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TESTING FOR A GENERAL CLASS OF FUNCTIONAL INEQUALITIES

  • Sokbae Lee (a1), Kyungchul Song (a2) and Yoon-Jae Whang (a3)
Abstract

In this article, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of L p functionals of kernel-type estimators (1 ≤ p < ∞) and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on the nonparametric bootstrap applied to kernel-based test statistics, with an option of estimating “contact sets.” We provide regularity conditions under which the bootstrap test is asymptotically valid uniformly over a large class of distributions, including cases where the limiting distribution of the test statistic is degenerate. Our bootstrap test is shown to exhibit good power properties in Monte Carlo experiments, and we provide a general form of the local power function. As an illustration, we consider testing implications from auction theory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We supplement this example with the second empirical illustration in the context of wage inequality.

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Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
*Address correspondence to Sokbae Lee, Department of Economics, Columbia University, 1022 International Affairs Building 420 West 118th Street, New York, NY 10027, USA; e-mail: sl3841@columbia.edu.
Footnotes
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We would like to thank Editor, Peter C.B. Phillips, Co-Editor, Liangjun Su, three anonymous referees, Emmanuel Guerre and participants at numerous seminars and conferences for their helpful comments. We also thank Kyeongbae Kim, Koohyun Kwon and Jaewon Lee for capable research assistance. Lee’s work was supported by the European Research Council (ERC-2014-CoG-646917-ROMIA). Song acknowledges the financial support of Social Sciences and Humanities Research Council of Canada. Whang’s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2011-342-B00004) and the SNU Creative Leading Researcher Grant.

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