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Comparative Statics for Difference-in-Differences

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  • Finn Christensen

    (Department of Economics, Towson University)

Abstract

In a model with spillovers, consider the difference in the impact of a positive shock on average outcomes among the treated and untreated, or the marginal average effect of treatment among the treated with spillovers (MATTS). MATTS is positive if, and only if, the negated Jacobian inverse of the equilibrium system is a B-matrix by columns. This condition is also sufficient to answer traditional comparative statics questions. I also give several sufficient, and sometimes nec- essary, conditions on the noninverted Jacobian–which correspond to common modeling assumptions–under which its inverse is a B-matrix by columns. Sign restrictions on MATTS are testable because the sample difference-in-differences is an unbiased estimator for MATTS from a superpopulation perspective. I also show that MATTS generalizes the ATT and the ATE when interference, or spillovers, are present. I apply the results to oligopoly and contests.

Suggested Citation

  • Finn Christensen, 2023. "Comparative Statics for Difference-in-Differences," Working Papers 2023-08, Towson University, Department of Economics, revised Apr 2025.
    • Handle: RePEc:tow:wpaper:2023-08
    as

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    File URL: http://webapps.towson.edu/cbe/economics/workingpapers/2023-08.pdf
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    References listed on IDEAS

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    4. Jordan Norris & Charles Johnson & Ilya Spitkovsky, 2023. "Bounding Comparative Statics under Diagonal Dominance," Working Papers 20230082, New York University Abu Dhabi, Department of Social Science, revised Jan 2023.
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    6. Imbens,Guido W. & Rubin,Donald B., 2015. "Causal Inference for Statistics, Social, and Biomedical Sciences," Cambridge Books, Cambridge University Press, number 9780521885881, July.
    7. Roth, Jonathan & Sant’Anna, Pedro H.C. & Bilinski, Alyssa & Poe, John, 2023. "What’s trending in difference-in-differences? A synthesis of the recent econometrics literature," Journal of Econometrics, Elsevier, vol. 235(2), pages 2218-2244.
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    More about this item

    Keywords

    Comparative statics; difference-in-differences; interference; spillovers; profit maximization hypothesis; B-matrix; symmetric games.;
    All these keywords.

    JEL classification:

    • C31 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models; Quantile Regressions; Social Interaction Models
    • C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D21 - Microeconomics - - Production and Organizations - - - Firm Behavior: Theory
    • L21 - Industrial Organization - - Firm Objectives, Organization, and Behavior - - - Business Objectives of the Firm

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