This paper develops a concrete formula for the asymptotic distribution of two-step,
possibly non-smooth semiparametric M-estimators under general misspecification. Our
regularity conditions are relatively straightforward to verify and also weaker than those
available in the literature. The first-stage nonparametric estimation may depend on
finite dimensional parameters. We characterize: (1) conditions under which the first-stage
estimation of nonparametric components do not affect the asymptotic distribution,
(2) conditions under which the asymptotic distribution is affected by the derivatives of
the 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 estimation
of a single index quantile regression model, (2) semiparametric least squares estimation
under model misspecification, and (3) a smoothed matching estimator.
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