What's the Magic Formula Instrument?

Peizan Sheng and Alexander Torgovitsky
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Peizan Sheng: University of Chicago, Harris School of Public Policy
Alexander Torgovitsky: University of Chicago, Kenneth C. Griffin Department of Economics

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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.

Date: 2026-06
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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2606.21569

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