Local Estimation in AR Models with Nonparametric ARCH Errors

Abstract

In this article, we consider the problem of estimating the volatility function of a parametric AR(1) model with nonparametric ARCH(1) errors. Consistency and asymptotic normality of local constant and local log-linear estimators are established. Our simulation study and an application to finance lead to superior performance of the local log-linear estimator compared with the conventional treatments in the literature. A possible extension of the estimation procedure is described.

Keywords:

• ARCH errors
• Autoregressive
• Kernel estimation
• Local quasi-likelihood

Mathematics Subject Classification:

• 62M10
• 62G05

Acknowledgments

We thank a referee for a careful reading of the article and very useful comments. We also thank Hongzhi An, Xia Chen, Tiefeng Jiang, and Xiaojing Shen for their help. This work is supported by National Natural Science Foundation of China (No. 10571073) and 985 project of Jilin University.