Semiparametric models with single-index nuisance parameters

In many semiparametric models, the parameter of interest is identified through conditional expectations, where the conditioning variable involves a single-index that is estimated in the first step. Among the examples are sample selection models and propensity score matching estimators. When the first-step estimator follows cube-root asymptotics, no method of analyzing the asymptotic variance of the second step estimator exists in the literature. This paper provides nontrivial sufficient conditions under which the asymptotic variance is not affected by the first step single-index estimator regardless of whether it is root-n or cube-root consistent. The finding opens a way to simple inference procedures in these models. Results from Monte Carlo simulations show that the procedures perform well in finite samples.