Comparing null models for testing multifractality in time series

Abstract

The behaviors of fat-tailed distribution, linear long memory, and nonlinear long memory are considered as possible sources of apparent multifractality. Which behavior should be preserved in null models plays an important role in statistical tests of empirical multifractality. In this paper, we compare the performance of two null models on testing the existence of multifractality in fractional Brownian motions (fBm), Markov-switching multifractal (MSM) model, and financial returns. One null model is obtained by shuffling the original data, which keeps the distribution unchanged. The other null model is generated by the iterative amplitude adjusted Fourier transform (IAAFT) algorithm, which insures that the surrogate data and the original data sharing the same distribution and linear long memory behavior. We find that the tests based on the shuffle null model only reject the multifractality in fBm with H = 0.5 and the tests based on the IAAFT null model reject the multifractality in fBms (except for H = 0.1). And the multifractality in MSM and financial returns are significantly supported by the tests based on both null models. Our findings also shed light on the necessity of choosing suitable null models to test multifractality in other complex systems.