The Hybrid Wild Bootstrap for Time Series

Abstract

We introduce a new and simple bootstrap procedure for general linear processes, called the hybrid wild bootstrap. The hybrid wild bootstrap generates frequency domain replicates of the periodogram that imitate asymptotically correct the first- and second-order properties of the ordinary periodogram including its weak dependence structure at different frequencies. As a consequence, the hybrid wild bootstrapped periodogram succeeds in approximating consistently the distribution of statistics that can be expressed as functionals of the periodogram, including the important class of spectral means for which all so far existing frequency domain bootstrap methods generally fail. Moreover, by inverting the hybrid wild bootstrapped discrete Fourier transform, pseudo-observations in the time domain are obtained. The generated time domain pseudo-observations can be used to approximate correctly the random behavior of statistics, the distribution of which depends on the first-, second-, and, to some extent, on the fourth-order structure of the underlying linear process. Thus, the proposed hybrid wild bootstrap procedure applied to general time series overcomes several of the limitations of standard linear time domain bootstrap methods.

KEY WORDS:

• Bootstrap
• Linear process
• Nonparametric kernel estimation
• Periodogram
• Spectral means

Acknowledgments

2000 Mathematics Subject Classification: Primary 62M10, 62M15; Secondary 62G09.

The authors express their sincere gratitude to the Associate Editor and the two anonymous referees for their careful reading of the previous versions of the article and their thorough reports that led to a considerable improvement in the present article.