Broken Symmetry

R.I.P. Paul Samuelson, the Isaac Newton of Economics

From Ed Glaeser:

Mathematics doesn’t substitute for genuine insight or knowledge of the real world, but it is necessary if economists are going to craft logically coherent models of complex phenomena. While many mathematical models failed to predict the current downturn, plenty of non-mathematical seers also failed. If we ever do come to understand the vicissitudes of asset markets, that understanding will come in the form of formal mathematical analysis, ultimately inspired by Samuelson.

Glaeser's eulogy also hints at how the time may be ripe for a revision to Samuelson's Foundations:

If people can rank outcomes in terms of their desirability, and if they choose one outcome over another, then people have preferences. (Who can disagree with that?) And if, as Samuelson taught us, those preferences follow a few simple rules, then human behavior can be described with a utility function. That conclusion allows economists to turn to Newton and optimization theory.

The principle of revealed preferences results in a mean-field approximation of individual preferences. The incremental step beyond mean-field theory is gauge theory, and this is why the Coase Theorem and New Institutional Economics are more relevant now than neoclassical theory. What NIE has lacked thus far is a theoretical foundation equal in rigor to the foundations Samuelson laid for neoclassical theory.

Samuelson was like Newton also in that he brought his gifts to bear on solving practical problems for his country. Newton completely redesigned the Royal Mint, caught some of the worst counterfeiters, and advised on how to end a 17th century version of the carry trade, which was draining England's royal coffers. Samuelson led the Federal Reserve through several financial crises and advised countless students in high places on economic issues that affected billions of lives.

Not everybody makes this big a dent in history.

George Soros Explains His Worldview

And it is surprisingly fascinating:

While I was reading Popper I was also studying economic theory and I was struck by the contradiction between Popper’s emphasis on imperfect understanding and the theory of perfect competition in economics which postulated perfect knowledge. This led me to start questioning the assumptions of economic theory. These were the two major theoretical inspirations of my philosophy. It is also deeply rooted in my personal history.

The FT has a whole website up with transcripts from each of his lectures, which are ongoing this week. It's little things like this that make my day.

Transcripts of first three lectures are here, here, and here.

For an earlier post on epistemlogical and ontological questions that offers one version of reflexivity, see here.

In reponse to Soros's proposal that we dichotomize the natural and social sciences, I have this to say: His misinterpretation of Heisenberg's uncertainty principle kills his proposal. In fact it is true of both natural and social scientific theories that our theories influence how reality evolves. It is true too that bad natural science may not have as catastrophic or immediate consequences for us as bad social science. But what we perceive depends in part on our theory of what exists, and those theories are not social. Soros's dichotomy is in fact a continuum.

To wit, from whence did Soros derive his understanding of positive and negative feedback loops? By analogy to the natural sciences. Our exploration of nature is the ultimate origin of all of our theories, bot natural and social.

Mandelbrot and the Glassy State of the Economy

From Fractals and Scaling in Finance, page 367:

Everyone agrees that the distribution of firm sizes is far from Gaussian, hence the proper metaphor is not a "soft," but a "hard" object. This leaves open the question of whether it is a real solid or a glass. Physics describes glasses as being extremely viscous liquids that mimic solids and happen to be particularly difficult to study. In the same way, the lognormal distribution averages mildly in the very long run, but in the short run it mimics the wild randomness of the scaling in a very treacherous fashion. One can call it a sheep in wolf's clothing.

Why is Mandelbrot skeptical of lognnormal distributions of firm size? Because market concentration appears to be a scale invariant quantity -- i.e., the amount of concentration appears not to vary as much with the number of firms in the market as one would expect from distributions with finite variance.

Distinguishing Exogenous from Endogenous Market Crashes

Simoleon Sense links to what could be a useful bit of anlaysis from Sornette et al.:

[W]e have provided an objective test of the hypothesis that the largest negative markets moves are special and form two distinct populations. The exogenous crashes can be attributed to extraordinary important external perturbations and/or news impacting the market. The endogenous crashes can be seen as the natural deaths of self-organized self-reinforcing speculative bubbles giving rise to specific precursory signatures in the form of log-periodic power laws accelerating super-exponentially.

Sornette applied a similar approach to analyzing YouTube video traffic here.

I'm slowly warming up to Sornette's idea of log-periodic scaling. The mathematical part of this model, at bottom, arises from a treatment of fractional dimensions as complex rather than real-valued quantities.

What Mandelbrot and Fama Both Got Wrong: Buyers and Sellers Can Synchronize

I've been working through a paper by Eugene Fama, titled "Madelbrot and the Stable Paretian Hypothesis." The paper is from 1963. I found it in Modern Developments in Investment Management, edited by Lorie and Brealey and published in 1976. It was in the free books bin outside a bookstore in Noe Valley!

As a historical record, this is a wonderful article. In the article, Fama finds the data inconclusive as to whether the Pareto distribution applies better as a fit to the distribution of price changes during a given period of time. The punchline, of course, is that Fama's efficient market hypothesis (which he developed in the decade after this paper was published) gave the confidence at least to Black and Scholes and many others to base their models of price changes on Gaussian distributions. At least at one point, Fama was open to another possibility.

But what is more interesting is the evidence from this paper, which suggests that both Mandelbrot and Fama may have made the same mistake in analyzing time series of prices. In particular, it seems that both Mandelbrot and Fama wanted to maintain the assumption of independence in price changes in successive intervals. They seem to have had the same reason too. Both wanted whatever distribution that might apply in a shorter interval to apply also in longer or successive intervals -- the very condition that fails when the price changes are not independent. Here's how Fama puts it:

If the changes between transactions are independent, identically distributed, stable Paretian variables, daily, weekly, and monthly changes will follow stable Paretian distributions of exactly the same form, except for origin and scale.

Mandelbrot, recognizing that a Gaussian was not a good fit, picked the Pareto distribution as an alternative. Why did he choose the Pareto distribution? I'm not sure, but there is a good chance that he did so because while its infinite variance permitted wider swings in price than the Gaussian, the Pareto distribution is still stationary in the sense that it treats the price changes during particular intervals as wholly independent.

Both Mandelbrot and Fama seem to have let themselves be guided too strongly by their theoretical ambitions. If we observe what actually happens in crowded marketplaces (such as a crowded trading floor), it is easy to observe how buying and selling behavior is correlated. This kind of correlation (which Keynes attributed to "animal spirits") destroys the very independence that is necessary to the assumption of stationarity made by both Mandelbrot and Fama in their models of market prices!

What alternative models might we use to understand price changes? We have to start from models like the 1D Ising model described in the last post, models which show how correlations at the microeconomic scale can aggregate up into macroeconomic fluctuations. We must begin with the assumption that buying and selling decisions are at least weakly coupled, so that market price changes reflect the large swings that can result from synchronization. In future posts, I shall attempt an explanation for how the Kuramoto Model of synchronization could be used to model the non-stationary distribution(s) of market prices.

UPDATE: My sincere apologies to Mandelbrot. I find in this early work that he did not assume stationarity, and was aware of the non-stationarity in these distributions all along.