Sieve Wald and QLR Inferences on Semi/nonparametric Conditional Moment Models

TL;DR

This paper develops simple, unified inference methods for semi/nonparametric conditional moment models, capable of handling ill-posed problems and nonsmooth residuals, with theoretical validation and empirical illustration.

Contribution

It introduces new asymptotic results and inference procedures for functionals in semi/nonparametric models, including sieve Wald and QLR tests, applicable regardless of root-$n$ estimability.

Findings

Asymptotic normality of plug-in PSMD estimators

Consistency of sieve variance estimators and chi-square distribution of Wald statistic

Validity of bootstrap sieve Wald and QLR tests

Abstract

This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. These models are often ill-posed and hence it is difficult to verify whether a (possibly nonlinear) functional is root-$n$ estimable or not. We provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is root-$n$ estimable or not. We establish the following new useful results: (1) the asymptotic normality of a plug-in penalized sieve minimum distance (PSMD) estimator of a (possibly nonlinear) functional; (2) the consistency of simple sieve variance estimators for the plug-in PSMD estimator, and hence the asymptotic chi-square distribution of the sieve Wald statistic; (3) the asymptotic chi-square distribution of an optimally weighted sieve quasi likelihood ratio (QLR) test under the null hypothesis; (4) the asymptotic tight distribution of a non-optimally weighted sieve QLR statistic under the null; (5) the consistency of generalized residual bootstrap sieve Wald and QLR tests; (6) local power properties of sieve Wald and QLR tests and of their bootstrap versions; (7) asymptotic properties of sieve Wald and SQLR for functionals of increasing dimension. Simulation studies and an empirical illustration of a nonparametric quantile IV regression are presented.