Distribution-free tests of stochastic monotonicity

Abstract

This article proposes a nonparametric test of monotonicity for conditional distributions and its moments.Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves. Distinguishing features of our approach are that critical values are pivotal under the null in finite samples and that the test is invariant to any monotonic continuous transformation of the explanatory variable. The test statistic is the sup-norm of the difference between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate. The resulting test is able to detect local alternatives converging to the null at the parametric rate n−1/2, with n the sample size. The finite sample performance of the test is examined by means of a Monte Carlo experiment and an application to testing intergenerational income mobility.