Efficient estimation of the Hurst parameter in high frequency financial data with seasonalities using wavelets

S&P 500 Index data taken at one-minute intervals over the course of 11.5 years (January 1989- May 2000) is analyzed, and in particular the Hurst parameter over segments of stationarity (the time period over which the Hurst parameter is almost constant) is estimated. (The segments of stationarity are a byproduct of our analysis, no prior assumption about it is made.) An asymptotically efficient estimator using the log-scale spectrum is employed. This estimator is robust to additive non-stationarities, and it is shown to be robust to multiplicative non-stationarities, i.e. seasonalities, as well. Analyzing cumulative sums of returns, rather than the returns themselves, is essential in removing the effect of seasonalities. It is shown that it is necessary to use wavelets with at least two vanishing moments for the analysis in order to achieve this robustness. This analysis shows that the market has become more efficient since 1997.