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TESTING FOR A GENERAL CLASS OF FUNCTIONAL INEQUALITIES

Published online by Cambridge University Press:  01 December 2017

Sokbae Lee*
Affiliation:
Columbia University Institute for Fiscal Studies
Kyungchul Song
Affiliation:
University of British Columbia
Yoon-Jae Whang
Affiliation:
Seoul National University
*
*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.
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Abstract

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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 Lp 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.

Type
ARTICLES
Creative Commons
Creative Common License - CCCreative Common License - BY
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.
Copyright
Copyright © Cambridge University Press 2017

Footnotes

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.

References

Acemoglu, D. & Autor, D. (2011) Skills, tasks and technologies: Implications for employment and earnings. In Card, D. & Ashenfelter, O. (eds.), Handbook of Labor Economics, vol. 4, part B, pp. 10431171. Elsevier.Google Scholar
Anderson, G., Linton, O., & Whang, Y.-J. (2012) Nonparametric estimation and inference about the overlap of two distributions. Journal of Econometrics 171(1), 123.CrossRefGoogle Scholar
Andrews, D.W.K. (1994) Empirical process method in econometrics. In Engle, R.F. & McFadden, D.L. (eds.), The Handbook of Econometrics, vol. 4, pp. 22472294. North-Holland.Google Scholar
Andrews, D.W.K. & Shi, X. (2013) Inference based on conditional moment inequalities. Econometrica 81(2), 609666.Google Scholar
Andrews, D.W.K. & Shi, X. (2014) Nonparametric inference based on conditional moment inequalities. Journal of Econometrics 179(1), 3145.CrossRefGoogle Scholar
Andrews, D.W.K. & Shi, X. (2017) Inference based on many conditional moment inequalities. Journal of Econometrics 196(2), 275287.CrossRefGoogle Scholar
Aradillas-López, A., Gandhi, A., & Quint, D. (2013) Identification and inference in ascending auctions with correlated private values. Econometrica 81(2), 489534.Google Scholar
Aradillas-López, A., Gandhi, A., & Quint, D. (2016) A simple test for moment inequality models with an application to English auctions. Journal of Econometrics 194(1), 96115.CrossRefGoogle Scholar
Armstrong, T.B. (2014a) On the choice of test statistic for conditional moment inequalities. ArXiv:1410.4718.Google Scholar
Armstrong, T.B. (2014b) Weighted KS statistics for inference on conditional moment inequalities. Journal of Econometrics 181(2), 92116.CrossRefGoogle Scholar
Armstrong, T.B. (2015) Asymptotically exact inference in conditional moment inequalitymodels. Journal of Econometrics 186(1), 5165.CrossRefGoogle Scholar
Armstrong, T.B. & Chan, H.P. (2016) Multiscale adaptive inference on conditional moment inequalities. Journal of Econometrics 194(1), 2443.CrossRefGoogle Scholar
Baraud, Y., Huet, S., & Laurent, B. (2005) Testing convex hypotheses on the mean of a Gaussian vector. Application to testing qualitative hypotheses on a regression function. Annals of Statistics 33(1), 214257.CrossRefGoogle Scholar
Beare, B.K. & Moon, J.-M. (2015) Nonparametric tests of density ratio ordering. Econometric Theory 31(3), 471492.CrossRefGoogle Scholar
Biau, G., Cadre, B., Mason, D., & Pelletier, B. (2009) Asymptotic normality in density support estimation. Electronic Journal of Probability 14(91), 26172635.CrossRefGoogle Scholar
Bickel, P.J., Ritov, Y., & Stoker, T.M. (2006) Tailor-made tests for goodness of fit to semiparametric hypotheses. Annals of Statistics 34(2), 721741.CrossRefGoogle Scholar
Chang, M., Lee, S., & Whang, Y.-J. (2015) Nonparametric tests of conditional treatment effects with an application to single-sex schooling on academic achievements. Econometrics Journal 18(3), 307346.CrossRefGoogle Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71(5), 15911608.CrossRefGoogle Scholar
Chernozhukov, V., Lee, S., & Rosen, A. (2013) Intersection bounds: Estimation and inference. Econometrica 81(2), 667737.Google Scholar
Chetverikov, D. (2012) Testing Regression Monotonicity in Econometric Models. Working paper, MIT.Google Scholar
Chetverikov, D. (2017) Adaptive tests of conditional moment inequalities. Econometric Theory, forthcoming. doi:10.1017/S0266466617000184.Google Scholar
Delgado, M.A. & Escanciano, J.C. (2012) Distribution-free tests of stochastic monotonicity. Journal of Econometrics 170(1), 6875.CrossRefGoogle Scholar
Delgado, M.A. & Escanciano, J.C. (2013) Conditional stochastic dominance testing. Journal of Business & Economic Statistics 31(1), 1628.CrossRefGoogle Scholar
Dümbgen, L. & Spokoiny, V.G. (2001) Multiscale testing of qualitative hypotheses. Annals of Statistics 29(1), 124152.CrossRefGoogle Scholar
Fan, Y. & Park, S. (2014) Nonparametric inference for counterfactual means: Bias-correction, confidence sets, and weak IV. Journal of Econometrics 178(Part 1), 4556.CrossRefGoogle Scholar
Ghosal, S., Sen, A., & van der Vaart, A.W. (2000) Testing monotonicity of regression. Annals of Statistics 28, 10541082.Google Scholar
Gimenes, N. & Guerre, E. (2013) Augmented Quantile Regression Methods for First-Price Auction. Working paper, Queen Mary University of London.Google Scholar
Giné, E., Mason, D.M., & Zaitsev, A.Y. (2003) The L1-norm density estimator process. Annals of Probability 31, 719768.Google Scholar
Guerre, E., Perrigne, I., & Vuong, Q. (2009) Nonparametric identification of risk aversion in first-price auctions under exclusion restrictions. Econometrica 77(4), 11931227.Google Scholar
Haile, P., Hong, H., & Shum, M. (2003) Nonparametric Tests for Common Values at First- Price Sealed-Bid Auctions. Working paper, Yale University.Google Scholar
Haile, P.A. & Tamer, E. (2003) Inference with an incomplete model of English auctions. Journal of Political Economy 111(1), 151.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21(4), 19261947.CrossRefGoogle Scholar
Horváth, L. (1991) On L p-norms of multivariate density estimators. Annals of Statistics 19(4), 19331949.CrossRefGoogle Scholar
Hsu, Y.-C. (2017) Consistent tests for conditional treatment effects. Econometrics Journal 20, 122.CrossRefGoogle Scholar
Juditsky, A. & Nemirovski, A. (2002) On nonparametric tests of positivity/monotonicity/convexity. Annals of Statistics 30(2), 498527.CrossRefGoogle Scholar
Jun, S.J., Pinkse, J., & Wan, Y. (2010) A consistent nonparametric test of affiliation in auction models. Journal of Econometrics 159(1), 4654.CrossRefGoogle Scholar
Khan, S. & Tamer, E. (2009) Inference on endogenously censored regression models using conditional moment inequalities. Journal of Econometrics 152(2), 104119.CrossRefGoogle Scholar
Krasnokutskaya, E., Song, K., & Tang, X. (2016) The Role of Quality in On-Line Service Markets. Working paper, Johns Hopkins University.Google Scholar
Lee, S., Linton, O., & Whang, Y.-J. (2009) Testing for stochastic monotonicity. Econometrica 77(2), 585602.Google Scholar
Lee, S., Song, K., & Whang, Y.-J. (2013) Testing functional inequalities. Journal of Econometrics 172(1), 1432.CrossRefGoogle Scholar
Linton, O., Song, K., & Whang, Y.-J. (2010) An improved bootstrap test of stochastic dominance. Journal of Econometrics 154(2), 186202.CrossRefGoogle Scholar
Lu, J. & Perrigne, I. (2008) Estimating risk aversion from ascending and sealed-bid auctions: The case of timber auction data. Journal of Applied Econometrics 23(7), 871896.CrossRefGoogle Scholar
Mammen, E., Van Keilegom, I., & Yu, K. (2013) Expansion for moments of regression quantiles with application to nonparametric testing. Working paper, arXiv:1306.6179.Google Scholar
Marmer, V., Shneyerov, A., & Xu, P. (2013) What model for entry in first-price auctions? A nonparametric approach. Journal of Econometrics 176(1), 4658.CrossRefGoogle Scholar
Mason, D.M. & Polonik, W. (2009) Asymptotic normality of plug-in level set estimates. Annals of Applied Probability 19, 11081142.CrossRefGoogle Scholar
Menzel, K. (2014) Consistent estimation with many moment inequalities. Journal of Econometrics 182(2), 329350.CrossRefGoogle Scholar
Nikitin, Y. (1995) Asymptotic Efficiency of Nonparametric Tests. Cambridge University Press.CrossRefGoogle Scholar
Proschan, M.A. & Presnell, B. (1998) Expect the unexpected from conditional expectation. American Statistician 52(3), 248252.Google Scholar
Rosenblatt, M. (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Annals of Statistics 3(1), 114.CrossRefGoogle Scholar
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