This paper develops a concrete formula for the asymptotic distribution of two-step,possibly non-smooth semiparametric M-estimators under general misspecification. Ourregularity conditions are relatively straightforward to verify and also weaker than thoseavailable in the literature. The first-stage nonparametric estimation may depend onfinite dimensional parameters. We characterize: (1) conditions under which the first-stageestimation of nonparametric components do not affect the asymptotic distribution,(2) conditions under which the asymptotic distribution is affected by the derivatives ofthe first-stage nonparametric estimator with respect to the finite-dimensional parameters,and (3) conditions under which one can allow non-smooth objective functions.Our framework is illustrated by applying it to three examples: (1) profiled estimationof a single index quantile regression model, (2) semiparametric least squares estimationunder model misspecification, and (3) a smoothed matching estimator.
Centre for microdata methods and practice