Optimal Sup-norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression

Understand

This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function $h_0$ and its functionals.

• First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of $h_0$ and its derivatives.
• Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating $h_0$ and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when $h_0$ is approximated via a spline or wavelet sieve.
• Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean-squared rates for mildly ill-posed problems.