# Testing for Quantile Sample Selection

## TL;DR

This paper develops nonparametric tests for detecting sample selection in quantile functions, using propensity scores and identification at infinity, without requiring parametric assumptions, and demonstrates their effectiveness with simulations and real data.

## Contribution

It introduces two novel nonparametric tests for sample selection in quantile functions that do not rely on parametric models or continuous exclusion restrictions.

## Key findings

- First test shows good finite sample performance in simulations.
- Second test provides supporting evidence for selection causes.
- Application to UK wage data demonstrates practical utility.

## Abstract

This paper provides tests for detecting sample selection in nonparametric conditional quantile functions. The first test is an omitted predictor test with the propensity score as the omitted variable. As with any omnibus test, in the case of rejection we cannot distinguish between rejection due to genuine selection or to misspecification. Thus, we suggest a second test to provide supporting evidence whether the cause for rejection at the first stage was solely due to selection or not. Using only individuals with propensity score close to one, this second test relies on an `identification at infinity' argument, but accommodates cases of irregular identification. Importantly, neither of the two tests requires parametric assumptions on the selection equation nor a continuous exclusion restriction. Data-driven bandwidth procedures are proposed, and Monte Carlo evidence suggests a good finite sample performance in particular of the first test. Finally, we also derive an extension of the first test to nonparametric conditional mean functions, and apply our procedure to test for selection in log hourly wages using UK Family Expenditure Survey data as \citet{AB2017}.