Multiple Testing of a Function's Monotonicity

Instead of having a "yes" or "no" result from a test of the global null hypothesis that a function is increasing, I propose a multiple testing procedure to test at multiple points. If the global null is rejected, then this multiple testing provides more information about why. If the global null is not rejected, then multiple testing can provide stronger evidence in favor of increasingness, by rejecting the null hypotheses that the function is decreasing. With high-level assumptions that apply to a wide array of models, this approach can be used to test for monotonicity of a function in a broad class of structural and descriptive econometric models. By inverting the proposed multiple testing procedure that controls the familywise error rate, I also equivalently generate "inner" and "outer" confidence sets for the set of points at which the function is increasing. With high asymptotic probability, the inner confidence set is contained within the true set, whereas the outer confidence set contains the true set. I also improve power with stepdown and two-stage procedures. Simulated and empirical examples (income-education conditional mean, and IV Engel curve) illustrate the methodology.