Spline confidence bands for variance functions in nonparametric time series regressive models

Abstract

For nonparametric time series regression, we propose to apply polynomial splines to squared residuals to develop the variance function estimation. Furthermore, we obtain and use simultaneous confidence bands to detect certain parametric forms for entire variance curves. The proposed method is extremely fast. Asymptotic results are established under the assumption that observations are from a strictly stationary $\alpha$-mixing process. Simulations and a financial data set application are provided to illustrate the performance of the proposed method numerically.

Keywords:

• B-splines
• variance function estimation
• confidence bands
• α-mixing process
• time series
• nonparametric autoregression

AMS Subject Classification::

• 62G05
• 62G15
• 62M10
• 62G08

Acknowledgements

The authors are grateful for the many constructive suggestions from two referees, the associate editor and the editor.