Estimation of Possibly Misspecified Semiparametric Conditional Moment Restriction Models with Different Conditioning Variables Chunrong Ai a , Xiaohong Chen b, 1 a Department of Economics, University of Florida, Gainesville, FL 32611, USA b Department of Economics, New York University, 269 Mercer Street, New York, NY 10003, USA First version: March 2003, Revised version: November 2005 ––––––––––––––––––––––––––––––––––––––––––– Abstract Newey and Powell (2003) and Ai and Chen (2003) propose the sieve minimum distance (SMD) estimation of both finite dimensional parameter (θ) and infinite dimensional parameter (h) that are identified through a conditional moment restriction model. This paper modifies their SMD procedure to allow for different conditioning variables to be used in different equations, and derives the asymptotic results when the model may be misspecified. Under low-level sufficient conditions, we show that: (i) the SMD estimators of both θ and h converge to some pseudo-true values in probability; (ii) the SMD estimators of smooth functionals, including the θ estimator and the average derivative estimator, are asymptotically normally distributed; and (iii) the estimators for the asymptotic covariances of the SMD estimators of smooth functionals are consistent and easy to compute. These results allow for asymptotically valid tests of various hypotheses on the smooth functionals regardless of whether the semiparametric model is correctly specified or not. JEL Classification: C14; C22 Keywords: Misspecification; Sieve minimum distance; Conditional moment models with different conditioning sets; Nonparametric endogeneity; Weighted average derivatives ––––––––––––––––––––––––––––––––––––––––––– 1 Corresponding author. Tel.: +1 212 998 8970; Fax: +1 212 995 4186. E-mail addresses: chunrong.ai@cba.ufl.edu (C. Ai), xiaohong.chen@nyu.edu (X. Chen).