Document Type
Discussion Paper
Publication Date
11-1-2013
CFDP Number
1923R
CFDP Revision Date
2015-04-01
CFDP Pages
39
Abstract
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.
Recommended Citation
Chen, Xiaohong and Christensen, Timothy M., "Optimal Sup-norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation" (2013). Cowles Foundation Discussion Papers. 2314.
https://elischolar.library.yale.edu/cowles-discussion-paper-series/2314