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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.
Chen, Xiaohong and Pouzo, Demian, "Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Moments" (2008). Cowles Foundation Discussion Papers. 1951.