Inference under Shape Restrictions

We propose a uniformly valid inference method for an unknown function or parameter vector satisfying certain shape restrictions. The method applies very generally, namely to a wide range of nite dimensional and nonparametric problems, such as regressions or instrumental variable estimation, to both kernel or series estimators, and to many dierent shape restrictions. One application of our inference method is to construct uniform condence bands for an unknown function of interest. These bands are build around a shape restricted estimator and the upper and lower bound functions are consistent with the shape restrictions. Moreover, the bands are asymptotically equivalent to standard unrestricted condence bands if the true function strictly satises all shape restrictions, but they can be much smaller if some of the shape restrictions are binding or close to binding. We illustrate these sizable width gains as well as the wide applicability of our method in Monte Carlo simulations and in an empirical application. Keywords: Shape restrictions, inference, nonparametric, uniform condence bands. Joint with Brandon Reeves.