What's the Magic Formula Instrument?

Peizan Sheng; Alexander Torgovitsky

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arXiv · 2606.21569

What's the Magic Formula Instrument?

Peizan Sheng·Alexander Torgovitsky

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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.

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Peizan Sheng, Alexander Torgovitsky. 2026-06-19. What's the Magic Formula Instrument?. https://arxiv.org/abs/2606.21569

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