Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals

This paper considers semiparametric efficient estimation of conditional moment models with possibly nonsmooth residuals in unknown parametric components (θ) and unknown functions (h) of endogenous variables. We show that: (1) the penalized sieve minimum distance(PSMD) estimator (ˆθ,ˆh) can simultaneously achieve root-n asymptotic normality of ˆθ and nonparametric optimal convergence rate of h, allowing for noncompact function parameter spaces; (2) a simple weighted bootstrap procedure consistently estimates the limiting distribution of the PSMD ˆθ ; (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 centered, profiled optimally weighted PSMD criterion is asymptotically chi-square distributed. We illustrate our theories using a partially linear quantile instrumental variables (IV) regression, a Monte Carlo study, and an empirical estimation of the shape-invariant quantile IV Engel curves.