A Projection Framework for Testing Shape Restrictions That Form Convex Cones

TL;DR

This paper introduces a new projection-based testing framework for shape restrictions forming convex cones, providing a simple, valid, and bootstrap-based method applicable to various models, with a data-driven tuning parameter.

Contribution

It develops a novel, geometrically motivated test for convex cone restrictions that avoids complex local parameter estimation and is applicable to nonparametric and structural models.

Findings

Test performs well in Monte Carlo simulations.

Framework accommodates nonparametric and structural models.

Data-driven tuning parameter is valid.

Abstract

This paper develops a uniformly valid and asymptotically nonconservative test based on projection for a class of shape restrictions. The key insight we exploit is that these restrictions 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 construct a bootstrap procedure that, unlike many studies in nonstandard settings, dispenses with estimation of local parameter spaces, and the critical values are obtained in a way as simple as computing the test statistic. Moreover, by appealing to strong approximations, our framework accommodates nonparametric regression models as well as distributional/density-related and structural settings. Since the test entails a tuning parameter (due to the nonstandard nature of the problem), we propose a data-driven choice and prove its validity. Monte Carlo simulations confirm that our test works well.