Efficiency bounds for estimating linear functionals of nonparametric regression models with endogenous regressors

Let Y=μ^∗(X)+ε, where μ^∗ is unknown and E[ε|X]≠0 with positive probability but there exist instrumental variables W such that E[ε|W]=0 w.p.1. It is well known that such nonparametric regression models are generally “ill-posed” in the sense that the map from the data to μ^∗ is not continuous. In this paper, we derive the efficiency bounds for estimating certain linear functionals of μ^∗ without assuming μ^∗ itself to be identified.