COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1650 Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Moments Xiaohong Chen and Demian Pouzo April 2008 This paper studies nonparametric estimation of conditional moment models in which the
residual functions could be nonsmooth with respect to the unknown functions of endogenous
variables. It is a problem of nonparametric nonlinear instrumental variables (IV)
estimation, and a difficult nonlinear ill-posed inverse problem with an unknown operator.
We first propose a penalized sieve minimum distance (SMD) estimator of the unknown
functions that are identified via the conditional moment models. We then establish its
consistency and convergence rate (in strong metric), allowing for possibly non-compact
function parameter spaces, possibly non-compact finite or infinite dimensional sieves with
flexible lower semicompact or convex penalty, or finite dimensional linear sieves without
penalty. Under relatively low-level sufficient conditions, and for both mildly and
severely ill-posed problems, we show that the convergence rates for the nonlinear
ill-posed inverse problems coincide with the known minimax optimal rates for the
nonparametric mean IV regression. We illustrate the theory by two important applications:
root-n asymptotic normality of the plug-in penalized SMD estimator of a weighted average
derivative of a nonparametric nonlinear IV regression, and the convergence rate of a
nonparametric additive quantile IV regression. We also present a simulation study and an
empirical estimation of a system of nonparametric quantile IV Engel curves. |
