IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2606.21569.html

What's the Magic Formula Instrument?

Author

Listed:
  • Peizan Sheng

    (University of Chicago, Harris School of Public Policy)

  • Alexander Torgovitsky

    (University of Chicago, Kenneth C. Griffin Department of Economics)

Abstract

Two recent papers by Borusyak and Hull (2023, 2026) propose using known formulas to adjust linear instrumental variable estimators for confounding covariates. Implementing this "formula instrument" approach requires making a parametric assumption on the distribution of the unobserved shocks that generated the instrument. We develop a method for systematically evaluating the sensitivity of formula instrument estimates to this parametric assumption. The method is straightforward to implement using our companion R package formulaiv. We use our method to reanalyze the applications in both Borusyak and Hull (2023) and Borusyak and Hull (2026). In both applications, we find that a variety of estimates of different signs and magnitudes can be recovered by slightly changing the shock distribution.

Suggested Citation

  • Peizan Sheng & Alexander Torgovitsky, 2026. "What's the Magic Formula Instrument?," Papers 2606.21569, arXiv.org.
    • Handle: RePEc:arx:papers:2606.21569
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2606.21569
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Timothy Christensen & Benjamin Connault, 2023. "Counterfactual Sensitivity and Robustness," Econometrica, Econometric Society, vol. 91(1), pages 263-298, January.
    2. Timothy J. Bartik, 1991. "Who Benefits from State and Local Economic Development Policies?," Books from Upjohn Press, W.E. Upjohn Institute for Employment Research, number wbsle.
    3. Stéphane Bonhomme & Martin Weidner, 2022. "Minimizing sensitivity to model misspecification," Quantitative Economics, Econometric Society, vol. 13(3), pages 907-954, July.
    4. Bagues, Manuel & Makany, Milan & Vattuone, Giulia & Zinovyeva, Natalia, 2026. "Female Promotions and the Academic Pipeline: Evidence from a Natural Experiment," IZA Discussion Papers 18477, IZA Network @ LISER.
    5. Robinson, Peter M, 1988. "Root- N-Consistent Semiparametric Regression," Econometrica, Econometric Society, vol. 56(4), pages 931-954, July.
    6. Patrick Kline & Andres Santos, 2013. "Sensitivity to missing data assumptions: Theory and an evaluation of the U.S. wage structure," Quantitative Economics, Econometric Society, vol. 4(2), pages 231-267, July.
    7. Gruber, Jon & Saez, Emmanuel, 2002. "The elasticity of taxable income: evidence and implications," Journal of Public Economics, Elsevier, vol. 84(1), pages 1-32, April.
    8. Melissa Dell & Benjamin A Olken, 2020. "The Development Effects of the Extractive Colonial Economy: The Dutch Cultivation System in Java," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 87(1), pages 164-203.
    9. Chaurey, Ritam & Nayyar, Guarav & Sharma, Siddharth & Verhoogen, Eric, 2025. "Social Learning among Urban Manufacturing Firms: Energy-Efficient Motors in Bangladesh," IZA Discussion Papers 18183, IZA Network @ LISER.
    10. Moroni, Gloria & Nicoletti, Cheti & Salvanes, Kjell Gunnar & Tominey, Emma, 2025. "Gender equality through marriage," Discussion Paper Series in Economics 20/2025, Norwegian School of Economics, Department of Economics.
    11. Martin Eckhoff Andresen, 2018. "Exploring marginal treatment effects: Flexible estimation using Stata," Stata Journal, StataCorp LLC, vol. 18(1), pages 118-158, March.
    12. Ritam Chaurey & Gaurav Nayyar & Siddharth Sharma & Eric Verhoogen, 2025. "Social Learning among Urban Manufacturing Firms: Energy-Efficient Motors in Bangladesh," NBER Working Papers 34296, National Bureau of Economic Research, Inc.
    13. Conley, T. G., 1999. "GMM estimation with cross sectional dependence," Journal of Econometrics, Elsevier, vol. 92(1), pages 1-45, September.
    14. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney Newey & James Robins, 2018. "Double/debiased machine learning for treatment and structural parameters," Econometrics Journal, Royal Economic Society, vol. 21(1), pages 1-68, February.
    15. Pedro Carneiro & James J. Heckman & Edward J. Vytlacil, 2011. "Estimating Marginal Returns to Education," American Economic Review, American Economic Association, vol. 101(6), pages 2754-2781, October.
    16. Aviv Nevo & Adam M. Rosen, 2012. "Identification With Imperfect Instruments," The Review of Economics and Statistics, MIT Press, vol. 94(3), pages 659-671, August.
    17. Xiaohong Chen & Elie Tamer & Alexander Torgovitsky, 2011. "Sensitivity Analysis in Semiparametric Likelihood Models," Cowles Foundation Discussion Papers 1836, Cowles Foundation for Research in Economics, Yale University.
    18. A. Charnes & W. W. Cooper, 1962. "Programming with linear fractional functionals," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 9(3‐4), pages 181-186, September.
    19. Chaurey, Ritam & Nayyar, Gaurav & Sharma, Siddharth & Verhoogen, Eric, 2025. "Social Learning among Urban Manufacturing Firms: Energy-Efficient Motors in Bangladesh," CEPR Discussion Papers 20713, Centre for Economic Policy Research.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Michael Pollmann, 2020. "Causal Inference for Spatial Treatments," Papers 2011.00373, arXiv.org, revised Apr 2026.
    2. Mogstad, Magne & Torgovitsky, Alexander, 2024. "Instrumental variables with unobserved heterogeneity in treatment effects," Handbook of Labor Economics,, Elsevier.
    3. Stéphane Bonhomme & Martin Weidner, 2022. "Minimizing sensitivity to model misspecification," Quantitative Economics, Econometric Society, vol. 13(3), pages 907-954, July.
    4. Vitor Possebom, 2026. "Crime and Mismeasured Punishment: Marginal Treatment Effect with Misclassification," The Review of Economics and Statistics, MIT Press, vol. 108(1), pages 44-56, January.
    5. Ober-Reynolds, Daniel, 2026. "Robustness to missing data: breakdown point analysis," Journal of Econometrics, Elsevier, vol. 253(C).
    6. Simon Calmar Andersen & Louise Beuchert & Phillip Heiler & Helena Skyt Nielsen, 2023. "A Guide to Impact Evaluation under Sample Selection and Missing Data: Teacher's Aides and Adolescent Mental Health," Papers 2308.04963, arXiv.org.
    7. Juan Carlos Escanciano & Lin Zhu, 2013. "Set inferences and sensitivity analysis in semiparametric conditionally identified models," CeMMAP working papers 55/13, Institute for Fiscal Studies.
    8. Dong, Chaohua & Gao, Jiti & Linton, Oliver, 2023. "High dimensional semiparametric moment restriction models," Journal of Econometrics, Elsevier, vol. 232(2), pages 320-345.
    9. Sasaki, Yuya & Ura, Takuya, 2023. "Estimation and inference for policy relevant treatment effects," Journal of Econometrics, Elsevier, vol. 234(2), pages 394-450.
    10. Spanos, Yiannis E., 2021. "Exploring heterogeneous returns to collaborative R&D: A marginal treatment effects perspective," Research Policy, Elsevier, vol. 50(5).
    11. Giesecke, Matthias & Schuss, Eric, 2019. "Heterogeneity in marginal returns to language training of immigrants," IAB-Discussion Paper 201919, Institut für Arbeitsmarkt- und Berufsforschung (IAB), Nürnberg [Institute for Employment Research, Nuremberg, Germany].
    12. Hsiao, Cheng & Zhou, Qiankun, 2025. "Statistical inference for the low dimensional parameters of linear regression models in the presence of high-dimensional data: An orthogonal projection approach," Journal of Econometrics, Elsevier, vol. 252(PB).
    13. Céline Grislain-Letrémy & Bertrand Villeneuve & Marc Yeterian, 2024. "Don't bet the Farm on Crop Insurance Subsidies: A Marginal Treatment Effect Analysis of French Farms," Working papers 956, Banque de France.
    14. Samuel Bazzi & Martin Fiszbein & Mesay Gebresilasse, 2020. "Frontier Culture: The Roots and Persistence of “Rugged Individualism” in the United States," Econometrica, Econometric Society, vol. 88(6), pages 2329-2368, November.
    15. Nathan Canen & Kyungchul Song, 2021. "Counterfactual analysis under partial identification using locally robust refinement," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 36(4), pages 416-436, June.
    16. Ertian Chen, 2025. "Robust Structural Estimation under Misspecified Latent-State Dynamics," Papers 2510.22347, arXiv.org, revised Nov 2025.
    17. Victor Chernozhukov & Juan Carlos Escanciano & Hidehiko Ichimura & Whitney K. Newey & James M. Robins, 2022. "Locally Robust Semiparametric Estimation," Econometrica, Econometric Society, vol. 90(4), pages 1501-1535, July.
    18. Jieun Lee & Esfandiar Maasoumi, 2026. "Informativeness under Model Uncertainty: Shadow Prices and Ridge Penalties," Papers 2604.15571, arXiv.org.
    19. Kirill Borusyak & Peter Hull & Xavier Jaravel, 2025. "Design-based identification with formula instruments: a review," The Econometrics Journal, Royal Economic Society, vol. 28(1), pages 83-108.
    20. Hup, Mark, 2024. "Labor coercion, fiscal modernization, and state capacity: Evidence from colonial Indonesia," Explorations in Economic History, Elsevier, vol. 94(C).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2606.21569. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.