Discussion Paper

2-1-2008

1640R

2009-07-01

33

#### Abstract

This paper considers semiparametric eﬀicient estimation of conditional moment models with possibly nonsmooth residuals in unknown parametric components ( theta ) and unknown functions ( h ) of endogenous variables. We show that: (1) the penalized sieve minimum distance (PSMD) estimator ( theta\hat,h\hat ) can simultaneously achieve root- n asymptotic normality of theta\hat and nonparametric optimal convergence rate of h\hat , allowing for noncompact function parameter spaces; (2) a simple weighted bootstrap procedure consistently estimates the limiting distribution of the PSMD theta\hat ; (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.

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