Optimal Formula Instruments

When estimating the effects of treatments defined by complex formulas, researchers often use simple functions of exogenous shocks as instruments. A leading example is “simulated instruments” for public policy eligibility, which capture variation in state-level policy generosity. We show how more powerful instruments can be constructed by incorporating heterogeneous shock exposure while using a recentering procedure to avoid bias. We characterize the asymptotically efficient instruments in this class and propose an algorithm for constructing feasible approximations to them. Compared to a simulated instrument approach, our approach yields a 44% smaller standard error on the private insurance crowd-out effect of Medicaid enrollment from the 2014 Affordable Care Act expansions.