## Electronic Journal of Statistics

### Significance testing in quantile regression

#### Abstract

We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^{2}$-distances - the new test is able to detect local alternatives which converge to the null hypothesis with any rate $a_{n}\to 0$ such that $a_{n}\sqrt{n}\to\infty$ (here $n$ denotes the sample size). We also present a small simulation study illustrating the finite sample properties of a bootstrap version of the corresponding Kolmogorov-Smirnov test.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 105-145.

Dates
First available in Project Euclid: 24 January 2013

https://projecteuclid.org/euclid.ejs/1359041587

Digital Object Identifier
doi:10.1214/12-EJS765

Mathematical Reviews number (MathSciNet)
MR3020416

Zentralblatt MATH identifier
1337.62084

#### Citation

Volgushev, Stanislav; Birke, Melanie; Dette, Holger; Neumeyer, Natalie. Significance testing in quantile regression. Electron. J. Statist. 7 (2013), 105--145. doi:10.1214/12-EJS765. https://projecteuclid.org/euclid.ejs/1359041587

#### References

• Belloni, A. and Chernozhukov, V. (2011). $\ell_1$-penalized quantile regression in high-dimensional sparse models., Annals of Statistics, 39(1):82–130.
• Bondell, H. D., Reich, B. J., and Wang, H. (2010). Noncrossing quantile regression curve estimation., Biometrika, 97(4):825–838.
• Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation., Annals of Statistics, 19:760–777.
• Chernozhukov, V., Fernandéz-Val, I., and Galichon, A. (2010). Quantile and probability curves without crossing., Econometrica, 78(3):1093–1125.
• Delgado, M. A. and González-Manteiga (2001). Significance testing in nonparametric regression., Annals of Statistics, 29:1469–1507.
• Dette, H., Neumeyer, N., and Pilz, K. F. (2006). A simple nonparametric estimator of a strictly monotone regression function., Bernoulli, 12:469–490.
• Dette, H. and Volgushev, S. (2008). Non-crossing nonparametric estimates of quantile curves., Journal of the Royal Statistical Society, Ser. B, 70(3):609–627.
• Fan, J. and Gijbels, I. (1996)., Local Polynomial Modelling and its Applications. Chapman & Hall.
• Fan, Y. and Li, Q. (1996). Consistent model specification tests: Omitted variables and semiparametric functional forms., Econometrica, 64:865–890.
• Feng, X., He, X., and Hu, J. (2011). Wild bootstrap for quantile regression., Biometrika, 98(4):995–999.
• Gasser, T., Müller, H., and Mammitzsch, V. (1985). Kernels for nonparametric curve estimation., Journal of the Royal Statistical Society. Series B (Methodological), pages 238–252.
• Gozalo, P. L. (1993). A consistent model specification test for nonparametric estimation of regression function models., Econometric Theory, 9:451–577.
• Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods., Econometrica, 37:424–438.
• He, X. and Zhu, L. (2003). A lack-of-fit test for quantile regression., Journal of the American Statistical Association, 98:1013–1022.
• Jeong, K., Härdle, W. K., and Song, S. (2012). A consistent nonparametric test for causality in quantile., Econometric Theory, 3:1–27.
• Lavergne, P. and Vuong, Q. H. (1996). Nonparametric selection of regressors: The nonnested case., Econometrica, 64:207–219.
• Lavergne, P. and Vuong, Q. H. (2000). Nonparametric significance testing., Econometric Theory, 16:576–601.
• Neumeyer, N. and Van Keilegom, I. (2010). Estimating the error distribution in nonparametric multiple regression with applications to model testing., Journal of Multivariate Analysis, 101:1067–1078.
• Racine, J. (1997). Feasible cross-validatory model selection for general stationary processes., Journal of Applied Econometrics, 12:169–179.
• Rice, J. (1984). Bandwidth choice for nonparametric regression., The Annals of Statistics, 12(4):1215–1230.
• Sun, Y. (2006). A consistent nonparametric equality test of conditional quantile functions., Econometric Theory, 22:614–632.
• Takeuchi, I., Le, Q. V., Sears, T. D., and Smola, A. J. (2006). Nonparametric quantile regression., Journal of Machine Learning Research, 7:1231–1264.
• van der Vaart, A. W. and Wellner, J. A. (1996)., Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer, New York.
• Volgushev, S. (2006)., Nonparametric quantile regression for censored data. PhD thesis, Fakultät für Mathematik, Ruhr-Universität Bochum, Germany.
• Wu, Y. and Liu, Y. (2009). Variable selection in quantile regression., Statistica Sinica, 19:801–817.
• Yatchew, A. J. (1992). Nonparametric regression tests based on least squares., Econometric Theory, 8:435–451.
• Yu, K. and Jones, M. C. (1997). A comparison of local constant and local linear regression quantile estimators., Computational Statistics and Data Analysis, 25:159–166.
• Yu, K. and Jones, M. C. (1998). Local linear quantile regression., Journal of the American Statistical Association, 93:228–237.
• Zheng, J. X. (1998). A consistent test of parametric regression models under conditional quantile restrictions., Econometric Theory, 14:123–138.
• Zou, H. and Yuan, M. (2008). Composite quantile regression and the oracle model selection theory., Annals of Statistics, 36:1108–1126.