number of pre-treatment periods is sufficiently large, making it a natural alternative to the Difference-in-Differences.

Suggested Citation

  • Dmitry Arkhangelsky & David Hirshberg, 2023. "Large-Sample Properties of the Synthetic Control Method under Selection on Unobservables," Papers 2311.13575, arXiv.org, revised Dec 2023.
    • Handle: RePEc:arx:papers:2311.13575
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    1. Card, David & Krueger, Alan B, 1994. "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania," American Economic Review, American Economic Association, vol. 84(4), pages 772-793, September.
    2. Clément de Chaisemartin & Xavier D'Haultfœuille, 2020. "Two-Way Fixed Effects Estimators with Heterogeneous Treatment Effects," American Economic Review, American Economic Association, vol. 110(9), pages 2964-2996, September.
    3. Hainmueller, Jens, 2012. "Entropy Balancing for Causal Effects: A Multivariate Reweighting Method to Produce Balanced Samples in Observational Studies," Political Analysis, Cambridge University Press, vol. 20(1), pages 25-46, January.
    4. Ashenfelter, Orley C, 1978. "Estimating the Effect of Training Programs on Earnings," The Review of Economics and Statistics, MIT Press, vol. 60(1), pages 47-57, February.
    5. Javier Alvarez & Manuel Arellano, 2003. "The Time Series and Cross-Section Asymptotics of Dynamic Panel Data Estimators," Econometrica, Econometric Society, vol. 71(4), pages 1121-1159, July.
    6. Dmitry Arkhangelsky & Susan Athey & David A. Hirshberg & Guido W. Imbens & Stefan Wager, 2021. "Synthetic Difference-in-Differences," American Economic Review, American Economic Association, vol. 111(12), pages 4088-4118, December.
    7. Holtz-Eakin, Douglas & Newey, Whitney & Rosen, Harvey S, 1988. "Estimating Vector Autoregressions with Panel Data," Econometrica, Econometric Society, vol. 56(6), pages 1371-1395, November.
    8. Alberto Abadie & Javier Gardeazabal, 2003. "The Economic Costs of Conflict: A Case Study of the Basque Country," American Economic Review, American Economic Association, vol. 93(1), pages 113-132, March.
    9. Timothy B. Armstrong & Michal Kolesár, 2021. "Finite‐Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness," Econometrica, Econometric Society, vol. 89(3), pages 1141-1177, May.
    10. Bryan S. Graham & Cristine Campos De Xavier Pinto & Daniel Egel, 2012. "Inverse Probability Tilting for Moment Condition Models with Missing Data," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 79(3), pages 1053-1079.
    11. Nikolay Doudchenko & Guido W. Imbens, 2016. "Balancing, Regression, Difference-In-Differences and Synthetic Control Methods: A Synthesis," NBER Working Papers 22791, National Bureau of Economic Research, Inc.
    12. Jinyong Hahn & Guido Kuersteiner, 2002. "Asymptotically Unbiased Inference for a Dynamic Panel Model with Fixed Effects when Both "n" and "T" Are Large," Econometrica, Econometric Society, vol. 70(4), pages 1639-1657, July.
    13. Daron Acemoglu & Suresh Naidu & Pascual Restrepo & James A. Robinson, 2019. "Democracy Does Cause Growth," Journal of Political Economy, University of Chicago Press, vol. 127(1), pages 47-100.
    14. Keisuke Hirano & Guido W. Imbens & Geert Ridder, 2003. "Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score," Econometrica, Econometric Society, vol. 71(4), pages 1161-1189, July.
    15. Guido W. Imbens & Richard H. Spady & Phillip Johnson, 1998. "Information Theoretic Approaches to Inference in Moment Condition Models," Econometrica, Econometric Society, vol. 66(2), pages 333-358, March.
    16. Alberto Abadie, 2005. "Semiparametric Difference-in-Differences Estimators," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 72(1), pages 1-19.
    17. Damon Jones & Ioana Marinescu, 2022. "The Labor Market Impacts of Universal and Permanent Cash Transfers: Evidence from the Alaska Permanent Fund," American Economic Journal: Economic Policy, American Economic Association, vol. 14(2), pages 315-340, May.
    18. Eduardo Cavallo & Sebastian Galiani & Ilan Noy & Juan Pantano, 2013. "Catastrophic Natural Disasters and Economic Growth," The Review of Economics and Statistics, MIT Press, vol. 95(5), pages 1549-1561, December.
    19. Victor Chernozhukov & Kaspar Wüthrich & Yinchu Zhu, 2021. "An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1849-1864, October.
    20. Dmitry Arkhangelsky & Guido W Imbens, 2022. "Doubly robust identification for causal panel data models [Sufficient statistics for unobserved heterogeneity in structural dynamic logit models]," The Econometrics Journal, Royal Economic Society, vol. 25(3), pages 649-674.
    21. Ashenfelter, Orley & Card, David, 1985. "Using the Longitudinal Structure of Earnings to Estimate the Effect of Training Programs," The Review of Economics and Statistics, MIT Press, vol. 67(4), pages 648-660, November.
    22. Hyungsik Roger Moon & Martin Weidner, 2015. "Linear Regression for Panel With Unknown Number of Factors as Interactive Fixed Effects," Econometrica, Econometric Society, vol. 83(4), pages 1543-1579, July.
    23. Susan Athey & Guido W. Imbens & Stefan Wager, 2018. "Approximate residual balancing: debiased inference of average treatment effects in high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(4), pages 597-623, September.
    24. Yixin Wang & Jose R Zubizarreta, 2020. "Minimal dispersion approximately balancing weights: asymptotic properties and practical considerations," Biometrika, Biometrika Trust, vol. 107(1), pages 93-105.
    25. Jushan Bai, 2009. "Panel Data Models With Interactive Fixed Effects," Econometrica, Econometric Society, vol. 77(4), pages 1229-1279, July.
    26. Jaap H. Abbring & Gerard J. van den Berg, 2003. "The Nonparametric Identification of Treatment Effects in Duration Models," Econometrica, Econometric Society, vol. 71(5), pages 1491-1517, September.
    27. Abadie, Alberto & Diamond, Alexis & Hainmueller, Jens, 2010. "Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 493-505.
    28. Julius J. Andersson, 2019. "Carbon Taxes and CO2 Emissions: Sweden as a Case Study," American Economic Journal: Economic Policy, American Economic Association, vol. 11(4), pages 1-30, November.
    29. Dmitry Arkhangelsky & Guido W. Imbens, 2019. "Doubly Robust Identification for Causal Panel Data Models," Papers 1909.09412, arXiv.org, revised Feb 2022.
    30. Lea Bottmer & Guido Imbens & Jann Spiess & Merrill Warnick, 2021. "A Design-Based Perspective on Synthetic Control Methods," Papers 2101.09398, arXiv.org, revised Jul 2023.
    31. Alberto Abadie & Susan Athey & Guido W. Imbens & Jeffrey M. Wooldridge, 2020. "Sampling‐Based versus Design‐Based Uncertainty in Regression Analysis," Econometrica, Econometric Society, vol. 88(1), pages 265-296, January.
    32. Moon, Hyungsik Roger & Weidner, Martin, 2017. "Dynamic Linear Panel Regression Models With Interactive Fixed Effects," Econometric Theory, Cambridge University Press, vol. 33(1), pages 158-195, February.
    33. Blundell, Richard & Bond, Stephen, 1998. "Initial conditions and moment restrictions in dynamic panel data models," Journal of Econometrics, Elsevier, vol. 87(1), pages 115-143, August.
    34. Kosuke Imai & Marc Ratkovic, 2014. "Covariate balancing propensity score," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(1), pages 243-263, January.
    35. Goodman-Bacon, Andrew, 2021. "Difference-in-differences with variation in treatment timing," Journal of Econometrics, Elsevier, vol. 225(2), pages 254-277.
    36. Sun, Liyang & Abraham, Sarah, 2021. "Estimating dynamic treatment effects in event studies with heterogeneous treatment effects," Journal of Econometrics, Elsevier, vol. 225(2), pages 175-199.
    37. Jushan Bai, 2003. "Inferential Theory for Factor Models of Large Dimensions," Econometrica, Econometric Society, vol. 71(1), pages 135-171, January.
    38. Arellano, Manuel, 2003. "Panel Data Econometrics," OUP Catalogue, Oxford University Press, number 9780199245291, Decembrie.
    39. Nickell, Stephen J, 1981. "Biases in Dynamic Models with Fixed Effects," Econometrica, Econometric Society, vol. 49(6), pages 1417-1426, November.
    40. Susanne M. Schennach, 2007. "Point estimation with exponentially tilted empirical likelihood," Papers 0708.1874, arXiv.org.
    41. Alberto Abadie & Jérémy L’Hour, 2021. "A Penalized Synthetic Control Estimator for Disaggregated Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1817-1834, October.
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