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Example 2: Evaluating the In-Sample Tests
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How can a system that has been fit to a data sample by the repeated adjustment of parameters (i.e., an optimized system) be evaluated Traders frequently optimize systems to obtain good results. In this instance, the use of statistics is more important than ever since the results can be analyzed, compensating for the multiplicity of tests being performed as part of the process of optimization. Table 4-2 contains the profit/loss figures and a variety of statistics for the in-sample trades (those
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taken on the data sample used to optimize the system). The system was optimized on data from l/1/90 through l/2/95. Most of the statistics in Table 4-2 are identical to those in Table 4-1, which was associated with Example 1. Two additional statistics (that differ from those in the first example) are labeled Optimization Tests Run and Adjusted for Optimization. The first statistic is simply the number of different parameter combinations tried, i.e., the total number of times the system was run on the data, each time using a different set of parameters. Since the lunar system parameter, LI, was stepped from 1 to 20 in increments of 1, 20 tests were performed; consequently, there were 20 t-statistics, one for each test. The number of tests mn is used to make an adjustment to the probability or significance obtained from the best t-statistic
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TABLE 4-2
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Trades from the S&P 500 Data Sample on Which the Lunar Model Was Optimized
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computed on the sample: Take 1, and subtract from it the statistical significance obtained for the best-performing test. Take the resultant number and raise it to the mth power (where m = the number of tests mn). Then subtract that number from 1. This provides the probability of finding, in a sample of m tests (in this case, 20), at least one t-statistic as good as the one actually obtained for the optimized solution. The uncorrected probability that the profits observed for the best solution were due to chance was less than 2%, a fairly significant result, Once adjusted for multiple tests, i.e., optimization, the statistical significance does not appear anywhere near as good. Results at the level of those observed could have been obtained for such an optimized system 3 1% of the time by chance alone. However, things are not quite as bad as they seem. The adjustment was extremely conservative and assumed that every test was completely independent of every other test. In actual fact, there will be a high serial correlation between most tests since, in many trading systems, small changes in the parameters produce relatively small changes in the results. This is exactly like serial dependence in data samples: It reduces the effective population size, in this case, the effective number of tests run. Because many of the tests are correlated, the 20 actual tests probably correspond to about 5 to 10 independent tests. If the serial dependence among tests is considered, the adjusted-for-optimization probability would most likely be around 0.15, instead of the 0.3 104 actually calculated. The nature and extent of serial dependence in the multiple tests are never known, and therefore, a less conservative adjustment for optimization cannot be directly calculated, only roughly reckoned. Under certain circumstances, such as in multiple regression models, there are exact mathematical formulas for calculating statistics that incorporate the fact that parameters are being tit, i.e., that optimization is occurring, making corrections for optimization unnecessary.
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