Revisiting the relations between Hurst exponent and fractional differencing parameter for long memory

Abstract

This study aims to verify the efficiency of the estimating methods of long memory process based on the linear relationship between the Hurst exponent (H) and the fractional differencing parameter (d), which are the two approaches used to identify the long memory process. By using the Monte Carlo simulation and empirical examinations, the results show that there is a distinct linear relationship between the Hurst exponent and the fractional differencing parameter. This linear relationship may provide an efficient estimation of the range of Hurst exponent and fractional differencing parameter with each other when the stable index of the stable distribution is close to 2. However, according to the findings, the detailed form of linear equation is impacted by the fat tailed distribution, which becomes more volatile as the stable index of stable distribution is less than 2. Therefore, researchers should be cautious when using the linear relation to estimate H and d them by each other when the distribution has fat tail.