

Fourth order cumulant study with more FX rates and time windows 
Written by Forex Automaton  
Wednesday, 13 October 2010 14:13  
The previous study revealed the positiveness of the fourthorder cumulant among the logarithmic increments for 24hour highs and lows in EUR/USD and the respective ForexAutomaton forecasts. By expanding the scope of the study to include all of the 14 most popular exchange rates, and by splitting the time span of the simulated trading into five independent intervals, I demonstrate that the result is not just a feature of EUR/USD and is stable in time. The data hint at a correlation between the fourthorder cumulant under study and predictability of close (measured by Pearson correlation coefficient between predicted and actual logarithmic returns). However, the signal strength for these figures of merit, defined as the ratio of the value in question to the estimated precision of its measurement, appears to lend more credibility to the cumulant. As always, the quantities we are going to look at are not the actual low, high and close. Since prices are always positive, they are trivially correlated; this feature is absent in the correlations of the socalled logarithmic returns (or logarithmic increments) which are the ratios of price levels (low, high, close) to the values they had during the previous 24hour interval. In this study, the 24hour intervals (days) are defined to begin and end at 9am Eastern time (New York). The fourth order cumulant, explained before, measures the socalled genuine correlation of four quantities. These quantities are the actual logarithmic returns for day's low, high, and the respective predicted logarithmic returns. A cumulant can be positive, negative, or zero. Given the way these four quantities are defined, a positive cumulant means that the model operates in a regime when there is a "correlation" between its stoploss not being hit and its profit target being hit within the same trade, if these are set at the past lows and highs as has been discussed numerous times and implemented in Demi. This is different from a situation when there is a positive correlation ("predictability") between a change in the day's low and its forecast, and there is a separate and independent positive correlation between a change in day's high and its forecast. These latter ones are secondorder correlations measured by Pearson correlation coefficients. One of the key observations related to the Pearson correlation coefficients was the high stability of such coefficients in time for daily high and low. In the same study, predictability of close has been found to be a lot less stable. The study was done by splitting simulated trading time into five equal intervals and comparing the results. Now with four more months of data, we redefine the intervals. See Table 1 for new definitions.
The 14 popular exchange rates are the same as those tracked by the Danica and Heidi forecasting systems: AUD/USD, EUR/USD, GBP/USD, USD/CAD, USD/CHF, USD/JPY, AUD/JPY, CHF/JPY, EUR/AUD, EUR/CHF, EUR/GBP, EUR/JPY, GBP/CHF, and GBP/JPY. The data are aggregated and visualized as profile histograms. These profile histograms are shown below. The data are shown as colored bands where the middle of the band corresponds to the measured value and the halfwidth of the band  to the precision of its measurement, estimated by comparing measurements for the 14 exchange rates. Plotted along the horizontal axis is the quantity nicknamed Fred, the only adjustable parameter in the system responsible for the forecasting quality. 1.1 1.2 1.3 Stability of the plotted figures of merit can be estimated visually. For the 24hour extremes (only data for highs are shown in Fig.1.1), the Pearson correlation coefficient is both positive with very high degree of certainty and highly stable. The plots for close (Fig.1.2) and the 4cumulant (Fig.1.3) look similar: positiveness is not guaranteed. Time interval number 3, the one which happens to contain the financial panic of 2008, truly stands out in the plots due to its highly predictable dynamics driven by global deleveraging. There are some similarities between Fig.1.2 and Fig.1.3: time interval 3 is the best in both cases, intervals 2 and 4 (in this order in both cases) are the next best ones, while 1 and 5 seem to be the worst. This is interesting given that the Pearson coefficient for daily close and the 4cumulant for daily low and high deal with different aspects of the day candle. The way these correlaton measures rise and fall with time (as see from their dependence on the time interval number) looks consistent with being cyclic, with the panic of 2008 certainly marking one of the extremes of the cycle. Whether this is an accidental feature or some peculiar reflection of the global business cycle, remains an open question. It is also not clear what the period of this cycle is  what we are seeing is a chunk of data which looks like it could belong to a cycle, with periods 1 and 5 roughly corresponding to each other. A moneymaking strategy can not be formulated on the basis of either high or low alone (at least in the spot market), therefore Fig.1.1 is largely academic. A strategy can be formulated for low and high together, and a portfolio model implementing such a strategy, Demi, is currently accessible to registered clients. Based on the above remarks concerning the meaning of the 4cumulant, performance of such a strategy should be tied to the positiveness of the 4cumulant. Likewise, a strategy can be formulated on the basis of daily close and its performance will be tied to the positiveness of the Pearson for close. An important question is: which one  Pearson for close or the 4cumulant  is more reliably positive? Which strategy choice minimizes the risk of picking a wrong value of Fred in the course of its optimization? 2.1 2.2 To address these questions, I divide the data in panels 1.2 and 1.3 by the statistical error estimate. In the new units, the statistical error is always 1. Then I look at the newly renormalized quantities, Fig.2. The impression is that the cumulant gives a more consistent performance. This impression is mainly due to the sharp difference between cumulant and Pearson performance for time interval 1, and somewhat different results for time interval 2  both cases favoring the cumulant. In case of cumulant, there is also more consistency in the way various values of Fred perform, especially for the "good" time intervals 2 and 3. Thus, with the data so far, Demi looks more sound than a hypothetic strategy based on staying in the market from close to close. 

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