  This paper develops theory for feasible estimators of finite-dimensional
parameters identified by general conditional quantile restrictions, under much
weaker assumptions than previously seen in the literature. This includes
instrumental variables nonlinear quantile regression as a special case. More
specifically, we consider a set of unconditional moments implied by the
conditional quantile restrictions, providing conditions for local
identification. Since estimators based on the sample moments are generally
impossible to compute numerically in practice, we study feasible estimators
based on smoothed sample moments. We propose a method of moments estimator for
exactly identified models, as well as a generalized method of moments estimator
for over-identified models. We establish consistency and asymptotic normality
of both estimators under general conditions that allow for weakly dependent
data and nonlinear structural models. Simulations illustrate the finite-sample
properties of the methods. Our in-depth empirical application concerns the
consumption Euler equation derived from quantile utility maximization.
Advantages of the quantile Euler equation include robustness to fat tails,
decoupling of risk attitude from the elasticity of intertemporal substitution,
and log-linearization without any approximation error. For the four countries
we examine, the quantile estimates of discount factor and elasticity of
intertemporal substitution are economically reasonable for a range of quantiles
above the median, even when two-stage least squares estimates are not
reasonable.
