Journal of Mathematics and Statistics


Wei Xiong and Maozai Tian

DOI : 10.3844/jmssp.2014.169.185

Journal of Mathematics and Statistics

Volume 10, Issue 2

Pages 169-185


One of the greatest values of Quantile Regression (QR) is that it provides a good procedure in the sense that QR could be much more efficient and sometimes arbitrarily more efficient in recovering the mean function than the Least Squares (LS) even when without moment conditions. However, heteroscedasticity definitely causes conditional variances of parametric or nonparametric estimates of mean functions to be large, sometimes this may lead to a great loss of efficiency of estimators and affect the goodness-of-fit test substantially and pratically conditional variance of data is of more concerned in statistical analysis these days, thus detecting heteroscedasticity before further analysis becomes essential. The virtue of QR as well as the limitation of LS motivates us to develop a new robust detecting tool for heteroscedasticity. Main contributions of this study include three aspects: First of all, a new Dynamic Quantile Regression (DQR) is introduced. Based on this method estimators for mean function, heteroscedastic function and the error distribution can be obtained simultaneously. Second, a novel diagnostic tool is developed for checking heteroscedasticity by employing the hybrid of QR and DQR. Theoretical properties of the procedure are investigated and we also demonstrate the performance of the new tool on small sample power properties. Third, further estimator of the conditional variance can be obtained based on improved DQR, when heteroscedasticity is detected. Finally these methods are illustrated with some simulated examples. Compared with the classical testing procedures, Monte Carlo simulations indicate that the new tool is more effective, powerful and easy to implement. Applications to a real data analysis is also discussed.


© 2014 Wei Xiong and Maozai Tian. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.