Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals


  • Xiaohong Chen,

    1. Cowles Foundation for Research in Economics, Yale University, 30 Hillhouse, Box 208281, New Haven, CT 06520, U.S.A.;
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  • Demian Pouzo

    1. Dept. of Economics, UC at Berkeley, 508-1 Evans Hall 3880, Berkeley, CA 94704-3880, U.S.A.;
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    • This paper is a substantially revised version of Cowles Foundation Discussion Paper 1650. We are grateful to two co-editors, three anonymous referees, R. Blundell, V. Chernozhukov, T. Christensen, K. Evdokimov, H. Hong, J. Horowitz, J. Huang, S. Lee, Z. Liao, O. Linton, D. Nekipelov, J. Powell, A. Santos, E. Tamer, and A. Torgovitsky for their constructive comments that led to a much improved revision. Earlier versions were presented in August 2006 at the European Meeting of the Econometric Society, in March 2007 at the Oberwolfach Workshop on Semi/nonparametrics, in June 2007 at the Cemmap Conference on Measurement Matters, in 2008 at the Cowles Summer Conference, and at econometric workshops at many universities. We thank participants of these conferences and workshops for helpful suggestions. Chen acknowledges financial support from the National Science Foundation under Grant SES0631613 and SES0838161. The usual disclaimer applies.


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.