Econometrica: Sep 2021, Volume 89, Issue 5

A Projection Framework for Testing Shape Restrictions that Form Convex Cones

https://doi.org/10.3982/ECTA17764
p. 2439-2458

Zheng Fang, Juwon Seo

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.



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Supplement to "A Projection Framework for Testing Shape Restrictions that Form Convex Cones"

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Supplement to "A Projection Framework for Testing Shape Restrictions that Form Convex Cones"

This supplement is organized as follows. Appendix B discusses particular shape restrictions with the convex cone property, Appendix C specializes our test to the regular case where rn{ˆθ_n − θ0} converges, Appendix D collects additional proofs and auxiliary results, and Appendix E presents additional simulation studies and an empirical application. Appendix F verifies the main assumptions for our examples, Appendix G provides proofs for Appendix C, while Appendix H contains simulation results omitted from the main text and Appendix E, all of which are relegated to the arXiv version of this paper (https://arxiv.org/abs/1910.07689) due to space limitation.

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