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.