Statistica Sinica 22 (2012), 1075-1101

NEW EFFICIENT AND ROBUST ESTIMATION IN

VARYING-COEFFICIENT MODELS

WITH HETEROSCEDASTICITY

Jie Guo$^1$, Maozai Tian$^1$ and Kai Zhu$^2$

$^1$Renmin University of China and $^2$Chinese Academy of Sciences

Abstract: Varying-coefficient models with heteroscedasticity are considered in this paper. Based on local composite quantile regression, we propose a new estimation method to estimate the coefficient functions and heteroscedasticity simultaneously. Moreover, we can get the estimated conditional quantile curves of the error part. The conditional biases, variances, and asymptotic normalities of these estimators are studied explicitly. A simple and quick plug-in bandwidth selector is employed to select the optimal bandwidth. The estimators of the coefficient functions perform efficiently and robustly regardless of the error distributions. When the error $\varepsilon$ follows a non-normal distribution, the proposed estimators of the coefficient functions are much more efficient than local polynomial weighted least squares estimators and almost as efficient for normal random errors. The estimator of heteroscedasticity also outperforms other classical estimators in the literature. A goodness-of-fit test based on a bootstrap procedure is proposed to test whether the coefficient functions are actually varying. Both simulations and data analysis are used to illustrate the proposed method.

Key words and phrases: Goodness-of-fit test, heteroscedasticity, local composite quantile regression, plug-in bandwidth selector, varying-coefficient models.