Testing Monotonicity of Mean Potential Outcomes in a Continuous Treatment

While most treatment evaluations focus on binary interventions, a growing literature also considers continuously distributed treatments, e.g. hours spent in a training program to assess its effect on labor market outcomes. In this paper, we propose a Cram\'er-von Mises-type test for testing whether the mean potential outcome given a specific treatment has a weakly monotonic relationship with the treatment dose under a weak unconfoundedness assumption. This appears interesting for testing shape restrictions, e.g. whether increasing the treatment dose always has a non-negative effect, no matter what the baseline level of treatment is. We formally show that the proposed test controls asymptotic size and is consistent against any fixed alternative. These theoretical findings are supported by the method's finite sample behavior in our Monte-Carlo simulations. As an empirical illustration, we apply our test to the Job Corps study and reject a weakly monotonic relationship between the treatment (hours in academic and vocational training) and labor market outcomes like earnings or employment.