Power laws, baby

Thanks to reader GS for the link to this excellent article by statistical physicist Eugene Stanley. There's much more to the article, but I decided to excerpt this discussion of security price fluctuations and the observation that they are far from log normal.

Stanley: ...

When they analyzed these data–200 million of them–in exactly the same fashion that Bachelier had analyzed data almost a century earlier, they made a startling discovery. The pdf of price changes was not Gaussian plus outliers, as previously believed. Rather, all the data–including data previously termed outliers–conformed to a single pdf encompassing both everyday fluctuations and “once in a century” fluctuations. Instead of a Gaussian or some correction to a Gaussian, they found a power law pdf with exponent -4, a sufficiently large exponent that the difference from a Gaussian is not huge; however, the probability of a “once in a century” event of, say, 100 standard deviations is exp(-10,000) for the Gaussian, but simply 10-8 for an inverse quartic law. If one analyzes a data set containing 200 million data in two years, this means there are only two such events–in two years!

Now which is better, the concept of “everyday fluctuations” which can be modeled with a drunkard’s walk, complemented by a few “once in a century” outliers? Or a single empirical law with no outliers but for which a complete theory does not exist despite promising progress by Xavier Gabaix of NYU’s Stern School of Management and his collaborators? Here we come to one of the most salient differences between traditional economics and the econophysicists: economists are hesitant to put much stock in laws that have no coherent and complete theory supporting them, while physicists cannot afford this reluctance. There are so many phenomena we do not understand. Indeed, many physics “laws” have proved useful long before any theoretical underpinning was developed . . . Newton’s laws and Coulomb’s law to name but two.

And all of us are loathe to accept even a well-documented empirical law that seems to go against our own everyday experience. For stock price fluctuations, we all experience calm periods of everyday fluctuations, punctuated by highly volatile periods that seem to cluster. So we would expect the pdf of stock price fluctuations to be bimodal, with a broad maximum centered around, say, 1-3 standard deviations and then a narrow peak centered around, say, 50 standard deviations. And it is easy to show that if we do not have access to “all the data” but instead sample only a small fraction of the 200 million data recently analyzed, then this everyday experience is perfectly correct, since the rare events are indeed rare and we barely recall those that are “large but not that large”.

The same is true for earthquakes: our everyday experience teaches us that small quakes are going on all the time but are barely noticeable except by those who work at seismic detection stations. And every so often occurs a “once in a century” truly horrific event, such as the famous San Francisco earthquake. Yet when seismic stations analyze all the data, they find not the bimodal distribution of everyday experience but rather a power law, the Gutenberg-Richter law, describing the number of earthquakes of a given magnitude.