Constructing an Instrument as a Function of Covariates

Researchers often use instrumental variables (IV) models to investigate the causal relationship between an endogenous variable and an outcome while controlling for covariates. When an exogenous variable is unavailable to serve as the instrument for an endogenous treatment, a recurring empirical practice is to construct one from a nonlinear transformation of the covariates. We investigate how reliable these estimates are under mild forms of misspecification. Our main result shows that for instruments constructed from covariates, the IV estimand can be arbitrarily biased under mild forms of misspecification, even when imposing constant linear treatment effects. We perform a semi-synthetic exercise by calibrating data to alternative models proposed in the literature and estimating the average treatment effect. Our results show that IV specifications that use instruments constructed from covariates are non-robust to nonlinearity in the true structural function.