Polynomial spline estimation for nonparametric (auto-)regressive models

In this paper we consider nonparametric (auto-)regressive models with conditional variance function. Based on polynomial spline estimation, we construct the estimators of conditional mean function and conditional variance function. Consistency of these estimators is proved under the alpha-mixing condition. Meanwhile, uniform and global rates of convergence of these estimators are established, and global rates of convergence can attain to the optimal ones. Moreover, a method for selecting knot number is developed based on Bayes Information Criterion. The methodology is illustrated by a simulation study and is applied to CNY/USD middle exchange rate.