Tests of Stochastic Monotonicity with Improved Size and Power Properties

We develop improved statistical procedures for testing the null hypothesis of stochastic monotonicity. Stochastic monotonicity can be reformulated in terms of the concavity of cross-sections of a copula function; our test statistic is based on a empirical measure of departures from concavity. While existing tests of stochastic monotonicity deliver a limiting rejection rate equal to the nominal significance level at one point and below the nominal significance level elsewhere in the null, our test raises the limiting rejection rate to the nominal significance level over a wide region of the null. This improves power against relevant local alternatives. Implementation of our procedure is based on preliminary estimation of a contact set, similar to procedures developed recently in other contexts. To show the validity of our approach we draw on recent results on the directional differentiability of the least concave majorant operator, and on bootstrap inference when smoothness conditions sufficient to apply the functional delta method for the bootstrap are not satisfied. An application to intergenerational income mobility is provided.