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Small Sample Size: The Wrong Way to Trade for Profit

In the excerpt below Matthew Rabin’s quotes from his white paper are in italics. He is responding to the bold news briefs:

Source: Inference by Believers in the Law of Small Numbers, Matthew Rabin, Department of Economics University of California, Berkeley

FROM NEWS REPORTS: Sudden Reversal for Coles: Warriors guard is now hitting jumpers with confidence. The body filling out that No. 12 in a Warriors uniform is the same one as in the last three years, the one prone to clanking jump shots and running the offense at a glacial pace…Ever since Bimbo Coles got a release from the deepest part of the bench a week ago…the nine-year point guard has been a changed player…Considering Coles’ shooting percentage has declined in each of the last five years, to a career-low of 37.9 in 1997-98, the turnaround is stunning. In the last four games, Coles has shot 15-for-26 (57.7 percent)…The mind is a crazy thing, Coles said. When you totally lose your confidence, you’re not going to play well. I’m starting to regain my confidence and going out there and having fun…The confidence was especially evident on the late jumpers, the last of which tied the game 87-87 with 1:33 to play.

In the four games following the stunning four-game turnaround, Coles went 7 for 21 for 33%. After regained confidence led to his stunning turnaround, Coles apparently either re-lost his confidence, or became over-confident. It is very plausible that a player can have significantly different success from season to season or team to team (Coles had switched teams the previous season), or even game to game because of changed team compositions or changed position, or different opponents (this headline was written after the Warriors defeated the L.A. Clippers, who were 0-17 at the time, and finished the season with the leagues worst record).

Indeed, for the remainder of the season Coles did improve over the previous year, and finished the season by shooting 137 for 303 (45.2%) following the turnaround, for a year end total of 156 for 348 (44.8%). But the article made an inference that was statistically unwarranted, yet typical of such articles. The chance of a 37.9% shooter making 15 out of a given 26 shots is about 2%. If (say) 200 NBA players a week have enough shots to warrant such a headline were they to perform comparably stunning four-game turnarounds, about one player per day would warrant this headline. It could also be noted that had Coles remained a 37.9%, i.i.d. shooter, then the chance of him having had a 15-for-26 streak at some point in his 348 shot season would be about 76%. Roughly speaking, the standards by which a four-game performance gets labeled a stunning turnaround are such that if most players don’t experience turnarounds in either direction throughout their career, they will be deemed to have had a stunning turnaround at least once a season.

While this example of over-inference is typical of sports and financial media analysis, a more gruesome and more worrying example of the gamblers fallacy comes from an article in the NY Times Magazine titled How Not to Get Killed on Deadline. [It is reported] that:

FROM NEWS REPORTS: In hostile-environment school, foreign correspondents learn how to improve their chances of surviving kidnappings, cross-fires and other perils of the workplace. The article is about advice given to journalists by a company called Centurion Risk Assessment Services Ltd. Centurion gives the following advice to journalists paying for war-survival training: In mortar attacks…lie down. If you can, crawl into one of the holes made by a previous shell because lightning rarely strikes twice in the same place.

I do not know if, within a journalist’s crawling range, the pattern of mortar attacks exhibits positive or negative correlation. But the metaphor chosen to be persuasive is a commonplace metaphor used to convey intuition for a commonplace misguided belief in a law of averages that says that once a rare event occurs, it becomes less likely it will reoccur, because such recurrence will throw averages out of whack. The lightening metaphor is striking: In actuality, heading for the same spot where lightning struck earlier is a bad idea in a thunderstorm, since lightening is more likely to hit a spot it has hit before than to hit a spot for the first time. Lightening rarely strikes twice, but only because it rarely strikes once.

What is the point from a trend following trading perspective (or trading in general)? When someone speaks of a great 3-month trading run and that’s all there is…be very wary.

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