  This paper considers estimation and inference for a weighted average
derivative (WAD) of a nonparametric quantile instrumental variables regression
(NPQIV). NPQIV is a non-separable and nonlinear ill-posed inverse problem,
which might be why there is no published work on the asymptotic properties of
any estimator of its WAD. We first characterize the semiparametric efficiency
bound for a WAD of a NPQIV, which, unfortunately, depends on an unknown
conditional derivative operator and hence an unknown degree of ill-posedness,
making it difficult to know if the information bound is singular or not. In
either case, we propose a penalized sieve generalized empirical likelihood
(GEL) estimation and inference procedure, which is based on the unconditional
WAD moment restriction and an increasing number of unconditional moments that
are implied by the conditional NPQIV restriction, where the unknown quantile
function is approximated by a penalized sieve. Under some regularity
conditions, we show that the self-normalized penalized sieve GEL estimator of
the WAD of a NPQIV is asymptotically standard normal. We also show that the
quasi likelihood ratio statistic based on the penalized sieve GEL criterion is
asymptotically chi-square distributed regardless of whether or not the
information bound is singular.
