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Fun with Benford’s Law, one of those cool mathematical principles whose name I can never remeber.
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"Reviewing Chinese Democracy is not like reviewing music. It’s more like reviewing a unicorn. Should I primarily be blown away that it exists at all? Am I supposed to compare it to conventional horses? To a rhinoceros? Does its pre-existing mythology impact its actual value, or must it be examined inside a cultural vacuum, as if this creature is no more (or less) special than the remainder of the animal kingdom?"
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In the parallel universe where I have copious free time, I bet I read this.
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For those who want information to go with lurid rumors of sinister deeds off the Horn of Africa
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Fun times at the Zeno Bar and Grille.
7 thoughts on “links for 2008-11-20”
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Funny thing is, though, that stock market data actually don’t follow Benford’s law very well. For the Dow Jones and the S&P 500, there’s definitely a surplus of 8s and 9s, while the Nikkei has about twice as many 1s for leading digits as expected. I wrote about my analysis here.
Aleph null bottles of beer on the wall
Aleph null bottles of beer
Take one down, pass it around…
Aleph null bottles of beer on the wall.
@thm: You have made the assumption that Benford’s Law applies to time series data. I see no reason to think that should be true, because these time series you used have a nonzero correlation length. If the Dow is 8000 on one day, the probability that it will be 1000 the next day is zero, if for no other reason than the fact that circuit breakers will prevent a drop of that magnitude.
The lists of numbers that were described in the video as satisfying Benford’s Law are lists in which the individual entries are uncorrelated. The population of County A in 2002 does not depend in any significant way on the population of County B in 2002. The size of Lake X is entirely independent of the size of Lake Y. Et cetera. So there is no reason to expect numbers with a particular leading digit to cluster near other numbers with the same leading digit, assuming you have not attempted to sort the list. Not so with stock market indices: the closing levels seldom change by more than a few percent, so if the level is 9500 one day the leading digit on the next day’s close is overwhelmingly likely to be a 9. Also be sure to factor in psychology: round numbers are significant milestones. The correct way to do the comparison is to take the closing level of every stock market index in the world on a given day–this will give you an uncorrelated data set, so it should obey Benford’s Law. Alternatively, take the absolute value of the change from one day to the next of some index, because the change from one day to the next should not (at least to first order) be correlated with the change from the second day to the third day.
thm: That sounds like something explainable by psychological reactions — the herd of US stockbrokers might be reluctant to bid prices up past the 10,000 boundary, for instance.
In the Grading Education post, the author says the following: “Second, the authors make a revolutionary suggestion about the role of school boards, which particularly resonated with me after two recent local events. The less absurd of these was the last round of budget cuts, which consumed our local school board’s time and energy. For the key meetings each member brought in his or her own list of exactly what to cut, then they traded till the cuts worked out. It struck me at the time that nobody could seriously believe that the upshot was going to be superior in any way to a budget that the Superintendent would have recommended, and that the discussion was simply a waste of time; none of the board members is a fool, and I imagine that none of them thought their list was much better than anyone else’s, and that most of their lists were probably better than any compromise that they would forge; wouldn’t it have been better to pick a list out of the hat than to engage in endless detailed discussion?”
Actually, I would expect the result arrived at by discussion and negotiation to be considerably better than what any one board member had selected. And really, if Crooked Timber doesn’t believe that, he should support abolishing the the boards entirely, since discussion and negotiation is what they DO. This dynamic tends to produce middle-of-the-road solutions, and if he perpetually prefers something more extreme, then he is really a bit of a radical. Perhaps a single elected schools official would be more to his liking.
@Eric: The linked video (at 0:40) does state, albeit offhandedly, that stock prices follow Benford’s Law. My original inquiry was sparked by a post by Brad DeLong in which many commenters suggested that Benford’s Law was the reason for his observation that the Dow Jones average seemed to cluster near 1000 then near 10000. thought DeLong’s analysis was a bit sloppy, especially considering how easy it is to get the actual values, and also that the matter needed more than a reflexive invocation of Benford’s Law, for the reasons you and Zack suggest.
The linked video (at 0:40) does state, albeit offhandedly, that stock prices follow Benford’s Law.
You are right. They were not clear about this, but I assumed that they meant prices of individual stocks, as opposed to closing levels of an index as a function of time.
I took another look at your post, and I can make the following comment about the Nikkei: IIRC this index has traded in the range 7k to 40k throughout the entire time you covered, thus accounting for the lack of 4, 5, and 6 as leading digits in the series. There was a Japanese stock bubble in the late 1980s; after it popped prices settled around a third to a half of the peak value(i.e., somewhere between 10k and 20k), and things basically went sideways for most of the next two decades (earlier this year it fell below 10k again).