What's the Magic Formula Instrument?

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.