Wavelet-based bootstrap for time series analysis

The bootstrap is a popular data resampling method for statistical analysis. Although it was originally developed for independent and identically distributed data, in recent years several bootstrap methods for short-range dependent data have been introduced. However, more complex statistical characteristics such as long-range dependence, the mixture of short-range dependence and long-range dependence, and non-Gaussian marginal probability density functions found in random processes pose challenges for these bootstrap methods and call for new bootstrap schemes.; In this research, based on 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 time series with either Gaussian or non-Gaussian marginal probability density functions. Our idea was motivated by the observation that the discrete wavelet transform is capable of decorrelating long-range dependence in the time domain into short-range dependence in the wavelet domain. Hence we can use simple Markov models to model the short-range dependence in the wavelet domain. Once a suitable wavelet model is identified and its parameters are properly estimated, we can use the established model 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. Our bootstrap scheme can be easily adapted to time series with non-Gaussian marginal probability density functions.; We compare our wavelet-based bootstrap with the moving block bootstrap in estimating the standard error of the unit lag sample autocorrelation and the standard error of the sample standard deviation under short-range dependence and long-range dependence. Our results show that the wavelet-based bootstrap can achieve performance similar to that of other bootstrap schemes under short-range dependence and has better performance for long-range dependent processes.; We also apply the wavelet-based bootstrap to financial risk analysis. In particular, we develop new simulation schemes with better performance for Value-at-Risk estimation and options pricing.