A Unifying Framework for Testing Shape Restrictions

This paper makes the following econometric contributions. First, we develop a unifying framework for testing shape restrictions based on the Wald principle. Second, we examine the applicability and usefulness of some prominent shape enforcing operators in implementing our test, including rearrangement and the greatest convex minorization (or the least concave majorization). In particular, the influential rearrangement operator is inapplicable due to a lack of convexity, while the greatest convex minorization is shown to enjoy the analytic properties required to employ our framework. The importance of convexity in establishing size control has been noted elsewhere in the literature. Third, we show that, despite that the projection operator may not be well-defined/behaved in general non-Hilbert parameter spaces (e.g., ones defined by uniform norms), one may nonetheless devise a powerful distance-based test by applying our framework. The finite sample performance of our test is evaluated through Monte Carlo simulations, and its empirical relevance is showcased by investigating the relationship between weekly working hours and the annual wage growth in the high-end labor market.