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Treatment Effects in Interactive Fixed Effects Models with a Small Number of Time Periods

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  • Brantly Callaway
  • Sonia Karami

Abstract

This paper considers identifying and estimating the Average Treatment Effect on the Treated (ATT) when untreated potential outcomes are generated by an interactive fixed effects model. That is, in addition to time-period and individual fixed effects, we consider the case where there is an unobserved time invariant variable whose effect on untreated potential outcomes may change over time and which can therefore cause outcomes (in the absence of participating in the treatment) to follow different paths for the treated group relative to the untreated group. The models that we consider in this paper generalize many commonly used models in the treatment effects literature including difference in differences and individual-specific linear trend models. Unlike the majority of the literature on interactive fixed effects models, we do not require the number of time periods to go to infinity to consistently estimate the ATT. Our main identification result relies on having the effect of some time invariant covariate (e.g., race or sex) not vary over time. Using our approach, we show that the ATT can be identified with as few as three time periods and with panel or repeated cross sections data.

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  • Brantly Callaway & Sonia Karami, 2020. "Treatment Effects in Interactive Fixed Effects Models with a Small Number of Time Periods," Papers 2006.15780, arXiv.org, revised Feb 2022.
    • Handle: RePEc:arx:papers:2006.15780
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    1. Cragg, John G. & Donald, Stephen G., 1997. "Inferring the rank of a matrix," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 223-250.
    2. Isaiah Andrews & Timothy B. Armstrong, 2017. "Unbiased instrumental variables estimation under known first‐stage sign," Quantitative Economics, Econometric Society, vol. 8(2), pages 479-503, July.
    3. Benjamin Williams, 2018. "Identification of the Linear Factor Model," Working Papers 2018-002, The George Washington University, Department of Economics, H. O. Stekler Research Program on Forecasting.
    4. Eli Ben-Michael & Avi Feller & Jesse Rothstein, 2021. "The Augmented Synthetic Control Method," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1789-1803, October.
    5. John Gardner, 2020. "Identification and estimation of average causal effects when treatment status is ignorable within unobserved strata," Econometric Reviews, Taylor & Francis Journals, vol. 39(10), pages 1014-1041, November.
    6. M. Hashem Pesaran, 2006. "Estimation and Inference in Large Heterogeneous Panels with a Multifactor Error Structure," Econometrica, Econometric Society, vol. 74(4), pages 967-1012, July.
    7. 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.
    8. Laurent Gobillon & Thierry Magnac, 2016. "Regional Policy Evaluation: Interactive Fixed Effects and Synthetic Controls," The Review of Economics and Statistics, MIT Press, vol. 98(3), pages 535-551, July.
    9. Ahn, Seung C. & Lee, Young H. & Schmidt, Peter, 2013. "Panel data models with multiple time-varying individual effects," Journal of Econometrics, Elsevier, vol. 174(1), pages 1-14.
    10. Simon Freyaldenhoven & Christian Hansen & Jesse M. Shapiro, 2019. "Pre-event Trends in the Panel Event-Study Design," American Economic Review, American Economic Association, vol. 109(9), pages 3307-3338, September.
    11. 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.
    12. Sanderson, Eleanor & Windmeijer, Frank, 2016. "A weak instrument F-test in linear IV models with multiple endogenous variables," Journal of Econometrics, Elsevier, vol. 190(2), pages 212-221.
    13. Kleibergen, Frank & Paap, Richard, 2006. "Generalized reduced rank tests using the singular value decomposition," Journal of Econometrics, Elsevier, vol. 133(1), pages 97-126, July.
    14. Jushan Bai & Serena Ng, 2002. "Determining the Number of Factors in Approximate Factor Models," Econometrica, Econometric Society, vol. 70(1), pages 191-221, January.
    15. 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.
    16. Pedro Carneiro & Karsten T. Hansen & James J. Heckman, 2003. "Estimating Distributions of Treatment Effects with an Application to the Returns to Schooling and Measurement of the Effects of Uncertainty on College," NBER Working Papers 9546, National Bureau of Economic Research, Inc.
    17. Guido Imbens & Nathan Kallus & Xiaojie Mao, 2021. "Controlling for Unmeasured Confounding in Panel Data Using Minimal Bridge Functions: From Two-Way Fixed Effects to Factor Models," Papers 2108.03849, arXiv.org.
    18. Richard Blundell & Monica Costa Dias, 2009. "Alternative Approaches to Evaluation in Empirical Microeconomics," Journal of Human Resources, University of Wisconsin Press, vol. 44(3).
    19. Jacobson, Louis S & LaLonde, Robert J & Sullivan, Daniel G, 1993. "Earnings Losses of Displaced Workers," American Economic Review, American Economic Association, vol. 83(4), pages 685-709, September.
    20. Artūras Juodis & Vasilis Sarafidis, 2022. "A Linear Estimator for Factor-Augmented Fixed-T Panels With Endogenous Regressors," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 40(1), pages 1-15, January.
    21. Alberto Abadie, 2005. "Semiparametric Difference-in-Differences Estimators," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 72(1), pages 1-19.
    22. Kosuke Imai & Marc Ratkovic, 2015. "Robust Estimation of Inverse Probability Weights for Marginal Structural Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 1013-1023, September.
    23. Susan Athey & Mohsen Bayati & Nikolay Doudchenko & Guido Imbens & Khashayar Khosravi, 2021. "Matrix Completion Methods for Causal Panel Data Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1716-1730, October.
    24. Louis S. Jacobson & Robert J. LaLonde & Daniel G. Sullivan, 1993. "Long-term earnings losses of high-seniority displaced workers," Economic Perspectives, Federal Reserve Bank of Chicago, vol. 17(Nov), pages 2-20.
    25. Heckman, J.J. & Hotz, V.J., 1988. "Choosing Among Alternative Nonexperimental Methods For Estimating The Impact Of Social Programs: The Case Of Manpower Training," University of Chicago - Economics Research Center 88-12, Chicago - Economics Research Center.
    26. Hansen, Christian & Kozbur, Damian, 2014. "Instrumental variables estimation with many weak instruments using regularized JIVE," Journal of Econometrics, Elsevier, vol. 182(2), pages 290-308.
    27. Carneiro, Pedro & Hansen, Karsten T. & Heckman, James J., 2003. "Estimating Distributions of Treatment Effects with an Application to the Returns to Schooling and Measurement of the Effects of Uncertainty on College Choice," IZA Discussion Papers 767, Institute of Labor Economics (IZA).
    28. Ricardo Mora & Iliana Reggio, 2019. "Alternative diff-in-diffs estimators with several pretreatment periods," Econometric Reviews, Taylor & Francis Journals, vol. 38(5), pages 465-486, May.
    29. Robertson, Donald & Sarafidis, Vasilis, 2015. "IV estimation of panels with factor residuals," Journal of Econometrics, Elsevier, vol. 185(2), pages 526-541.
    30. Kline Patrick & Santos Andres, 2012. "A Score Based Approach to Wild Bootstrap Inference," Journal of Econometric Methods, De Gruyter, vol. 1(1), pages 23-41, August.
    31. Benjamin Williams, 2020. "Identification of the linear factor model," Econometric Reviews, Taylor & Francis Journals, vol. 39(1), pages 92-109, January.
    32. Guido W. Imbens & Jeffrey M. Wooldridge, 2009. "Recent Developments in the Econometrics of Program Evaluation," Journal of Economic Literature, American Economic Association, vol. 47(1), pages 5-86, March.
    33. Joachim Freyberger, 2018. "Non-parametric Panel Data Models with Interactive Fixed Effects," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 85(3), pages 1824-1851.
    34. 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.
    35. Li, Kathleen T. & Bell, David R., 2017. "Estimation of average treatment effects with panel data: Asymptotic theory and implementation," Journal of Econometrics, Elsevier, vol. 197(1), pages 65-75.
    36. Jushan Bai & Serena Ng, 2021. "Matrix Completion, Counterfactuals, and Factor Analysis of Missing Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1746-1763, October.
    37. Bruno Ferman & Cristine Pinto, 2021. "Synthetic controls with imperfect pretreatment fit," Quantitative Economics, Econometric Society, vol. 12(4), pages 1197-1221, November.
    38. Pedro Carneiro & Karsten T. Hansen & James J. Heckman, 2003. "2001 Lawrence R. Klein Lecture Estimating Distributions of Treatment Effects with an Application to the Returns to Schooling and Measurement of the Effects of Uncertainty on College Choice," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 44(2), pages 361-422, May.
    39. James Heckman & Hidehiko Ichimura & Jeffrey Smith & Petra Todd, 1998. "Characterizing Selection Bias Using Experimental Data," Econometrica, Econometric Society, vol. 66(5), pages 1017-1098, September.
    40. Goodman-Bacon, Andrew, 2021. "Difference-in-differences with variation in treatment timing," Journal of Econometrics, Elsevier, vol. 225(2), pages 254-277.
    41. Seung‐Hyun Hong, 2013. "Measuring The Effect Of Napster On Recorded Music Sales: Difference‐In‐Differences Estimates Under Compositional Changes," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 28(2), pages 297-324, March.
    42. Callaway, Brantly & Sant’Anna, Pedro H.C., 2021. "Difference-in-Differences with multiple time periods," Journal of Econometrics, Elsevier, vol. 225(2), pages 200-230.
    43. Cheng Hsiao & H. Steve Ching & Shui Ki Wan, 2012. "A Panel Data Approach For Program Evaluation: Measuring The Benefits Of Political And Economic Integration Of Hong Kong With Mainland China," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 27(5), pages 705-740, August.
    44. Lechner, Michael, 2008. "A note on endogenous control variables in causal studies," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 190-195, February.
    45. Sant’Anna, Pedro H.C. & Zhao, Jun, 2020. "Doubly robust difference-in-differences estimators," Journal of Econometrics, Elsevier, vol. 219(1), pages 101-122.
    46. James J. Heckman & Jora Stixrud & Sergio Urzua, 2006. "The Effects of Cognitive and Noncognitive Abilities on Labor Market Outcomes and Social Behavior," Journal of Labor Economics, University of Chicago Press, vol. 24(3), pages 411-482, July.
    47. Stéphane Bonhomme & Ulrich Sauder, 2011. "Recovering Distributions in Difference-in-Differences Models: A Comparison of Selective and Comprehensive Schooling," The Review of Economics and Statistics, MIT Press, vol. 93(2), pages 479-494, May.
    48. Marianne Bertrand & Esther Duflo & Sendhil Mullainathan, 2004. "How Much Should We Trust Differences-In-Differences Estimates?," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 119(1), pages 249-275.
    49. Ahn, Seung Chan & Hoon Lee, Young & Schmidt, Peter, 2001. "GMM estimation of linear panel data models with time-varying individual effects," Journal of Econometrics, Elsevier, vol. 101(2), pages 219-255, April.
    50. Xu, Yiqing, 2017. "Generalized Synthetic Control Method: Causal Inference with Interactive Fixed Effects Models," Political Analysis, Cambridge University Press, vol. 25(1), pages 57-76, January.
    51. Jushan Bai, 2009. "Panel Data Models With Interactive Fixed Effects," Econometrica, Econometric Society, vol. 77(4), pages 1229-1279, July.
    52. Brantly Callaway & Tong Li, 2019. "Quantile treatment effects in difference in differences models with panel data," Quantitative Economics, Econometric Society, vol. 10(4), pages 1579-1618, November.
    53. Dukpa Kim & Tatsushi Oka, 2014. "Divorce Law Reforms And Divorce Rates In The Usa: An Interactive Fixed‐Effects Approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 29(2), pages 231-245, March.
    54. 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.
    55. Jeffrey M. Wooldridge, 2005. "Fixed-Effects and Related Estimators for Correlated Random-Coefficient and Treatment-Effect Panel Data Models," The Review of Economics and Statistics, MIT Press, vol. 87(2), pages 385-390, May.
    56. A. Colin Cameron & Douglas L. Miller, 2015. "A Practitioner’s Guide to Cluster-Robust Inference," Journal of Human Resources, University of Wisconsin Press, vol. 50(2), pages 317-372.
    57. Cheng Hsiao & Qiankun Zhou, 2019. "Panel parametric, semiparametric, and nonparametric construction of counterfactuals," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(4), pages 463-481, June.
    58. Blackwell, Matthew & Glynn, Adam N., 2018. "How to Make Causal Inferences with Time-Series Cross-Sectional Data under Selection on Observables," American Political Science Review, Cambridge University Press, vol. 112(4), pages 1067-1082, November.
    59. Maxwell Kellogg & Magne Mogstad & Guillaume A. Pouliot & Alexander Torgovitsky, 2021. "Combining Matching and Synthetic Control to Tradeoff Biases From Extrapolation and Interpolation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1804-1816, October.
    60. Cheng Hsiao & Qiankun Zhou, 2018. "Panel Parametric, Semi-parametric and Nonparametric Construction of Counterfactuals - California Tobacco Control Revisited," Departmental Working Papers 2018-02, Department of Economics, Louisiana State University.
    61. James Heckman & Jose Scheinkman, 1987. "The Importance of Bundling in a Gorman-Lancaster Model of Earnings," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 54(2), pages 243-255.
    62. José Luis Montiel Olea & Carolin Pflueger, 2013. "A Robust Test for Weak Instruments," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(3), pages 358-369, July.
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    Cited by:

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    2. Gregory Faletto, 2023. "Fused Extended Two-Way Fixed Effects for Difference-in-Differences with Staggered Adoptions," Papers 2312.05985, arXiv.org.
    3. Albert Chiu & Xingchen Lan & Ziyi Liu & Yiqing Xu, 2023. "What To Do (and Not to Do) with Causal Panel Analysis under Parallel Trends: Lessons from A Large Reanalysis Study," Papers 2309.15983, arXiv.org.
    4. Qi’ang Du & Hongbo Li & Yanyan Fu & Xintian Fu & Rui Wang & Tingting Jia, 2023. "More Green, Better Funding? Exploring the Dynamics between Corporate Bank Loans and Trade Credit," Sustainability, MDPI, vol. 15(13), pages 1-20, June.
    5. Zhenhao Gong & Min Seong Kim, 2024. "Improved Inference for Interactive Fixed Effects Model under Cross-Sectional Dependence," Working papers 2024-02, University of Connecticut, Department of Economics.
    6. Brantly Callaway, 2022. "Difference-in-Differences for Policy Evaluation," Papers 2203.15646, arXiv.org.

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    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models

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