Could Prediction Error Help in Fractal Analysis of Time Series?

Fractal analysis provides important characterization of any stock market. Time series can be considered as a sample of fractal Brownian motion (fBm). In the case of European stock market indices, logarithmic transform of daily values is necessary as preprocessing. Original fBm is traditionally converted to fractional Gaussian noise (fGn) by differentiation and then analysed in frequency and time domains. Resulting Hurst exponent is a measure of stock fractal behaviour. Novel approach to Hurst exponent estimation is based on fGn generation using forward or backward linear predictors or symmetric smoother, respectively. Prediction error is supposed to be fGn and analysed using various techniques. The paper evaluates efficiency of proposed methods and compares them with traditional one. The second aim is to compare individual stock market indices and interpret their fractal behaviour.