COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 0000 Sieve Quasi Likelihood Ratio Inference on Xiaohong Chen and Demian Pouzo May 2013 This paper considers inference on functionals of semi/nonparametric
conditional moment restrictions with possibly nonsmooth generalized residuals. These
models belong to the difficult (nonlinear) ill-posed inverse problems with unknown
operators, and include all of the (nonlinear) nonparametric instrumental variables (IV) as
special cases. For these models it is generally difficult to verify whether a functional
is regular (i.e., root-n estimable) or irregular (i.e., slower than root-n estimable). In
this paper we provide computationally simple, unified inference procedures that are
asymptotically valid regardless of whether a functional is regular or irregular. We
establish the following new results: (1) the asymptotic normality of the plug-in penalized
sieve minimum distance (PSMD) estimators of the (possibly irregular) functionals; (2) the
consistency of sieve variance estimators of the plug-in PSMD estimators; (3) the
asymptotic chi-square distribution of an optimally weighted sieve quasi likelihood ratio
(SQLR) statistic; (4) the asymptotic tight distribution of a possibly non-optimally
weighted SQLR statistic; (5) the consistency of the nonparametric bootstrap and the
weighted bootstrap (possibly non-optimally weighted) SQLR and sieve Wald statistics, which
are proved under virtually the same conditions as those for the original-sample
statistics. Small simulation studies and an empirical illustration of a nonparametric
quantile IV regression are presented. |
