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The following section is intended only to acquaint the reader with some other statistical techniques that are available. We strongly suggest that a more thorough study be undertaken by those serious about developing and evaluating trading systems.
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We develop many systems using genetic algorithms. A popular$fimessfunction (criterion used to determine whether a model is producing the desired outcome) is the total net profit of the system. However, net profit is not the best measure of system quality! A system that only trades the major crashes on the S&P 500 will yield a very high total net profit with a very high percentage of winning trades. But who knows if such a system would hold up Intuitively, if the system only took two or three trades in 10 years, the probability seems very low that it would continue to perform well in the future or even take any more trades. Part of the problem is that net profit does not consider the number of trades taken or their variability. An alternative fitness function that avoids some of the problems associated with net profit is the t-statistic or its associated probability. When using the t-statistic as a fitness function, instead of merely trying to evolve the most profitable systems, the intention is to genetically evolve systems that have the greatest likelihood of being profitable in the future or, equivalently, that have the least likelihood of being profitable merely due to chance or curve-fitting. This approach works fairly well. The t-statistic factors in profitability, sample size, and number of trades taken. All things being equal, the greater the number of trades a system takes, the greater the t-statistic and the more likely it will hold up in the future. Likewise, systems that produce more consistently profitable trades with less variation are more desirable than systems that produce wildly varying trades and will
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yield higher t-statistic values. The t-statistic incorporates many of the features that define the quality of a trading model into one number that can be maximized by a genetic algorithm.
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Multiple Regression
Another statistical technique frequently used is multiple regression. Consider intermarket analysis: The purpose of intermarket analysis is to find measures of behaviors in other markets that are predictive of the future behavior of the market being studied. Running various regressions is an appropriate technique for analyzing such potential relationships; moreover, there are excellent statistics to use for testing and setting confidence intervals on the correlations and regression (beta) weights generated by the analyses. Due to lack of space and the limited scope of this chapter, no examples are presented, but the reader is referred to Myers (1986), a good basic text on multiple regression. A problem with most textbooks on multiple regression analysis (including the one just mentioned) is that they do not deal with the issue of serial correlation in time series data, and its effect on the statistical inferences that can be made from regression analyses using such data. The reader will need to take the effects of serial correlation into account: Serial correlation in a data sample has the effect of reducing the effective sample size, and statistics can be adjusted (at least in a rough-and-ready manner) based on this effect. Another trick that can be used in some cases is to perform some transformations on the original data series to make the time series more stationary and to remove the unwanted serial correlations.
Monte Carlo Simulations
One powerful, unique approach to making statistical inferences is known as the Monte Carlo Simulation, which involves repeated tests on synthetic data that are constructed to have the properties of samples taken from a random population. Except for randomness, the synthetic data are constructed to have the basic characteristics of the population from which the real sample was drawn and about which inferences must be made. This is a very powerful method. The beauty of Monte Carlo Simulations is that they can be performed in a way that avoids the dangers of assumptions (such as that of the normal distribution) being violated, which would lead to untrustworthy results.
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