Improved Estimates For The Rescaled Range And Hurst Exponents

The error of statistical volatility of intra-daily quoted price changes observed over a time interval,Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis', Jounal of Banking and standard deviation to correct this bias when the observed time scale is large enough. When the time scale is small, however, the R= ~ S statistic introduces new estimation errors due to negative biases in ~ S. This eeect also results in downward errors in estimated Hurst exponents. Our proposed R=S statistic overcomes deeciencies in both R=S and R= ~ S correcting for short term dependencies in the time series without introducing additional biases on short time scales N. (1995b)), there exist very signiicant one or two tick anti-correlations in the returns of the DEM/USD series. When analyzing the price behavior and forecasting price changes on longer time scales, such short-term anti-correlations should be removed. We have demonstrated in this paper that not considering these eeects results in completely diierent conclusions about the behavior of the series as measured by the Hurst exponents on time scales of 10 to 100 ticks. As shown in (Moody & Wu (1995b)), simply down-sampling the price series cannot removed this short-term anti-correlation. However, our proposed R=S ](N) analysis connrms our previous results obtained by short term block averaging that the DEM/USD series is actually mildly trending on time scales of 10 to 100 ticks, and that the suggested mean-reversion in the R=S and R= ~ S analyses on these time scales is spurious. 7. Acknowledgements We thank Steve Rehfuss for valuable comments and discussions. We gratefully acknowledge support for this work from ARPA and ONR under grant N00014-92-J-4062 and NSF under grant CDA-9309728. On the trend-following behavior of intra-daily foreign exchange market and its relationship with the volatility,Preservation of the rescaled adjusted range. mean-reversion on these time scales in the R=S](N) curves is actually due to high frequency oscillations on time scales of a few ticks.