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 ( θ ˆ , h ˆ ) can simultaneously achieve root- n asymptotic normality of θ ˆ and nonparametric optimal convergence rate of h ˆ , allowing for noncompact function parameter spaces; (2) a simple weighted bootstrap procedure consistently estimates the limiting distribution of the PSMD θ ˆ ; (3) the semiparametric efficiency bound formula of [Ai, C., Chen, X., 2003. Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica, 71, 1795–1843] 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.