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AN ADAPTIVE TEST OF STOCHASTIC MONOTONICITY

Published online by Cambridge University Press:  16 June 2020

Denis Chetverikov
Affiliation:
University of California at Los Angeles
Daniel Wilhelm*
Affiliation:
University College London
Dongwoo Kim
Affiliation:
Simon Fraser University
*
Address correspondence to Daniel Wilhelm, Department of Economics, University College London, Gower Street, LondonWC1E 6BT, United Kingdom; e-mail: d.wilhelm@ucl.ac.uk.

Abstract

We propose a new nonparametric test of stochastic monotonicity which adapts to the unknown smoothness of the conditional distribution of interest, possesses desirable asymptotic properties, is conceptually easy to implement, and computationally attractive. In particular, we show that the test asymptotically controls size at a polynomial rate, is nonconservative, and detects certain smooth local alternatives that converge to the null with the fastest possible rate. Our test is based on a data-driven bandwidth value and the critical value for the test takes this randomness into account. Monte Carlo simulations indicate that the test performs well in finite samples. In particular, the simulations show that the test controls size and, under some alternatives, is significantly more powerful than existing procedures.

Type
ARTICLES
Copyright
© Cambridge University Press 2020

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Footnotes

*

We thank Ivan Canay and Whitney Newey for useful comments. Daniel Wilhelm gratefully acknowledges financial support from the ESRC Centre for Microdata Methods and Practice at IFS (RES-589-28-0001) and the European Research Council (ERC-2014-CoG-646917-ROMIA).

References

Adler, R. & Taylor, J. (2007) Random Fields and Geometry. Springer.Google Scholar
Andrews, D. & Shi, X. (2013) Inference based on conditional moment inequalities. Econometrica 81, 609666.Google Scholar
Armstrong, T. (2014) Weighted KS statistics for inference on conditional moment inequalities. Journal of Econometrics 181, 92116.CrossRefGoogle Scholar
Armstrong, T. (2015) Adaptive testing on a regression function at a point. Annals of Statistics 43, 20862101.CrossRefGoogle Scholar
Armstrong, T. & Chan, H. (2016) Multiscale adaptive inference on conditional moment inequalities. Journal of Econometrics 194, 2443.CrossRefGoogle Scholar
Banerjee, M., Mukherjee, D., & Mishra, S. (2009) Semiparametric binary regression models under shape constraints with an application to Indian schooling data. Journal of Econometrics 149(2), 101117.CrossRefGoogle Scholar
Belloni, A., Chernozhukov, V., Chetverikov, D., Hansen, C., & Kato, K. (2018) High-Dimensional Econometrics and Regularized GMM. Discussion Paper.Google Scholar
Boudoukh, J., Richardson, M., Smith, T., & Whitelaw, R.F. (1999) Ex ante bond returns and the liquidity preference hypothesis. The Journal of Finance 54(3), 11531167.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2013) Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. Annals of Statistics 41(6), 27862819.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2014) Testing Many Moment Inequalities. Discussion Paper.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2015) Comparison and anti-concentration bounds for maxima of Gaussian random vectors. Probability Theory and Related Fields 162, 47.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2017) Central limit theorems and bootstrap in high dimensions. The Annals of Probability 45, 144.CrossRefGoogle Scholar
Chernozhukov, V., Lee, S., & Rosen, A.M. (2013) Intersection bounds: Estimation and inference. Econometrica 81(2), 667737.Google Scholar
Chetverikov, D. (2012) Testing Regression Monotonicity in Econometric Models. Discussion Paper.CrossRefGoogle Scholar
Chetverikov, D. (2018) Adaptive tests of conditional moment inequalities. Econometric Theory 34, 186227.CrossRefGoogle Scholar
Chetverikov, D. & Wilhelm, D. (2017) Nonparametric instrumental variable estimation under monotonicity. Econometrica 85(4), 13031320.CrossRefGoogle 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. (2001) Multiscale testing of qualitative hypotheses. Annals of Statistics 29, 124152.CrossRefGoogle Scholar
Ellison, G. & Ellison, S.F. (2011) Strategic entry deterrence and the behavior of pharmaceutical incumbents prior to patent expiration. American Economic Journal: Microeconomics 3(1), 136.Google Scholar
Ericson, R. & Pakes, A. (1995) Markov-perfect industry dynamics: A framework for empirical work. The Review of Economic Studies 62(1), 5382.CrossRefGoogle Scholar
Fan, J. (1996) Test of significance based on wavelet thresholding and Neyman’s truncation. Journal of the American Statistical Association 91(434), 674688.CrossRefGoogle Scholar
Fan, Y. & Li, Q. (2000) Consistent model specification tests: Kernel-based tests versus Bierens’ ICM tests. Econometric Theory 16(6), 10161041.CrossRefGoogle Scholar
Ghosal, S., Sen, A., & Vaart, A.W.v.d. (2000) Testing monotonicity of regression. Annals of Statistics 28(4), 10541082.CrossRefGoogle Scholar
Hoderlein, S., Holzmann, H., Kasy, M., & Meister, A. (2016) Erratum instrumental variables with unrestricted heterogeneity and continuous treatment. The Review of Economic Studies 84, rdw027.CrossRefGoogle Scholar
Horowitz, J. & Spokoiny, V. (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599631.CrossRefGoogle Scholar
Hsu, Y.-C., Liu, C.-A., & Shi, X. (2019) Testing generalized regression monotonicity. Econometric Theory 35(6), 11461200.CrossRefGoogle Scholar
Kasy, M. (2014) Instrumental variables with unrestricted heterogeneity and continuous treatment. The Review of Economic Studies 81(4), 1614.CrossRefGoogle 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
Lee, S., Song, K., & Whang, Y.-J. (2018) Testing for a general class of functional inequalities. Econometric Theory 34(5), 10181064.CrossRefGoogle Scholar
Lewbel, A. & Linton, O. (2007) Nonparametric matching and efficient estimators of homothetically separable functions. Econometrica 75(4), 12091227.CrossRefGoogle Scholar
Matzkin, R.L. (1994) Restrictions of economic theory in nonparametric methods. In Engle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. IV, pp. 25232558. Elsevier Science B.V.Google Scholar
Olley, G.S. & Pakes, A. (1996) The dynamics of productivity in the telecommunications equipment industry. Econometrica 64(6), 12631297.CrossRefGoogle Scholar
Patton, A.J. & Timmermann, A. (2010) Monotonicity in asset returns: New tests with applications to the term structure, the CAPM, and portfolio sorts. Journal of Financial Economics 98(3), 605625.CrossRefGoogle Scholar
Piterbarg, V. (1996) Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society.Google Scholar
Richardson, M., Richardson, P., & Smith, T. (1992) The monotonicity of the term premium: Another look. Journal of Financial Economics 31(1), 97105.CrossRefGoogle Scholar
Seo, J. (2018) Tests of stochastic monotonicity with improved power. Journal of Econometrics 207(1), 5370.CrossRefGoogle Scholar
Stokey, N.L. & Lucas, R.E. Jr. (1989) Recursive Methods in Economic Dynamics. Harvard University Press.CrossRefGoogle Scholar
Talagrand, M. (2011) Mean Field Models for Spin Glasses. Springer.Google Scholar
van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes, with Applications to Statistics. Springer.CrossRefGoogle Scholar
Wilhelm, D. (2019) Testing for the Presence of Measurement Error. Working Paper CWP48/19, CeMMAP.CrossRefGoogle Scholar