Efficient Estimation of Semiparametric Conditional Moment Models with Possibly Nonsmooth Residuals

Abstract

This paper greatly extends the results of Ai and Chen (2003) on efficient estimation of semiparametric conditional moment models containing unknown parametric components (theta) and unknown functions of endogenous variables (h). We show that (1) the penalized sieve minimum distance (PSMD) estimator (hat{theta},hat{h}) can simultaneously achieve root-n asymptotic normality of hat{theta} and nonparametric optimal convergence rate of hat{h}, allowing for models with possibly nonsmooth residuals and noncompact infinite dimensional parameter spaces; (2) a simple weighted bootstrap procedure consistently estimates the limiting distribution of the PSMD hat{theta}; (3) the semiparametric efficiency bound formula of Ai and Chen (2003) remains valid for conditional models with nonsmooth residuals, and the optimally weighted PSMD estimator achieves the bound; (4) the profiled optimally weighted PSMD criterion is asymptotically chi-square distributed. 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.

Keywords: penalized sieve minimum distance, Nonsmooth generalized residuals, Nonlinear nonparametric endogeneity, Weighted bootstrap, Semiparametric efficiency, Confidence region, Partially linear quantile IV regression, Shape-invariant quantile Engel curves

JEL Classification: C14, C22