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In Example 1, the verification test was presented. The in-sample optimization run was presented in Example 2. In the discussion of results, we are returning to the natural order in which the tests were run, i.e., optimization first, verification second. Optimization Results. Table 4-2 shows the results for the in-sample period. Over the 5 years of data on which the system was optimized, there were 118 trades (n = 118). the mean or average trade yielded about $740.97, and the trades were highly variable, with a sample standard deviation of around +$3,811: i.e., there were many trades that lost several thousand dollars, as well as trades that made many thousands. The degree of profitability can easily be seen by looking at the profit/loss column, which contains many $2,500 losses (the stop got hit) and a significant number of wins, many greater than $5,000, some even greater than $10,000. The expected standard deviation of the mean suggests that if samples of this kind were repeatedly taken, the mean would vary only about one-tenth as
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much as the individual trades, and that many of the samples would have mean profitabilities in the range of $740 + $350. The t-statistic for the best-performing system from the set of optimization mns was 2.1118, which has a statistical significance of 0.0184. This was a fairly strong result. If only one test had been run (no optimizing), this good a result would have been obtained (by chance alone) only twice in 100 tests, indicating that the system is probably capturing some real market inefficiency and has some chance of holding up. However, be warned: This analysis was for the best of 20 sets of parameter values tested. If corrected for the fact that 20 combinations of parameter values were tested, the adjusted statistical significance would only be about 0.3 1, not very good; the performance of the system could easily have been due to chance. Therefore, although the system may hold up, it could also, rather easily, fail. The serial correlation between trades was only 0.0479, a value small enough in the present context, with a significance of only 0.6083. These results strongly suggest that there was no meaningful serial correlation between trades and that the statistical analyses discussed above are likely to be correct. There were 58 winning trades in the sample, which represents about a 49% win rate. The upper 99% confidence boundary was approximately 61% and the lower 99% confidence boundary was approximately 37%, suggesting that the true percentage of wins in the population has a 99% likelihood of being found between those two values. In truth, the confidence region should have been broadened by correcting for optimization; this was not done because we were not very concerned about the percentage of wins. Vetificution Results. Table 4-1, presented earlier, contains the data and statistics for the out-of-sample test for the model. Since all parameters were already fixed, and only one test was conducted, mere was no need to consider optimization or its consequences in any manner. In the period from M/95 to t/1/97, there were 47 trades. The average trade in this sample yielded about $974, which is a greater average profit per trade than in the optimization sample! The system apparently did maintain profitable behavior. At slightly over $6,000, the sample standard deviation was almost double that of the standard deviation in me optimization sample. Consequently, the standard deviation of the sample mean was around $890, a fairly large standard error of estimate; together with the small sample size, this yielded a lower t-statistic than found in the optimization sample and, therefore, a lowered statistical significance of only about 14%. These results were neither very good nor very bad: There is better than an 80% chance that the system is capitalizing on some real (non-chance) market inefficiency. The serial correlation in the test sample, however, was quite a bit higher than in the optimization sample and was significant, with a probability of 0.1572; i.e., as large a serial correlation as this would only be expected about 16% of the time by chance alone, if no true (population) serial correlation was present. Consequently, the t-test on the profit/loss figures has likely
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overstated the statistical significance to some degree (maybe between 20 and 30%). If the sample size was adjusted downward the right amount, the t-test probability would most likely be around 0.18, instead of the 0.1392 that was calculated. The confidence interval for the percentage of wins in the population ranged from about 17% to about 53%. Overall, the assessment is that the system is probably going to hold up in the future, but not with a high degree of certainty. Considering there were two independent tests--one showing about a 31% probability (corrected for optimization) that the profits were due to chance, the other showing a statistical significance of approximately 14% (corrected to 18% due to the serial correlation), there is a good chance that the average population trade is profitable and, consequently, that the system will remain profitable in the future.
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