  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.
