Fourth order cumulant study with more FX rates and time windows

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Written by Forex Automaton   
Wednesday, 13 October 2010 14:13

The previous study revealed the positiveness of the fourth-order cumulant among the logarithmic increments for 24-hour 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 fourth-order 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 so-called logarithmic returns (or logarithmic increments) which are the ratios of price levels (low, high, close) to the values they had during the previous 24-hour interval. In this study, the 24-hour intervals (days) are defined to begin and end at 9am Eastern time (New York).

The fourth order cumulant, explained before, measures the so-called 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 stop-loss 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 second-order 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.

PeriodBeginsEnds
12006-03-22 2007-02-15
22007-02-152008-01-08
32008-01-08 2008-12-01
42008-12-012009-10-26
52009-10-26 2010-09-24

Table 1.Time interval definition for the study of temporal stability.

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 half-width 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.

Pearson correlation coefficient for 24-hour high and its forecast, as a function of the optimization parameter 1.1 Pearson correlation coefficient for 24-hour high and its forecast, as a function of the optimization parameter 1.2 Pearson correlation coefficient for 24-hour low and its forecast, as a function of the optimization parameter 1.3

Fig.1. Correlation measures of 2nd and 4th order as a function of the optimization parameter responsible for the forecasting. 1.1: Pearson correlation coefficient between actual and predicted logarithmic increment in 24-hour high for the five independent intervals (see Table 1). 1.2: Pearson correlation coefficient between actual and predicted logarithmic increment in 24-hour close for the five independent intervals. 1.3: Fourth order cumulant among the actual and predicted logarithmic increments in high and low. Numbers from 1 to 5 refer to the time intervals defined in Table 1.

Stability of the plotted figures of merit can be estimated visually. For the 24-hour 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 4-cumulant (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 de-leveraging.

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 4-cumulant 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 money-making 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 4-cumulant, performance of such a strategy should be tied to the positiveness of the 4-cumulant. 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 4-cumulant -- is more reliably positive? Which strategy choice minimizes the risk of picking a wrong value of Fred in the course of its optimization?

Pearson correlation coefficient for 24-hour high and its forecast, as a function of the optimization parameter 2.1 4th order cumulant for 24-hour high, low, and their forecasts, as a function of the optimization parameter 2.2

Fig.2. Correlation measures of 2nd and 4th order as a function of the optimization parameter responsible for the forecasting. Data from Fig.1 are divided by the respective statistical errors (precision of the mean for the 14 forex time series) to compare strengths and stabilities of the predictability effects. Numbers from 1 to 5 refer to the time intervals defined in Table 1.

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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Last Updated ( Saturday, 24 March 2012 11:18 )