Nonparametric two‐step regression estimation when regressors and error are dependent

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Abstract
This paper considers estimation of the functiongin the modelYt=g(Xt) + ϵtwhen E(ϵt|Xt) ≠ 0 with nonzero probability. We assume the existence of aninstrumental variable Ztthat is independent of ϵt, and of aninnovationηt=Xt—E(Xt|Zt). We use a nonparametric regression ofXtonZtto obtain residuals ηt, which in turn are used to obtain a consistent estimator ofg. The estimator was first analyzed by Newey, Powell & Vella (1999) under the assumption that the observations are independent and identically distributed. Here we derive a sample mean‐squared‐error convergence result for independent identically distributed observations as well as a uniform‐convergence result under time‐series dependence.