Efficient Estimation of Semiparametric Conditional Moment Models with Possibly Nonsmooth Residuals

For semi/nonparametric conditional moment models containing unknown parametric components (theta) and unknown functions of endogenous variables (h), Newey and Powell (2003) and Ai and Chen (2003) propose sieve minimum distance (SMD) estimation of (theta, h) and derive the large sample properties. This paper greatly extends their results by establishing the followings: (1) The penalized SMD (PSMD) estimator (hat{theta}, hat{h}) can simultaneously achieve root-n asymptotic normality of theta hat and nonparametric optimal convergence rate of hat{h}, allowing for models with possibly nonsmooth residuals and/or noncompact infinite dimensional parameter spaces. (2) A simple weighted bootstrap procedure can consistently estimate the limiting distribution of the PSMD hat{theta}. (3) The semiparametric efficiency bound results of Ai and Chen (2003) remain valid for conditional models with nonsmooth residuals, and the optimally weighted PSMD estimator achieves the bounds. (4) The profiled optimally weighted PSMD criterion is asymptotically Chi-square distributed, which implies an alternative consistent estimation of confidence region of the efficient PSMD estimator of theta. All the theoretical results are stated in terms of any consistent nonparametric estimator of conditional mean functions. We illustrate our general theories using a partially linear quantile instrumental variables regression, a Monte Carlo study, and an empirical estimation of the shape-invariant quantile Engel curves with endogenous total expenditure.