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29 Chapter 9: Statistics t>9 you ewe 49 v* Figure 9.1 The normal distribution, (source: The Economist Numbers Guide by Richard Stutely, New York: John Wiley & Sons, 1998.) distributed and the statistical rules you would normally expect to hold may not hold. To get an accurate estimate of something, you may sample it. Take a bowl with a couple of hundred red and green marbles in it. If you pick 30 marbles at random from the bowl, the proportion of red to green in the sample you selected should reflect the proportion of the population-all the marbles in the bowl. The larger the sample, the better the reflection or estimate will be. If you have more than two colors, youll need a larger sample size to get a good estimate. Okay, if you can grasp that, the rest should be easy. The use of standard deviation to drive the width of Bollinger Bands naturally invites the use (abuse?) of statistical rules. While many inferences are inappropriate due to the non-normal distribution of stock prices and the small sample sizes typically used, some statistical concepts do appear to hold. The central limit theorem suggests that even when the data is not normally distributed-as is the case for stocks-a random sampling will produce a normally distributed subset for which the statistical rules will hold. This is thought to be true even at relatively small sample sizes. So we should not be surprised to
Part II: The Basics learn that the statistical expectations do hold to some extent, even if everything isnt strictly kosher. The statistical concept most often inquired about in relation to Bollinger Bands is regression to the mean, which says that all things will eventually come home; for statisticians home is the mean or average. Thus as prices depart from the average, we should expect them to move back toward the average. This is the statistical concept behind the technical terms overbought and oversold. Regression to the mean implies that prices at the edges of the distribution-at the upper or lower Bollinger Bands-will revert to the mean-the average, or middle, Bollinger Band. While there is some evidence of regression to the mean demonstrated by financial instruments, it is not as strong as it should be, so tags of the bands are not automatic buys or sells with the average as a target. This is precisely why the use of indicators to confirm tags of the bands is such a powerful concept. With indicators we can make rational judgments about whether to expect regression to the mean or a continuation of the trend. When the chosen indicator confirms a tag of the bands, you do not have a buy or sell signal; you have a continuation signal. When a tag is unconfirmed, expect regression to the mean. In this manner we combine information from statistics with information from technical analysis, relying on the strengths of each to improve our decision making. At sample sizes smaller than the minimum required for statistical significance, the basic statistical processes should still be relevant if the central limit theorem holds. Our testing confirms that this is the case for Bollinger Bands. While slight adjustments are desirable to maintain the proportion of data contained within the bands as the sample size changes (the number of days), the behavior exhibited in and around the bands is much the same whether the period is 10 days or 50 days. This is true even though only approximately 89 percent of the data is contained within 2 standard deviation bands when we would expect 95 percent. There are two possible reasons why we dont get as high a level of containment as we would expect-near 95 percent with 2 standard deviation bands. First, we are using the population calculation, which results in slightly tighter bands than the sample calculation.1 Second, the distribution of stock prices is not normal- there are more observations at the extremes than one would
Chapter 9: Statistics expect-so there are more data points outside the bands too. There are undoubtedly more factors, but these appear to be the main ones. What is a non-normal distribution again? And what has a fat tail? The graph in Figure 9.2 illustrates the concept nicely. The taller hump is a normal distribution, the way things ought to be. The shorter hump is a distribution like the stock markets, less small changes than one would expect and more large changes. The amount of difference between the two humps is known as kurtosis, and it is a significant quantity for stocks. Perhaps one of the most interesting aspects of Bollinger Bands is the rhythmic contraction and expansion of the bands you can see on the charts. This is especially clear in the bond market where a fairly regular 19-day volatility cycle can be observed (Figure 9.3). It turns out that there is a fair amount of academic research into this phenomenon. A search for papers on GARCH and ARCH2 will reveal the details for those so inclined. In general, the idea is that while price is neither cyclical-in a regular sense-nor forecastable using cycles, volatility is both. So  <- large declines - small declines - unchanged price - small advances - large advances -> Figure 9.2 Kurtosis. The stock market is not normally distributed-too many large changes.
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