Econometrica: Jan 2012, Volume 80, Issue 1

Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals

https://doi.org/10.3982/ECTA7888
p. 277-321

Xiaohong Chen, Demian Pouzo

This paper studies nonparametric estimation of conditional moment restrictions in which the generalized residual functions can be nonsmooth in the unknown functions of endogenous variables. This is a nonparametric nonlinear instrumental variables (IV) problem. We propose a class of penalized sieve minimum distance (PSMD) estimators, which are minimizers of a penalized empirical minimum distance criterion over a collection of sieve spaces that are dense in the infinite‐dimensional function parameter space. Some of the PSMD procedures use slowly growing finite‐dimensional sieves with flexible penalties or without any penalty; others use large dimensional sieves with lower semicompact and/or convex penalties. We establish their consistency and the convergence rates in Banach space norms (such as a sup‐norm or a root mean squared norm), allowing for possibly noncompact infinite‐dimensional parameter spaces. For both mildly and severely ill‐posed nonlinear inverse problems, our convergence rates in Hilbert space norms (such as a root mean squared norm) achieve the known minimax optimal rate for the nonparametric mean IV regression. We illustrate the theory with a nonparametric additive quantile IV regression. We present a simulation study and an empirical application of estimating nonparametric quantile IV Engel curves.

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Supplemental Material

Supplement to "Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Generalized Residuals"

This supplemental document first provides a brief summary of commonly used function spaces and sieve spaces.  It then provides mathematical proofs of all the theorems, corollaries, propositions, and lemmas that appear in the main text and appendix.

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Supplement to "Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Generalized Residuals"

A zip file containing replication files for the manuscript.

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