Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series

The detection of long range dependence (LRD) is an important task in time series analysis. LRD is often summarized by the well-known Hurst parameter (or exponent) H∈[0,1], which can be estimated by a number of methods. Some of these techniques are designed to be applied to signals behaving as a stationary fractional Gaussian noise (fGn), whereas others imply that the analyzed time series behave as a non-stationary fractional Brownian motion (fBm). Moreover, some estimators do not yield the Hurst parameter but indexes related to H and ranging outside the unit interval. Therefore, the fGn or fBm nature of the studied time series has to be preliminarily analyzed before applying any estimation method, and the relationships between H and the indexes resulting from the analyses have to be taken into account to obtain coherent results. Since fGn-like series represent the increments of fBm-like processes and both the signals are characterized by the same H value by definition, estimators designed for fGn-like series can be applied to fBm-like sequences after preventive differentiation, and conversely estimators designed for fBm-like processes can be applied to fGn-like series after preventive integration. The signal characterization is particularly important when H is estimated on financial time series because the returns represent the first difference of price time series, which are often assumed to behave like self-affine sequences. The analysis of simulated fGn and fBm time series shows that all the considered methods yield comparable H values when properly applied. The reanalysis of several market price time series already studied in the literature points out that a correct application of the estimators (supported by a preventive signal classification) yields homogeneous H values allowing for a useful cross-validation of results reported in different works. Moreover, some conclusions reported in the literature about the anti-persistence of some financial series are shown to be incorrect because of the inappropriate application of the estimation methods.