On Hurst exponent estimation under heavy-tailed distributions

Abstract

In this paper, we show how the sampling properties of the Hurst exponent methods of estimation change with the presence of heavy tails. We run extensive Monte Carlo simulations to find out how rescaled range analysis (R/S), multifractal detrended fluctuation analysis (MF-DFA), detrending moving average (DMA) and generalized Hurst exponent approach (GHE) estimate Hurst exponent on independent series with different heavy tails. For this purpose, we generate independent random series from stable distribution with stability exponent {\alpha} changing from 1.1 (heaviest tails) to 2 (Gaussian normal distribution) and we estimate the Hurst exponent using the different methods. R/S and GHE prove to be robust to heavy tails in the underlying process. GHE provides the lowest variance and bias in comparison to the other methods regardless the presence of heavy tails in data and sample size. Utilizing this result, we apply a novel approach of the intraday time-dependent Hurst exponent and we estimate the Hurst exponent on high frequency data for each trading day separately. We obtain Hurst exponents for S&P500 index for the period beginning with year 1983 and ending by November 2009 and we discuss the surprising result which uncovers how the market's behavior changed over this long period

A COMPARISON OF THREE HURST EXPONENT APPROACHES TO PREDICT NASCENT BUBBLES IN S&P500 STOCKS

“…. , τ max } is the number of considered periods (see [4,7,14]). Further, it is also satisfied that statistical properties for X(t) scale with the time window resolution.…”

Asymptotic scaling properties and estimation of the generalized Hurst exponents in financial data

“…Many estimation methods are present in the literature; in particular the most popular are Multifractal Detrended Fluctuation Analyisis (MFDFA) [37], the Generalized Hurst Exponent Method (GHE) [4,[38][39][40], and Wavelet Transform Modulus Maxima (WWTM) [41]. All of them have advantages and drawbacks: MFDFA, which measures the scaling of the so-called fluctuation function, is applicable to nonstationary time series but the degree of the detrending method is arbitrary; GHE computes directly the scaling of the moments with respect to the aggregation horizon but the measurements are aggregation horizon dependent; and WWTM has a deep mathematical formulation which makes a parallelism with the thermodynamic and computes the scaling of a partition function defined in terms of WTMM coefficients but the choice of the wavelet function is arbitrary again.…”

Multifractal Analysis of Hydrologic Data Using Wavelet Methods and Fluctuation Analysis

“…Long-range dependence is characterized by the Hurst parameter, which describes the intensity of long memory phenomena. Many studies [3][4][5][6] for estimating the Hurst parameter (or Hurst exponent) to judge the correlation effect have been made available in recent years. These studies reveal that, in many areas of applied sciences, such as climate change, stock markets, telecommunication network, and river flow, the phenomena appear as long-range behaviour, and the estimation of Hurst has been widely used in decisions and predictions.…”