Distribution-free Tests of Stochastic Monotonicity1

This article proposes an omnibus test for monotonicity of nonparametric conditional distributions and its moments. Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves and it can be implemented even when probability densities do not exist. In fact, we only require continuity of the marginal distributions under the null and fixed alternatives. Distinguishing features of our approach are that critical values are pivotal under the null in finite samples and 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 article also discusses several applications and extensions of the proposal. These include testing monotonicity of general conditional moments and the extension to multivariate explanatory variables. The finite sample performance of the test is examined by means of a Monte Carlo experiment.