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detected, avoided, or compensated for; at the very least, they can be understood. In short, we are fully aware of these violations and will discuss our acts of hubris and their ramifications after a foundation for understanding the issues has been laid.
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Fundamental to statistics and, therefore, important to understand, is the act of sampling, which is the extraction of a number of data points or trades (a sample) from a larger, abstractly defined set of data points or trades (a population). The central idea behind statistical analysis is the use of samples to make inferences about the populations from which they are drawn. When dealing with trading models, the populations will most often be defined as all raw data (past, present, and future) for a given tradable (e.g., all 5-minute bars on all futures on the S&P 500), all trades (past, present, and future) taken by a specified system on a given tradable, or all yearly, monthly, or even daily returns. All quarterly earnings (past, present, and future) of IBM is another example of a population. A sample could be the specific historical data used in developing or testing a system, the simulated trades taken, or monthly returns generated by the system on that data. When creating a trading system, the developer usually draws a sample of data from the population being modeled. For example, to develop an S&P 500 system based on the hypothesis If yesterday s close is greater than the close three days ago, then the market will rise tomorrow, the developer draws a sample of end-of-day price data from the S&P 500 that extends back, e.g., 5 years. The hope is that the data sample drawn from the S&P 500 is represenrative of that market, i.e., will accurately reflect the actual, typical behavior of that market (the population from which the sample was drawn), so that the system being developed will perform as well in the future (on a previously unseen sample of population data) as it did in the past (on the sample used as development data). To help determine whether the system will hold up, developers sometimes test systems on one or more out-of-sample periods, i.e., on additional samples of data that have not been used to develop or optimize the trading model. In our example, the S&P 500 developer might use 5 years of data--e.g., 1991 through 1995-to develop and tweak the system, and reserve the data from 1996 as the out-of-sample period on which to test the system. Reserving one or more sets of out-of-sample data is strongly recommended. One problem with drawing data samples from financial populations arises from the complex and variable nature of the markets: today s market may not be tomorrow s Sometimes the variations are very noticeable and their causes are easily discerned, e.g., when the S&P 500 changed in 1983 as a result of the introduction of futures and options. In such instances, the change may be construed as having created two distinct populations: the S&P 500 prior to 1983 and the S&P 500 after 1983. A sample drawn from the earlier period would almost certainly
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not be representative of the population defined by the later period because it was drawn from a different population! This is, of course, an extreme case. More often, structural market variations are due to subtle influences that are sometimes impossible to identify, especially before the fact. In some cases, the market may still be fundamentally the same, but it may be going through different phases; each sample drawn might inadvertently be taken from a different phase and be representative of that phase alone, not of the market as a whole. How can it be determined that the population from which a sample is drawn for the purpose of system development is the same as the population on which the system will be traded Short of hopping into a time machine and sampling the future, there is no reliable way to tell if tomorrow will be the day the market undergoes a systemkilling metamorphosis! Multiple out-of-sample tests, conducted over a long period of time, may provide some assurance that a system will hold up, since they may show that the market has not changed substantially across several sampling periods. Given a representative sample, statistics can help make accurate inferences about the population from which the sample was drawn. Statistics cannot, however, reveal whether tomorrow s market will have changed in some fundamental manner.
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