Nonparametric "rich covariates" without saturation

We consider two nonparametric approaches to ensure that linear instrumental variables estimators satisfy the rich-covariates condition emphasized by Blandhol et al. (2025), even when the instrument is not unconditionally randomly assigned and the model is not saturated. Both approaches start with a nonparametric estimate of the expectation of the instrument conditional on the covariates, and ensure that the rich-covariates condition is satisfied either by using as the instrument the difference between the original instrument and its estimated conditional expectation, or by adding the estimated conditional expectation to the set of regressors. We derive asymptotic properties when the first step uses kernel regression, and assess finite-sample performance in simulations where we also use neural networks in the first step. Finally, we present an empirical illustration that highlights some significant advantages of the proposed methods.