Skip to main content
×
×
Home

TESTING REGRESSION MONOTONICITY IN ECONOMETRIC MODELS

  • Denis Chetverikov (a1)
Abstract

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).

Copyright
Corresponding author
*Address correspondence to Denis Chetverikov, e-mail: chetverikov@econ.ucla.edu.
Footnotes
Hide All

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.

Footnotes
References
Hide All
Andrews, D.W.K. & Shi, X. (2013) Inference based on conditional moment inequalities. Econometrica 81, 609666.
Armstrong, T. (2014) Weighted KS statistics for inference on conditional moment inequalities. Journal of Econometrics 181, 92116.
Armstrong, T. & Chan, H. (2016) Multiscale adaptive inference on conditional moment inequalities. Journal of Econometrics 194, 2443.
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, 214257.
Bowman, A.W., Jones, M.C., & Gijbels, I. (1998) Testing monotonicity of regression. Journal of Computational and Graphical Statistics 7, 489500.
Cai, T. & Wang, L. (2008) Adaptive variance function estimation in heteroscedastic nonparametric regression. The Annals of Statistics 36, 20252054.
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, 27862819.
Chernozhukov, V., Chetverikov, D., & Kato, K. (2015) Comparison and anti-concentration bounds for maxima of Gaussian random vectors. Probability Theory and Related Fields 162, 4770.
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, 36323651.
Chernozhukov, V., Chetverikov, D., and Kato, K. (2017) Central limit theorems and bootstrap in high dimensions. The Annals of Probability 45, 23092352.
Chernozhukov, V., Lee, S., & Rosen, A. (2013) Intersection bounds: Estimation and inference. Econometrica 81, 667737.
Chetverikov, D. (2016) Adaptive test of conditional moment inequalities. Econometric Theory 34, 186227.
Delgado, M. & Escanciano, J. (2010) Distribution-free tests of stochastic monotonicity. Journal of Econometrics 170, 6875.
Dudley, R. (1999) Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics. Cambridge University Press.
Dumbgen, L. & Spokoiny, V. (2001) Multiscale testing of qualitative hypotheses. The Annals of Statistics 29, 124152.
Durot, C. (2003) A Kolmogorov-type test for monotonicity of regression. Statistics and Probability Letters 63, 425433.
Ellison, G. & Ellison, S. (2011) Strategic entry deterrence and the behavior of pharmaceutical incumbents prior to patent expiration. American Economic Journal: Microeconomics 3, 136.
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645660.
Ghosal, S., Sen, A., & van der Vaart, A. (2000) Testing monotonicity of regression. The Annals of Statistics 28, 10541082.
Gijbels, I., Hall, P., Jones, M., & Koch, I. (2000) Tests for monotonicity of a regression mean with guaranteed level. Biometrika 87, 663673.
Gutknecht, (2016) Testing for monotonicity under endogeneity - An application to the reservation wage function. Journal of Econometrics 190, 100114.
Hall, P. & Heckman, N. (2000) Testing for monotonicity of a regression mean by calibrating for linear functions. The Annals of Statistics 28, 2039.
Hardle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. The Annals of Statistics 21, 19261947.
Hardle, W. & Tsybakov, A. (2007) Local polinomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics 81, 233242.
Holm, S. (1979) A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics 6, 6570.
Horowitz, J.L. & Spokoiny, V. (2001) An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69, 599631.
Juditsky, A. & Nemirovski, A. (2002) On nonparametric tests of positivity/monotonicity/convexity. The Annals of Statistics 30, 498527.
Lehmann, E.L. & Romano, J. (2005) Testing Statistical Hypotheses. Springer.
Lee, S., Linton, O., & Whang, Y. (2009) Testing for stochastic monotonicity. Econometrica 27, 585602.
Lee, S., Song, K., and Whang, Y.-J. (2017) Testing for a general class of functional inequalities. Econometric Theory, 147.
Liu, R. (1988) Bootstrap procedures under niid models. The Annals of Statistics 16, 16961708.
Mammen, E. (1993) Bootstrap and wild bootstrap for high dimensional linear models. The Annals of Statistics 21, 255285.
Matzkin, R. (1994) Restrictions of economic theory in nonparametric methods. Handbook of Econometrics, Volume IV. Edited by Engle, R. and McFadden, D., Elsevier Science, 25232558.
Milgrom, P. & Shannon, C. (1994) Monotone comparative statics. Econometrica 62, 157180.
Muller, H. & Stadtmuller, U. (1987) Estimation of heteroscedasticity in regression analysis. The Annals of Statistics 15, 610625.
Rice, J. (1984) Bandwidth choice for nonparametric kernel regression. The Annals of Statistics 12, 12151230.
Romano, J. & Shaikh, A. (2010) Inference for the identified sets in partially identified econometric models. Econometrica 78, 169211.
Romano, J. & Wolf, M. (2005a) Exact and approximate stepdown methods for multiple hypothesis testing. Journal of American Statistical Association 100, 94108.
Romano, J. & Wolf, M. (2005b) Stepwise multiple testing as formalized data snooping. Econometrica 73, 12371282.
Romano, J. & Wolf, M. (2013) Testing for monotonicity in expected asset returns. Journal of Empirical Finance 23, 93116.
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, 823836.
Tsybakov, A. (2009) Introduction to Nonparametric Estimation. Springer.
van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes with Applications to Statistics. Springer.
Wang, J. & Meyer, M. (2011) Testing the monotonicity or convexity of a function using regression splines. The Canadian Journal of Statistics 39, 89107.
Wu, C. (1986) Jacknife, bootstrap, and other resampling methods in regression analysis. The Annals of Statistics 14, 12611295.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed