Optimal Sup-Norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. 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)] (with < 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. Terms of use: Documents in

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

“…2 The uniform strong approximation and the score bootstrap UCB results are in the second version (Chen and Christensen (2015a)); see Theorem B.1 and its proof in that version. 3 The pointwise inference results on exact CS and DL are in the second version (Chen and Christensen (2015a)).…”

“…Finally, as important applications, we establish first pointwise and uniform inference results for two leading nonlinear welfare functionals of a nonparametric demand function h 0 estimated via sieve NPIV, namely the exact consumer surplus (CS) and deadweight loss (DL) arising from price changes at different income levels when prices (and possibly income) are endogenous. 3 We present two 1 The optimal sup-norm rates for estimating h 0 are in the first version (Chen and Christensen (2013)); the optimal sup-norm rates for estimating derivatives of h 0 are in the second version (Chen and Christensen (2015a)). 2 The uniform strong approximation and the score bootstrap UCB results are in the second version (Chen and Christensen (2015a)); see Theorem B.1 and its proof in that version.…”

“…3 We present two 1 The optimal sup-norm rates for estimating h 0 are in the first version (Chen and Christensen (2013)); the optimal sup-norm rates for estimating derivatives of h 0 are in the second version (Chen and Christensen (2015a)). 2 The uniform strong approximation and the score bootstrap UCB results are in the second version (Chen and Christensen (2015a)); see Theorem B.1 and its proof in that version. 3 The pointwise inference results on exact CS and DL are in the second version (Chen and Christensen (2015a)).…”

Compactness of infinite dimensional parameter spaces

“…15 Chen and Christensen (2015a) derive the rate of convergence in the sup-norm when X has compact support.…”

On ill-posedness of nonparametric instrumental variable regression with convexity constraints

“…In particular, they point out that, even if well-posedness is not restored, those two monotonicity constraints improve the rate of convergence in shrinking neighbourhoods of the constraint boundary and can have a significant impact on the estimator finite sample behaviour. Chen and Christensen (2013) show that imposing shape restrictions only is not enough to improve convergence rates as long as the derivative constraints hold with strict inequality (i.e. in the interior of the constraint space).…”