Before revision:  Our testing procedure does not require any parametric assumptions on the selection equation , and all our results hold uniformly across quantile ranks in a compact set. We apply our procedure to test for selection in log hourly wages using UK Family Expenditure Survey data  .
After revision:  Our testing procedure does not require any 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 .