Optimal sup‐norm rates and uniform inference on nonlinear functionals of nonparametric IV regression

This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h0 and functionals of h0. First, we derive sup‐norm convergence rates for computationally simple sieve NPIV (series two‐stage least squares) estimators of h0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup‐norm rates of estimating h0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h0 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. Third, we use our sup‐norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h0 under primitive conditions, allowing for mildly and severely ill‐posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug‐in sieve t‐statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low‐level conditions when demand is estimated via sieve NPIV. Our real data application of UCBs for exact CS and DL functionals of gasoline demand reveals interesting patterns and is applicable to other goods markets.