Abstract: These exist non-linear trends, heteroscedasticity and dependent relations in some economic and financial data. The heteroscedastic non-parametric regression models with fixed design and dependent errors have been used in these fields because of its ability to reflect these data characteristics. The spline method is one of the commonly used non-parametric smoothing methods. In order to explore the application of the spline method to this kind of models, we discuss the pointwise consistency of polynomial spline estimators of the mean function and variance function under the $\alpha$-mixing condition, and obtain the pointwise rate of convergence. In addition, we carry out the numerical simulation, and the results show that the proposed spline method is feasible.

Key words: nonparametric regression models, spline estimation, consistency, rate of convergence