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Discussion Paper

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This paper makes several contributions to the literature on the important yet difficult problem of estimating functions nonparametrically using instrumental variables. First, we derive the minimax optimal sup-norm convergence rates for nonparametric instrumental variables (NPIV) estimation of the structural function h 0 and its derivatives. Second, we show that a computationally simple sieve NPIV estimator can attain the optimal sup-norm rates for h 0 and its derivatives when h 0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal L 2 -norm rates for severely ill-posed problems, and are only up to a [log( n )] ε (with ε < 1/2) factor slower than the optimal L 2 -norm rates for mildly ill-posed problems. Third, we introduce a novel data-driven procedure for choosing the sieve dimension optimally. Our data-driven procedure is sup-norm rate-adaptive: the resulting estimator of h 0 and its derivatives converge at their optimal sup-norm rates even though the smoothness of h 0 and the degree of ill-posedness of the NPIV model are unknown. Finally, we present two non-trivial applications of the sup-norm rates to inference on nonlinear functionals of h 0 under low-level conditions. The first is to derive the asymptotic normality of sieve t -statistics for exact consumer surplus and deadweight loss functionals in nonparametric demand estimation when prices, and possibly incomes, are endogenous. The second is to establish the validity of a sieve score bootstrap for constructing asymptotically exact uniform confidence bands for collections of nonlinear functionals of h 0 . Both applications provide new and useful tools for empirical research on nonparametric models with endogeneity.

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