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SAMPLE SIZE AND REPRESENTATIVENESS
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Although, for statistical reasons, the system developer should seek the largest sam ple possible, there is a trade-off between sample size and representativeness when dealing with the financial markets. Larger samples mean samples that go farther back in time, which is a problem because the market of years ago may be fundamentally different from the market of today-remember the S&P 500 in 1983 This means that a larger sample may sometimes be a less representative sample, or one that confounds several distinct populations of data! Therefore, keep in mind that, although the goal is to have the largest sample possible, it is equally important to try to make sure the period from which the sample is drawn is still representative of the market being predicted.
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EVALUATING A SYSTEM STATISTICALLY
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Now that some of the basics are out of the way, let us look at how statistics are used when developing and evaluating a trading system. The examples below employ a system that was optimized on one sample of data (the m-sample data) and then run (tested) on another sample of data (the out-of-sample data). The outof-sample evaluation of this system will be discussed before the in-sample one because the statistical analysis was simpler for the former (which is equivalent to the evaluation of an unoptimized trading system) in that no corrections for multiple tests or optimization were required. The system is a lunar model that trades the S&P 500; it was published in an article we wrote (see Katz with McCormick, June 1997). The TradeStation code for this system is shown below:
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Example 1: Evaluating the Out-of-Sample Test
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Evaluating an optimized system on a set of out-of-sample data that was never used during the optimization process is identical to evaluating an unoptimized system. In both cases, one test is run without adjusting any parameters. Table 4-1 illustrates the use of statistics to evaluate an unoptimized system: It contains the outof-sample or verification results together with a variety of statistics. Remember, in this test, a fresh set of data was used; this data was not used as the basis for adjustments in the system s parameters. The parameters of the trading model have already been set. A sample of data was drawn from a period in the past, in this specific case, l/1/95 through l/1/97; this is the out-of-sample or verification data. The model was then run on this outof-sample data, and it generated simulated trades. Forty-seven trades were taken. This set of trades can itself be considered a sample of trades, one drawn from the population of all trades that the system took in the past or will take in the future; i.e., it is a sample of trades taken from the universe or population of all trades for that system. At this point, some inference must be made regarding the average profit per trade in the population as a whole, based on the sample of trades. Could the performance obtained in the sample be due to chance alone To find the answer, the system must be statistically evaluated. To begin statistically evaluating this system, the sample mean (average) for n (the number of trades or sample size) must first be calculated. The mean is simply the sum of the profit/loss figures for the trades generated divided by n (in this case, 47). The sample mean was $974.47 per trade. The standard deviation (the variability in the trade profit/loss figures) is then computed by subtracting the sample mean from each of the profit/loss numbers for all 47 trades in the sample; this results in 47 (n) deviations. Each of the deviations is then squared, and then all squared deviations are added together. The sum of the squared deviations is divided hy n - I (in this case, 46). By taking the square root of the resultant number (the mean squared deviation), the sample standard deviation is obtained. Using the sample standard deviation, the expected standard deviation of the nean is computed: The sample standard deviation (in this case, $6,091.10) is divided by the square root of the sample size. For this example, the expected standard deviation of the mean was $888.48. To determine the likelihood that the observed profitability is due to chance alone, a simple t-test is calculated. Since the sample profitability is being compared with no profitability, zero is subtracted from the sample mean trade profit/loss (computed earlier). The resultant number is then divided by the sample standard deviation to obtain the value of the t-statistic, which in this case worked out to be 1.0968. Finally the probability of getting such a large t-statistic by chance alone (under the assumption that the system was not profitable in the population from which the sample was drawn) is calculated: The cumulative t-distribution for that t-statistic is cotnputed with the appropriate degrees of freedom, which in this case was n - 1, or 46.
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