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TESTING REGRESSION MONOTONICITY IN ECONOMETRIC MODELS

Published online by Cambridge University Press:  18 September 2018

Denis Chetverikov
Denis Chetverikov*
Affiliation:
Department of Economics, UCLA
*
*Address correspondence to Denis Chetverikov, e-mail: chetverikov@econ.ucla.edu.
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Abstract

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Monotonicity is a key qualitative prediction of a wide array of economic models derived via robust comparative statics. It is therefore important to design effective and practical econometric methods for testing this prediction in empirical analysis. This article develops a general nonparametric framework for testing monotonicity of a regression function. Using this framework, a broad class of new tests is introduced, which gives an empirical researcher a lot of flexibility to incorporate ex ante information she might have. The article also develops new methods for simulating critical values, which are based on the combination of a bootstrap procedure and new selection algorithms. These methods yield tests that have correct asymptotic size and are asymptotically nonconservative. It is also shown how to obtain an adaptive and rate optimal test that has the best attainable rate of uniform consistency against models whose regression function has Lipschitz-continuous first-order derivatives and that automatically adapts to the unknown smoothness of the regression function. Simulations show that the power of the new tests in many cases significantly exceeds that of some prior tests, e.g., that of Ghosal, Sen, and Van der Vaart (2000).

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 4 , August 2019 , pp. 729 - 776
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

Date: First version: March 2012. This version: July 9, 2018. Email: chetverikov@econ.ucla.edu. I thank Victor Chernozhukov for encouragement and guidance. I am also grateful to Anna Mikusheva, Isaiah Andrews, Andres Aradillas-Lopez, Moshe Buchinsky, Glenn Ellison, Jin Hahn, Bo Honore, Rosa Matzkin, Jose Montiel Olea, Ulrich Muller, Whitney Newey, Joris Pinkse, and Jack Porter for valuable comments. The first version of the article was presented at the Econometrics lunch at MIT in April, 2012.

References

REFERENCES

Andrews, D.W.K. & Shi, X. (2013) Inference based on conditional moment inequalities. Econometrica 81, 609–666.Google Scholar
Armstrong, T. (2014) Weighted KS statistics for inference on conditional moment inequalities. Journal of Econometrics 181, 92–116.10.1016/j.jeconom.2014.04.021CrossRefGoogle Scholar
Armstrong, T. & Chan, H. (2016) Multiscale adaptive inference on conditional moment inequalities. Journal of Econometrics 194, 24–43.10.1016/j.jeconom.2016.04.001CrossRefGoogle 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. The Annals of Statistics 33, 214–257.10.1214/009053604000000896CrossRefGoogle 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
Cai, T. & Wang, L. (2008) Adaptive variance function estimation in heteroscedastic nonparametric regression. The Annals of Statistics 36, 2025–2054.10.1214/07-AOS509CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2013) Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. The Annals of Statistics 41, 2786–2819.10.1214/13-AOS1161CrossRefGoogle 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, 47–70.10.1007/s00440-014-0565-9CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., & Kato, K. (2016a) Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings. Stochastic Processes and their Applications 126, 3632–3651.10.1016/j.spa.2016.04.009CrossRefGoogle Scholar
Chernozhukov, V., Chetverikov, D., and Kato, K. (2017) Central limit theorems and bootstrap in high dimensions. The Annals of Probability 45, 2309–2352.10.1214/16-AOP1113CrossRefGoogle Scholar
Chernozhukov, V., Lee, S., & Rosen, A. (2013) Intersection bounds: Estimation and inference. Econometrica 81, 667–737.Google Scholar
Chetverikov, D. (2016) Adaptive test of conditional moment inequalities. Econometric Theory 34, 186–227.10.1017/S0266466617000184CrossRefGoogle Scholar
Delgado, M. & Escanciano, J. (2010) Distribution-free tests of stochastic monotonicity. Journal of Econometrics 170, 68–75.10.1016/j.jeconom.2012.02.005CrossRefGoogle Scholar
Dudley, R. (1999) Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics. Cambridge University Press.10.1017/CBO9780511665622CrossRefGoogle Scholar
Dumbgen, L. & Spokoiny, V. (2001) Multiscale testing of qualitative hypotheses. The Annals of Statistics 29, 124–152.10.1214/aos/996986504CrossRefGoogle Scholar
Durot, C. (2003) A Kolmogorov-type test for monotonicity of regression. Statistics and Probability Letters 63, 425–433.10.1016/S0167-7152(03)00122-6CrossRefGoogle Scholar
Ellison, G. & Ellison, S. (2011) Strategic entry deterrence and the behavior of pharmaceutical incumbents prior to patent expiration. American Economic Journal: Microeconomics 3, 1–36.Google Scholar
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660.10.1093/biomet/85.3.645CrossRefGoogle Scholar
Ghosal, S., Sen, A., & van der Vaart, A. (2000) Testing monotonicity of regression. The 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, 663–673.10.1093/biomet/87.3.663CrossRefGoogle Scholar
Gutknecht, (2016) Testing for monotonicity under endogeneity - An application to the reservation wage function. Journal of Econometrics 190, 100–114.10.1016/j.jeconom.2015.09.002CrossRefGoogle Scholar
Hall, P. & Heckman, N. (2000) Testing for monotonicity of a regression mean by calibrating for linear functions. The Annals of Statistics 28, 20–39.Google Scholar
Hardle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. The Annals of Statistics 21, 1926–1947.10.1214/aos/1176349403CrossRefGoogle Scholar
Hardle, W. & Tsybakov, A. (2007) Local polinomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81, 233–242.Google Scholar
Holm, S. (1979) A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics 6, 65–70.Google Scholar
Horowitz, J.L. & Spokoiny, V. (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599–631.10.1111/1468-0262.00207CrossRefGoogle Scholar
Juditsky, A. & Nemirovski, A. (2002) On nonparametric tests of positivity/monotonicity/convexity. The Annals of Statistics 30, 498–527.10.1214/aos/1021379863CrossRefGoogle Scholar
Lehmann, E.L. & Romano, J. (2005) Testing Statistical Hypotheses. Springer.Google Scholar
Lee, S., Linton, O., & Whang, Y. (2009) Testing for stochastic monotonicity. Econometrica 27, 585–602.Google Scholar
Lee, S., Song, K., and Whang, Y.-J. (2017) Testing for a general class of functional inequalities. Econometric Theory, 1–47.Google Scholar
Liu, R. (1988) Bootstrap procedures under niid models. The Annals of Statistics 16, 1696–1708.10.1214/aos/1176351062CrossRefGoogle Scholar
Mammen, E. (1993) Bootstrap and wild bootstrap for high dimensional linear models. The Annals of Statistics 21, 255–285.10.1214/aos/1176349025CrossRefGoogle Scholar
Matzkin, R. (1994) Restrictions of economic theory in nonparametric methods. Handbook of Econometrics, Volume IV. Edited by Engle, R. and McFadden, D., Elsevier Science, 2523–2558.10.1016/S1573-4412(05)80011-XCrossRefGoogle Scholar
Milgrom, P. & Shannon, C. (1994) Monotone comparative statics. Econometrica 62, 157–180.10.2307/2951479CrossRefGoogle Scholar
Muller, H. & Stadtmuller, U. (1987) Estimation of heteroscedasticity in regression analysis. The Annals of Statistics 15, 610–625.10.1214/aos/1176350364CrossRefGoogle Scholar
Rice, J. (1984) Bandwidth choice for nonparametric kernel regression. The Annals of Statistics 12, 1215–1230.10.1214/aos/1176346788CrossRefGoogle Scholar
Romano, J. & Shaikh, A. (2010) Inference for the identified sets in partially identified econometric models. Econometrica 78, 169–211.Google Scholar
Romano, J. & Wolf, M. (2005a) Exact and approximate stepdown methods for multiple hypothesis testing. Journal of American Statistical Association 100, 94–108.10.1198/016214504000000539CrossRefGoogle Scholar
Romano, J. & Wolf, M. (2005b) Stepwise multiple testing as formalized data snooping. Econometrica 73, 1237–1282.10.1111/j.1468-0262.2005.00615.xCrossRefGoogle Scholar
Romano, J. & Wolf, M. (2013) Testing for monotonicity in expected asset returns. Journal of Empirical Finance 23, 93–116.10.1016/j.jempfin.2013.05.001CrossRefGoogle Scholar
Schlee, (1982) Nonparametric tests of the monotonicity and convexity of regression. Nonparametric Statistical Inference, Volume II. Edited by Gnedenko, B., Puri, M., and Vincze, I., North-Holland, 823–836.Google Scholar
Tsybakov, A. (2009) Introduction to Nonparametric Estimation. Springer.10.1007/b13794CrossRefGoogle Scholar
van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes with Applications to Statistics. Springer.10.1007/978-1-4757-2545-2CrossRefGoogle Scholar
Wang, J. & Meyer, M. (2011) Testing the monotonicity or convexity of a function using regression splines. The Canadian Journal of Statistics 39, 89–107.10.1002/cjs.10094CrossRefGoogle Scholar
Wu, C. (1986) Jacknife, bootstrap, and other resampling methods in regression analysis. The Annals of Statistics 14, 1261–1295.10.1214/aos/1176350142CrossRefGoogle Scholar