Semiparametric Models with Single-Index Nuisance Parameters

TL;DR

This paper develops conditions under which the asymptotic variance of second-step estimators in semiparametric models is unaffected by the first-step single-index estimator, simplifying inference procedures.

Contribution

It provides new sufficient conditions ensuring the second-step estimator's variance is unaffected by the first-step estimator, even with cube-root asymptotics.

Findings

Asymptotic variance is unaffected by the first-step estimator under certain conditions.

Proposed inference procedures perform well in finite samples.

Results apply to models like sample selection and propensity score matching.

Abstract

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.