Bootstrap-based Bias Reduction for the Estimation of the Self-similarity Exponents of Multivariate Time Series

Abstract

Self-similarity has become a well-established modeling framework in several fields of application and its multivariate formulation is of ever-increasing importance in the Big Data era. Multivariate Hurst exponent estimation has thus received a great deal of attention recently, with wavelet eigenvalue-based regression becoming a focal point. The present work tackles the issue of the presence of significant finite-sample bias in wavelet eigenvalue regression stemming from the eigenvalue repulsion effect, whose origin and impact are analyzed and quantified. Furthermore, an original wavelet domain bias reduction technique is developed assuming a single multivariate time series is available. The protocol consists of a bootstrap resampling scheme that preserves the joint covariance structure of multivariate wavelet coefficients. Extensive numerical simulations show that this proposed method is effective in counteracting the bias at the price of a small increase in variance. This leads to wavelet eigenanalysis-based estimation of multivariate Hurst exponents with significantly improved finite-sample performance than earlier state-of-the-art formulations.