Universal Power of Kolmogorov-Smirnov Tests of Under-Identifying Restrictions

We provide a test for the specification of a structural model without identifying assumptions. From the dual formulation of a null hypothesis of compatibility of the data generating process with the structure, we derive a Kolmogorov-Smirnov statistic for Choquet capacity functionals, which we use to construct our test. We show that our test is uniform in level over all the data-generating processes compatible with the structure, and has power against local alternatives defined in terms of total variation