A lack-of-fit test for quantile regression models with high-dimensional covariates

A new lack-of-fit test for quantile regression models, that is suitable even with high-dimensional covariates, is proposed. The test is based on the cumulative sum of residuals with respect to unidimensional linear projections of the covariates. To approximate the critical values of the test, a wild bootstrap mechanism convenient for quantile regression is used. An extensive simulation study was undertaken that shows the good performance of the new test, particularly when the dimension of the covariate is high. The test can also be applied and performs well under heteroscedastic regression models. The test is illustrated with real data about the economic growth of 161 countries.

[1]  Rand R. Wilcox,et al.  Quantile Regression: A Simplified Approach to a Goodness-of-fit Test , 2008, Journal of Data Science.

[2]  J. Zheng,et al.  A CONSISTENT NONPARAMETRIC TEST OF PARAMETRIC REGRESSION MODELS UNDER CONDITIONAL QUANTILE RESTRICTIONS , 1998, Econometric Theory.

[3]  Xuming He,et al.  A Lack-of-Fit Test for Quantile Regression , 2003 .

[4]  R. Koenker.  Quantile Regression: Fundamentals of Quantile Regression , 2005 .

[5]  Valentin Patilea,et al.  Breaking the curse of dimensionality in nonparametric testing , 2008 .

[6]  Yoon-Jae Whang,et al.  SMOOTHED EMPIRICAL LIKELIHOOD METHODS FOR QUANTILE REGRESSION MODELS , 2004, Econometric Theory.

[7]  Lixing Zhu,et al.  Model diagnosis for parametric regression in high-dimensional spaces , 2008 .

[8]  Chih-Ling Tsai,et al.  Model selection for least absolute deviations regression in small samples , 1990 .

[9]  Winfried Stute,et al.  Nonparametric model checks for regression , 1997 .

[10]  Ingrid Van Keilegom,et al.  Assessing model adequacy in possibly misspecified quantile regression , 2013, Comput. Stat. Data Anal..

[11]  Taisuke Otsu,et al.  Conditional empirical likelihood estimation and inference for quantile regression models , 2008 .

[12]  Eduardo Garc.  A goodness-of-fit test for the functional linear model with scalar response , 2013 .

[13]  E. Mammen.  Bootstrap and Wild Bootstrap for High Dimensional Linear Models , 1993 .

[14]  J. Escanciano.  A CONSISTENT DIAGNOSTIC TEST FOR REGRESSION MODELS USING PROJECTIONS , 2006, Econometric Theory.

[15]  Xingdong Feng,et al.  Wild bootstrap for quantile regression. , 2011, Biometrika.

[16]  R. Koenker,et al.  Goodness of Fit and Related Inference Processes for Quantile Regression , 1999 .

[17]  Manuel Febrero-Bande,et al.  A Goodness-of-Fit Test for the Functional Linear Model with Scalar Response , 2012, 1205.6167.

[18]  J. Zheng,et al.  A consistent test of functional form via nonparametric estimation techniques , 1996 .

[19]  Juan Carlos Escanciano,et al.  Specification Tests of Parametric Dynamic Conditional Quantiles , 2008 .

[20]  Juan Carlos Escanciano,et al.  Specification analysis of linear quantile models , 2014 .

[21]  Joel L Horowitz,et al.  An Adaptive, Rate-Optimal Test of Linearity for Median Regression Models , 2002 .