Rate-Optimal Estimation of the Intercept in a Semiparametric Sample-Selection Model

TL;DR

This paper introduces a rate-optimal estimator for the intercept in a semiparametric sample-selection model, achieving the best possible convergence rate and demonstrating consistency and asymptotic normality.

Contribution

It proposes a new estimator that attains the optimal convergence rate for the intercept in sample-selection models under mild conditions.

Findings

Estimator achieves the rate of $n^{-p/(2p+1)}$ convergence.

Estimator is consistent and asymptotically normal.

Simulation and empirical results support the theoretical properties.

Abstract

This paper presents a new estimator of the intercept of a linear regression model in cases where the outcome varaible is observed subject to a selection rule. The intercept is often in this context of inherent interest; for example, in a program evaluation context, the difference between the intercepts in outcome equations for participants and non-participants can be interpreted as the difference in average outcomes of participants and their counterfactual average outcomes if they had chosen not to participate. The new estimator can under mild conditions exhibit a rate of convergence in probability equal to $n^{-p/(2p+1)}$, where $p\ge 2$ is an integer that indexes the strength of certain smoothness assumptions. This rate of convergence is shown in this context to be the optimal rate of convergence for estimation of the intercept parameter in terms of a minimax criterion. The new estimator, unlike other proposals in the literature, is under mild conditions consistent and asymptotically normal with a rate of convergence that is the same regardless of the degree to which selection depends on unobservables in the outcome equation. Simulation evidence and an empirical example are included.