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The Chairman
Victor Niederhoffer

06/26/2004
The Regression Fallacy

This is a very deep subject that Steve Stigler covers best in his book Statistics on the Table as well as his psychology article on same. It has a million applicabilities to market phenomena.

As a layman's guide to the subject, the result of a subject on two consecutive realizations of some test or the move of a market over two consecutive periods may be thought to consist of a permanent part and a transitory part y = p + t. the permanent part persists and the transitory part does not. This is very different from the concept of mean reversion as popularly used or the prediction of one variable from another by regression analysis . Since the subject's score contains a p and t , there is a reversion of the subject's score back to the mean. The extent of the move back to the mean can be estimated by counting such things as the variability of the subject's performance on the consecutive realizations.

When a market goes up, there is no reason that an analysis free of the regression bias would predict it to go down . But that's what almost everyone means when they talk about the mean reverting nature of prices. I use the regression bias when a company comes out with a great new discovery, an improvement in speed or performance, or the success of a movie or toy, or beverage, or perhaps the report of a great or dismal earnings, or the performance of a fund manager or a market-beating system, or an insight that was particularly good in a particular year, or a market has a spectacular move. Or a CEO did a great or terrible job and was is replaced like Kozlowski.

Part of the result -- indeed, in most cases, the entire superiority or inferiority -- can be expected to vanish in the next period. When such a system or company or market is evaluated as if the superiority or inferiority is expected to persist, then substantial opportunity exists.

Stigler's technical paper on this subject details the biases and incorrect uses of this phenomenon in the social sciences. It's sobering to think that there are as many abuses of this concept in the sciences as there are in our field by such experts as the derivatives expert and such users of technical analysis as the rank and file who extend a line drawn between two points on a chart to predict, or those who note that the further you get away from the starting point the greater the variability about the mean and thus mistakenly confuse trend with a necessary characteristic of randomness.  

Also very helpful is to look at batting averages or golf scores at selected times, and to work with random numbers and try to simulate the performance of stocks or systems in consecutive periods.  

6/26/2004
Alex Castaldo, PhD, comments:

Re: "As a layman's guide to the subject, the result [from] a subject on two consecutive realizations of some test [...] may be thought to consist of a permanent part and a transitory part y = p + t. The permanent part persists and the transitory part does not.

An example from psychology:

One hundred children are given IQ tests and are then ranked into deciles based on the results. y[i] is each child's result on this test. As mentioned this consists of two components p and t, which only G_d knows, while the psychologists measure y. t by definition has expected value 0 in the long run. (Sometimes p is called the true score and t is called the measurement error, but IMHO the terminology permanent/transient is preferable). For example Freddie's score on this particular day may be 95, consisting of p=100 and t=-5; the next time he is tested it may be 102 (p=100, t=+2). In any case the psychologist selects the children in the lowest decile and ask them to come back a month later for another test.

Often the average score of this group increases between the first and the second test. THE REGRESSION FALLACY CONSISTS IN BELIEVING THAT THE IMPROVEMENT IS REAL AND WAS DUE TO WHATEVER "TREATMENT" THE SUBJECTS WERE GIVEN BETWEEN THE FIRST AND SECOND TEST. Actually the result is entirely due to chance.

The lowest decile contained unintelligent children (low p) as well as children who scored particularly badly on this test for ephemeral reason (negative t). When these children are tested a second time, their t's have a 50-50 chance of being positive or negative. As a result the scores of this group (y) tend to increase. And similarly of course the scores of the first decile children tend to drop slightly on the second test.

Re: "Stigler's technical paper on this subject ... details the biases and incorrect uses of this phenomenon in the social sciences. And it's sobering to think that there are as many abuses of this concept in the sciences."

You can see that the result can be pretty disastrous to psychology research. The psychologist gave a lollipop to the low IQ children and he now concludes that giving a lollipop significantly enhances the IQ of low performing children. But the increase was entirely due to the research design of using a first test to classify the children and a second test to measure their "progress." [A research design of using a first test to classify, a second test to identify the children's starting point and a third test to measure the result of lollipop administration would have avoided most of the bias.] But it is not just psychological research into individual change that is affected; the phenomenon is ubiquitous.

College students often talk about the "sophomore slump": students who did well in their first year often do not do as well the next term. Popular imagination attributes this to: they became overconfident and did not study as much, but a simpler explanation is that a great success in the first year is partly the result of luck, which may be expected to reverse.

The paper referred to is "Regression Toward the Mean and the Study of Change," S. Stigler, J. Nesselroade and Baltes. Psychological Bulletin, 1980, Vol. 87, pp. 622-637. The paper goes on to consider more complicated cases where N consecutive tests are given and discusses when regression continues and when it stops after the second test. If the test are independent and no further ranking is performed, the regression stops after the second test.

If my masterful summary above is not sufficient for you, if you do not have access to a college library and if you cannot find any other information on Google, by all means send your snail-mail address c/o c/o  the specs address at mantrade.com, and I will mail you a copy of Stigler's paper.