Testing for Quantile Sample Selection

This paper provides distribution free tests for detecting sample selection in conditional quantile functions. The first test is an omitted predictor test with the propensity score as the omitted variable. In the case of rejection we cannot distinguish between rejection due to genuine selection or to misspecification. Thus, we suggest a second test using only individuals with propensity score close to one. The latter relies on an `identification at infinity' argument, but accommodates cases of irregular identification. Importantly, neither of the two tests requires a continuous exclusion restriction. We apply our procedure to test for selection in log hourly wages using UK survey data, and derive an extension of the tests to the conditional mean