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A Linear Panel Model with Heterogeneous Coefficients and Variation in Exposure

Author

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  • Liyang Sun
  • Jesse M. Shapiro

Abstract

Linear panel models featuring unit and time fixed effects appear in many areas of empirical economics. An active literature studies the interpretation of the ordinary least squares estimator of the model, commonly called the two-way fixed effects (TWFE) estimator, in the presence of unmodeled coefficient heterogeneity. We illustrate some implications for the case where the research design takes advantage of variation across units (say, US states) in exposure to some treatment (say, a policy change). In this case, the TWFE can fail to estimate the average (or even a weighted average) of the units' coefficients. Under some conditions, there exists no estimator that is guaranteed to estimate even a weighted average. Building on the literature, we note that when there is a unit totally unaffected by treatment, it is possible to estimate an average effect by replacing the TWFE with an average of difference-in-differences estimators.

Suggested Citation

  • Liyang Sun & Jesse M. Shapiro, 2022. "A Linear Panel Model with Heterogeneous Coefficients and Variation in Exposure," NBER Working Papers 29976, National Bureau of Economic Research, Inc.
    • Handle: RePEc:nbr:nberwo:29976
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    1. repec:ags:aaea22:343893 is not listed on IDEAS
    2. Elizabeth C. Klee & Adair Morse & Chaehee Shin, 2024. "Auto Finance in the Electric Vehicle Transition," Finance and Economics Discussion Series 2024-065, Board of Governors of the Federal Reserve System (U.S.).
    3. Arindrajit Dube & Attila Lindner, 2024. "Minimum Wages in the 21st Century," RFBerlin Discussion Paper Series 2425, ROCKWOOL Foundation Berlin (RFBerlin).
    4. Sheida Teimouri & Joachim Zietz, 2025. "Housing prices and import competition," Empirical Economics, Springer, vol. 68(1), pages 253-280, January.
    5. Liu, Jhih-Yun, 2023. "The Dust Bowl and Occupational Persistence in Agriculture," 2023 Annual Meeting, July 23-25, Washington D.C. 335779, Agricultural and Applied Economics Association.
    6. Millimet, Daniel L. & Bellemare, Marc, 2023. "Fixed Effects and Causal Inference," IZA Discussion Papers 16202, Institute of Labor Economics (IZA).
    7. Posso, Alberto, 2025. "Contextualizing child labor reforms: Education policy, legal bans, and female child labor in rural South India," Journal of Economic Behavior & Organization, Elsevier, vol. 235(C).
    8. Adrienne M. Lucas & Patrick J. McEwan & David Torres Irribarra, 2025. "Targeted Education Transfers Reduced Long-Run and Intergenerational Ethnic Inequality in Chile," NBER Working Papers 33798, National Bureau of Economic Research, Inc.
    9. Khambai Khamjalas, 2024. "Nexuses between Food, Energy, and Water Consumption on Urban-Rural Income Gap in South-Eastern Asian Countries Using Difference in Difference in Modelling Technique," International Journal of Economics and Financial Issues, Econjournals, vol. 14(2), pages 186-195, March.
    10. Pourya Valizadeh & Bart L. Fischer & Henry L. Bryant, 2024. "SNAP enrollment cycles: New insights from heterogeneous panel models with cross‐sectional dependence," American Journal of Agricultural Economics, John Wiley & Sons, vol. 106(1), pages 354-381, January.
    11. repec:ags:aaea22:335779 is not listed on IDEAS
    12. Gregory Faletto, 2023. "Fused Extended Two-Way Fixed Effects for Difference-in-Differences With Staggered Adoptions," Papers 2312.05985, arXiv.org, revised Apr 2025.
    13. Sung Jae Jun & Sokbae (Simon) Lee, 2024. "Learning the effect of persuasion via difference-in-differences," CeMMAP working papers 24/24, Institute for Fiscal Studies.

    More about this item

    JEL classification:

    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models
    • C87 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Econometric Software

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