Testing symmetry of model errors for nonparametric regression models by using correlation coefficient¹

Abstract

In this paper, we propose a residuals based estimator of k-th correlation coefficient between the density function and distribution function of a continuous variable for nonparametric regression models, and further we use this k-th correlation coefficient to test whether the density function of the true model error is symmetric or not. First, we propose a moment based estimator of k-th correlation coefficient and present its asymptotic results. Second, we consider statistical inference of k-th correlation coefficient by using the empirical likelihood method, and the empirical likelihood statistic is shown to be asymptotically distributed as Chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators.

Keywords:

• Correlation coefficient
• Empirical likelihood
• Kernel smoothing
• Residuals

Additional information

Funding

The authors thank the editor, the associate editor and two referees for their constructive suggestions that helped them to improve the early manuscript. Jun Zhang’s research was supported by the National Natural Science Foundation of China (Grant No. 11871411). Yujie Gai’s research was supported by Program for Innovation Research in Central University of Finance and Economics. Xuehu Zhu’s research is supported by the National Natural Sciences Foundation of China (Grant Nos. 11601415, 61877049). Xuehu Zhu's research is also supported by the China Postdoctoral Science Foundation (Grant Nos. 2016M590934, 2017T100731), and supported by the Fundamental Research Funds for the Central Universities.