A General Framework for Inference on Shape Restrictions
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This paper presents a general and uniformly valid procedure for conducting inference on shape restrictions. The key insight we exploit is that common shape restrictions in economics often form convex cones, a simple and yet elegant structure that has been barely harnessed in the literature. Based on a monotonicity property afforded by such a geometric structure, we develop a bootstrap test that is computationally convenient to implement. In particular, unlike many studies in similar nonstandard settings, the procedure dispenses with set estimation, and the critical values are obtained in a way as simple as computing the test statistic. Moreover, by appealing to the machinery of strong approximations, we accommodate nonparametric settings where estimators of the parameter of interest may not admit asymptotic distributions. We establish asymptotic uniform size control of our test, and characterize classes of alternatives against which it has nontrivial power. Since the test entails a tuning parameter (due to the inherent nonstandard nature of the problem), we propose a data-driven choice and prove its validity. Monte Carlo simulations confirm that our test works well.
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