Wavelet-Based Bootstrap for Time Series Analysis

ABSTRACT

Using wavelet domain analysis and modeling of stochastic processes, we develop a unified bootstrap scheme that can be applied to both short-range dependent and long-range dependent stationary Gaussian time series. Our idea was motivated by the fact that the discrete wavelet transform is capable of converting long-range dependence in the time domain into short-range dependence in the wavelet domain. Hence we can use a simple Markov model to model the short-range dependence in the wavelet domain. The Markov model is used to generate a new version (the bootstrapped version) of the wavelet representation of the time series. The bootstrapped series is obtained by performing the inverse wavelet transform on the new wavelet representation of the original series. We compare our wavelet-based bootstrap with the moving block bootstrap for estimating the standard errors of the unit lag sample autocorrelation and the sample standard deviation. Our results show that the wavelet-based bootstrap can achieve performance better than the moving block bootstrap for both short-range dependent data and long-range dependent data.

Key Words:

• Long memory
• Long-range dependence
• Standard error
• Wavelet

Mathematics Subject Classification:

• 62M10
• 62F40
• 62P05

Acknowledgments

This work benefited from comments by Aparna Gupta and Albert Paulson. The work of the second author was partially supported by National Science Foundation grant DMI9813097.

Notes

See Percival et al. (2001) for a similar simulation study on the standard error of the unit lag sample autocorrelation