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The Chained Difference-in-Differences

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  • Christophe Bell'ego
  • David Benatia
  • Vincent Dortet-Bernardet

Abstract

This paper studies the identification, estimation, and inference of long-term (binary) treatment effect parameters when balanced panel data is not available, or consists of only a subset of the available data. We develop a new estimator: the chained difference-in-differences, which leverages the overlapping structure of many unbalanced panel data sets. This approach consists in aggregating a collection of short-term treatment effects estimated on multiple incomplete panels. Our estimator accommodates (1) multiple time periods, (2) variation in treatment timing, (3) treatment effect heterogeneity, (4) general missing data patterns, and (5) sample selection on observables. We establish the asymptotic properties of the proposed estimator and discuss identification and efficiency gains in comparison to existing methods. Finally, we illustrate its relevance through (i) numerical simulations, and (ii) an application about the effects of an innovation policy in France.

Suggested Citation

  • Christophe Bell'ego & David Benatia & Vincent Dortet-Bernardet, 2023. "The Chained Difference-in-Differences," Papers 2301.01085, arXiv.org, revised Jan 2025.
  • Handle: RePEc:arx:papers:2301.01085
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    1. Hansen, Lars Peter, 1982. "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, Econometric Society, vol. 50(4), pages 1029-1054, July.
    2. Almas Heshmati, 1998. "Efficiency measurement in rotating panel data," Applied Economics, Taylor & Francis Journals, vol. 30(7), pages 919-930.
    3. Irene Botosaru & Federico H. Gutierrez, 2018. "Difference‐in‐differences when the treatment status is observed in only one period," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 33(1), pages 73-90, January.
    4. Rachel Griffith & Stephen Redding & John Van Reenen, 2004. "Mapping the Two Faces of R&D: Productivity Growth in a Panel of OECD Industries," The Review of Economics and Statistics, MIT Press, vol. 86(4), pages 883-895, November.
    5. 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.
    6. David H. Autor & Susan N. Houseman, 2010. "Do Temporary-Help Jobs Improve Labor Market Outcomes for Low-Skilled Workers? Evidence from "Work First"," American Economic Journal: Applied Economics, American Economic Association, vol. 2(3), pages 96-128, July.
    7. de Chaisemartin, Clément & D’Haultfœuille, Xavier, 2023. "Two-way fixed effects and differences-in-differences estimators with several treatments," Journal of Econometrics, Elsevier, vol. 236(2).
    8. J Ignacio García-Pérez & Ioana Marinescu & Judit Vall Castello, 2019. "Can Fixed-term Contracts Put Low Skilled Youth on a Better Career Path? Evidence from Spain," The Economic Journal, Royal Economic Society, vol. 129(620), pages 1693-1730.
    9. Robert Moffitt & John Fitzgerald & Peter Gottschalk, 1999. "Sample Attrition in Panel Data: The Role of Selection on Observables," Annals of Economics and Statistics, GENES, issue 55-56, pages 129-152.
    10. Gross, Robert & Hanna, Richard & Gambhir, Ajay & Heptonstall, Philip & Speirs, Jamie, 2018. "How long does innovation and commercialisation in the energy sectors take? Historical case studies of the timescale from invention to widespread commercialisation in energy supply and end use technolo," Energy Policy, Elsevier, vol. 123(C), pages 682-699.
    11. Kahn, Lisa B., 2010. "The long-term labor market consequences of graduating from college in a bad economy," Labour Economics, Elsevier, vol. 17(2), pages 303-316, April.
    12. repec:ehl:wpaper:54518 is not listed on IDEAS
    13. Bhattacharya, Debopam, 2008. "Inference in panel data models under attrition caused by unobservables," Journal of Econometrics, Elsevier, vol. 144(2), pages 430-446, June.
    14. Cl'ement de Chaisemartin & Xavier D'Haultf{oe}uille, 2020. "Difference-in-Differences Estimators of Intertemporal Treatment Effects," Papers 2007.04267, arXiv.org, revised Dec 2024.
    15. Chaudhuri, Saraswata, 2020. "On Efficiency Gains From Multiple Incomplete Subsamples," Econometric Theory, Cambridge University Press, vol. 36(3), pages 488-525, June.
    16. 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.
    17. Bakker, Gerben, 2013. "Money for nothing: How firms have financed R&D-projects since the Industrial Revolution," Research Policy, Elsevier, vol. 42(10), pages 1793-1814.
    18. Sant’Anna, Pedro H.C. & Zhao, Jun, 2020. "Doubly robust difference-in-differences estimators," Journal of Econometrics, Elsevier, vol. 219(1), pages 101-122.
    19. Richard Blundell & Luigi Pistaferri & Ian Preston, 2008. "Consumption Inequality and Partial Insurance," American Economic Review, American Economic Association, vol. 98(5), pages 1887-1921, December.
    20. Michael Lechner & Ruth Miquel & Conny Wunsch, 2011. "Long‐Run Effects Of Public Sector Sponsored Training In West Germany," Journal of the European Economic Association, European Economic Association, vol. 9(4), pages 742-784, August.
    21. Clément de Chaisemartin & Xavier D’Haultfœuille, 2023. "Two-way fixed effects and differences-in-differences with heterogeneous treatment effects: a survey," The Econometrics Journal, Royal Economic Society, vol. 26(3), pages 1-30.
    22. Verbeek, Marno & Nijman, Theo, 1992. "Testing for Selectivity Bias in Panel Data Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 33(3), pages 681-703, August.
    23. Alberto Abadie, 2005. "Semiparametric Difference-in-Differences Estimators," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 72(1), pages 1-19.
    24. Nijman, Theo & Verbeek, Marno & van Soest, Arthur, 1991. "The efficiency of rotating-panel designs in an analysis-of-variance model," Journal of Econometrics, Elsevier, vol. 49(3), pages 373-399, September.
    25. Badi Baltagi & Seuck Song, 2006. "Unbalanced panel data: A survey," Statistical Papers, Springer, vol. 47(4), pages 493-523, October.
    26. Markus Frölich & Martin Huber, 2014. "Treatment Evaluation With Multiple Outcome Periods Under Endogeneity and Attrition," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(508), pages 1697-1711, December.
    27. Philip Oreopoulos & Till von Wachter & Andrew Heisz, 2012. "The Short- and Long-Term Career Effects of Graduating in a Recession," American Economic Journal: Applied Economics, American Economic Association, vol. 4(1), pages 1-29, January.
    28. Pierre Hoonhout & Geert Ridder, 2019. "Nonignorable Attrition in Multi-Period Panels With Refreshment Samples," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(3), pages 377-390, July.
    29. 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.
    30. Keisuke Hirano & Guido W. Imbens & Geert Ridder & Donald B. Rubin, 2001. "Combining Panel Data Sets with Attrition and Refreshment Samples," Econometrica, Econometric Society, vol. 69(6), pages 1645-1659, November.
    31. Chaudhuri, Saraswata & Frazier, David T. & Renault, Eric, 2018. "Indirect Inference with endogenously missing exogenous variables," Journal of Econometrics, Elsevier, vol. 205(1), pages 55-75.
    32. Stevens, Ann Huff, 1997. "Persistent Effects of Job Displacement: The Importance of Multiple Job Losses," Journal of Labor Economics, University of Chicago Press, vol. 15(1), pages 165-188, January.
    33. Athey, Susan & Imbens, Guido W., 2022. "Design-based analysis in Difference-In-Differences settings with staggered adoption," Journal of Econometrics, Elsevier, vol. 226(1), pages 62-79.
    34. Jeffrey M Wooldridge, 2010. "Econometric Analysis of Cross Section and Panel Data," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262232588, December.
    35. Howitt, Peter & Aghion, Philippe, 1998. "Capital Accumulation and Innovation as Complementary Factors in Long-Run Growth," Journal of Economic Growth, Springer, vol. 3(2), pages 111-130, June.
    36. Clément de Chaisemartin & Xavier d'Haultfoeuille, 2022. "Not all Differences-in-differences are Equally Compatible with Outcome-based Selection Models," Working Papers hal-03873930, HAL.
    37. Kenneth Arrow, 1962. "Economic Welfare and the Allocation of Resources for Invention," NBER Chapters, in: The Rate and Direction of Inventive Activity: Economic and Social Factors, pages 609-626, National Bureau of Economic Research, Inc.
    38. Goodman-Bacon, Andrew, 2021. "Difference-in-differences with variation in treatment timing," Journal of Econometrics, Elsevier, vol. 225(2), pages 254-277.
    39. Carrasco, Marine & Florens, Jean-Pierre & Renault, Eric, 2007. "Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 77, Elsevier.
    40. Joshua Angrist & Eric Bettinger & Michael Kremer, 2006. "Long-Term Educational Consequences of Secondary School Vouchers: Evidence from Administrative Records in Colombia," American Economic Review, American Economic Association, vol. 96(3), pages 847-862, June.
    41. 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.
    42. Richard R. Nelson, 1959. "The Simple Economics of Basic Scientific Research," Journal of Political Economy, University of Chicago Press, vol. 67(3), pages 297-297.
    43. Hausman, Jerry A & Wise, David A, 1979. "Attrition Bias in Experimental and Panel Data: The Gary Income Maintenance Experiment," Econometrica, Econometric Society, vol. 47(2), pages 455-473, March.
    44. Scherer, F M, 1982. "Inter-Industry Technology Flows and Productivity Growth," The Review of Economics and Statistics, MIT Press, vol. 64(4), pages 627-634, November.
    45. Thomas A. Mroz & Timothy H. Savage, 2006. "The Long-Term Effects of Youth Unemployment," Journal of Human Resources, University of Wisconsin Press, vol. 41(2).
    46. Frazier, David T. & Renault, Eric, 2017. "Efficient two-step estimation via targeting," Journal of Econometrics, Elsevier, vol. 201(2), pages 212-227.
    47. repec:adr:anecst:y:1999:i:55-56:p:05 is not listed on IDEAS
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    Cited by:

    1. Gayani Rathnayake & Akanksha Negi & Otavio Bartalotti & Xueyan Zhao, 2024. "Difference-in-Differences with Sample Selection," Papers 2411.09221, arXiv.org, revised Dec 2024.

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    More about this item

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models
    • O3 - Economic Development, Innovation, Technological Change, and Growth - - Innovation; Research and Development; Technological Change; Intellectual Property Rights
    • J38 - Labor and Demographic Economics - - Wages, Compensation, and Labor Costs - - - Public Policy

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