Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-11T05:55:42.788Z Has data issue: false hasContentIssue false
  • English
  • Français

TESTING GENERALIZED REGRESSION MONOTONICITY

Published online by Cambridge University Press:  18 December 2018

Yu-Chin Hsu ,
Chu-An Liu  and
Xiaoxia Shi
Yu-Chin Hsu
Affiliation:
Academia Sinica National Central University National Chengchi University
Chu-An Liu
Affiliation:
Academia Sinica
Xiaoxia Shi*
Affiliation:
University of Wisconsin at Madison
*
*Address correspondence to Xiaoxia Shi, Department of Economics, University of Wisconsin at Madison, 1180 Observatory Dr, Social Sciences Building #6428, Madison, WI 53706-1320, USA; e-mail: xshi@ssc.wisc.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a test for a generalized regression monotonicity (GRM) hypothesis. The GRM hypothesis is the sharp testable implication of the monotonicity of certain latent structures, as we show in this article. Examples include the monotonicity of the conditional mean function when only interval data are available for the dependent variable and the monotone instrumental variable assumption of Manski and Pepper (2000). These instances of latent monotonicity can be tested using our test. Moreover, the GRM hypothesis includes regression monotonicity and stochastic monotonicity as special cases. Thus, our test also serves as an alternative to existing tests for those hypotheses. We show that our test controls the size uniformly over a broad set of data generating processes asymptotically, is consistent against fixed alternatives, and has nontrivial power against some ${n^{ - 1/2}}$ local alternatives.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 6 , December 2019 , pp. 1146 - 1200
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Yu-Chin Hsu gratefully acknowledges the research support from Ministry of Science and Technology of Taiwan (MOST103-2628-H-001-001-MY4 and MOST107-2410-H-001 -034 -MY3) and Career Development Award of Academia Sinica, Taiwan. All errors and omissions are our own responsibility.

References

REFERENCES

Andrews, D.W.K. & Shi, X. (2013a) Inference based on conditional moment inequalities. Econometrica 81, 609–666.Google Scholar
Andrews, D.W.K. & Shi, X. (2013b) Supplement to ‘Inference based on conditional moment inequalities’. Econometrica Supplemental Material 81, 1–93.Google Scholar
Andrews, D.W.K. & Shi, X. (2014) Nonparametric inference based on conditional moment inequalities. Journal of Econometrics 179, 31–45.CrossRefGoogle Scholar
Andrews, D.W.K. & Shi, X. (2017) Inference based on many conditional moment inequalities. Journal of Econometrics 196, 275–287.CrossRefGoogle Scholar
Beraud, 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. The Annals of Statistics 33, 214–257.CrossRefGoogle Scholar
Berry, S. & Tamer, E. (2006) Identification in models of oligopoly entry. In Blundell, R., Newey, W.K., & Persson, T. (eds.), Advances in Economics and Econometrics, vol. 2, pp. 46–85. Cambridge University Press.CrossRefGoogle Scholar
Bowman, A.W., Jones, M.C., & Gijbels, I. (1998) Testing monotonicity of regression. Journal of Computational and Graphical Statistics 7, 489–500.Google Scholar
Bugni, F.A., Canay, I.A., & Shi, X. (2015) Specification tests for partially identified models defined by moment inequalities. Journal of Econometrics 185, 259–282.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2014) Anti-concentration and adaptive honest confidence band. The Annals of Statistics 42, 1787–1818.CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2015a) Comparison and anti-concentration Bounds for Maxima of Gaussian Random VectorsGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2015b) Testing many moment inequalities, Working paper.CrossRefGoogle Scholar
Chetverikov, D. (2013) Testing regression monotonicity in econometric models, Working paper.CrossRefGoogle Scholar
Chetverikov, D. & Wilhelm, D. (2017) Nonparametric instrumental variable estimation under monotonicity. Econometrica 85, 1303–1320.CrossRefGoogle Scholar
Delgado, M.A. & Escanciano, J.C. (2012) Distribution-free tests of stochastic monotonicity. Journal of Econometrics 170, 68–75.CrossRefGoogle Scholar
Donald, S.G. & Hsu, Y.-C. (2016) Improving the power of tests of stochastic dominance. Econometric Reviews 35, 553–585.CrossRefGoogle Scholar
Dumbgen, L. & Spokoiny, V. (2001) Multiscale testing of qualitative hypotheses. The Annals of Statistics 29, 124–152.CrossRefGoogle Scholar
Durot, C. (2003) A kolmogorov-type test for monotonicity of regression. Statistics and Probability Letters 63, 425–433.CrossRefGoogle Scholar
Ghosal, S., Sen, A., & van der Vaart, A. (2000) Testing monotonicity of regression. Annals of Statistics 28, 1054–1082.Google Scholar
Gijbels, I., Hall, P., Jones, M., & Koch, I. (2000) Tests for monotonicity of a regression mean with guaranteed level. Biometrika 87, 663–673.CrossRefGoogle Scholar
Gundersen, C., Kreider, B., & Pepper, J. (2012) The impact of the national school lunch program on child health: A nonparametric bounds analysis. Journal of Econometrics 166, 79–91.CrossRefGoogle Scholar
Hall, P. & Heckman, N. (2000) Testing for monotonicity of a regression mean by calibrating for linear functions. Annals of Statistics 28, 20–39.Google Scholar
Hsu, Y.-C. (2017) Consistent tests for conditional treatment effects. Econometrics Journal 20, 1–22.CrossRefGoogle Scholar
Kitagawa, T. (2015) A test for instrument validity. Econometrica 83, 2043–2063.CrossRefGoogle Scholar
Kolmogorov, A.N. & Fomin, S.V. (1957) Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Garylock Press.Google Scholar
Kreider, B. & Hill, S.C. (2009) Partially identifying treatment effects with an application to covering the uninsured. Journal of Human Resources 44, 409–449.CrossRefGoogle Scholar
Kreider, B. & Pepper, J.V. (2007) Disability and employment: reevaluating the evidence in light of reporting errors. Journal of American Statistical Association 102, 432–441.CrossRefGoogle Scholar
Lee, S., Linton, O., & Whang, Y.-J. (2009) Testing for stochastic monotonicity. Econometrica 77, 585–602.Google Scholar
Lee, S., Song, K., & Whang, Y.-J. (2013) Testing functional inequalities. Journal of Econometrics 172, 14–32.CrossRefGoogle Scholar
Lee, S., Song, K., & Whang, Y.-J. (2016) Testing for a general class of functional inequalities. Econometric Theory 34, 1018–1064.CrossRefGoogle Scholar
Manski, C.F. & Pepper, J.V. (2000) Monotone instrumental variables: With an application to the returns to schooling. Econometrica 68, 997–1010.CrossRefGoogle Scholar
Manski, C.F. & Pepper, J.V. (2009) More on monotone instrumental variables. Econometrics Journal 12, S200–S216.CrossRefGoogle Scholar
Manski, C.F. & Tamer, E. (2002) Inference on regressions with interval data on a regressor or outcome. Econometrica 70, 519–546.CrossRefGoogle Scholar
McCarthy, I., Millimet, D.L., & Roy, M. (2015) Bounding treatment effects: A command for the partial identification of the average treatment effect with endogenous and misreported treatment assignment. The Stata Journal 15, 411–436.CrossRefGoogle Scholar
Pollard, D. (1990) Empirical Processes: Theory and Application. CBMS Conference Series in Probability and Statistics, vol. 2. Institute of Mathematical Statistics.Google Scholar
Seo, J. (2015) Tests of Stochastic Monotonicity with Improved Size and Power Properties, Working paper.Google Scholar
Shen, S. (2016) Estimation and inference of distributional partial effects: Theory and applications. Journal of Business and Economic Statistics, forthcoming.Google Scholar
Schlee, W. (1982) Nonparametric tests of the monotony and convexity of regression. In Gnedenko, B.V., Puri, M.L., & Vincze, I. (eds.), Nonparametric Statistical Inference, vol. 2, pp. 110–123. Amsterdam.Google Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes: With Application to Statistics. Springer-Verlag.CrossRefGoogle Scholar
Wang, J. & Meyer, M. (2011) Testing the monotonicity or convexity of a function using regression splines. The Canadian Journal of Statistics 39, 89–107.CrossRefGoogle Scholar
Wolak, F.A. (1991) The local nature of hypothesis tests involving inequality constraints in nonlinear models. Econometrica 59, 981–995.CrossRefGoogle Scholar