Paul Samuelson and the ergodic hypothesis7 August, 2012 at 18:40 | Posted in Economics, Statistics & Econometrics, Theory of Science & Methodology | 2 Comments
But is it really tenable to assume that ergodicity is essential to economics?
The answer can only be – as I have argued
here – NO WAY!
Samuelson said that we should accept the ergodic hypothesis because if a system is not ergodic you cannot treat it scientifically. First of all, that’s incorrect, although I think I understand how he ended up with this impression: ergodicity means that a system is very insensitive to initial conditions or perturbations and details of the dynamics, and that makes it easy to make universal statements about such systems …
Another problem with Samuelson’s statement is the logic: we should accept this hypothesis because then we can make universal statements. But before we make any hypothesis—even one that makes our lives easier—we should check whether we know it to be wrong. In this case, there’s nothing to hypothesize. Financial and economic systems are non-ergodic. And if that means we can’t say anything meaningful, then perhaps we shouldn’t try to make meaningful claims. Well, perhaps we can speak for entertainment, but we cannot claim that it’s meaningful.
In what sense would saying something that’s patently false be “meaningful,” or “scientific” rather than “historical”? You can see where I’m going with this. Important models that economists use are not ergodic, so what’s this debate about?…
In finance or economics the situation is different. Take the most basic model of a stock market, Louis Bachelier’s random walk. Is that model ergodic? No. A little later, in the 1950s, maybe starting with M. F. M. Osborne, the popular model in finance became geometric Brownian motion—basically a random walk in log-space …
Since geometric Brownian motion is a mathematical model, you can answer the question of whether that’s ergodic by scribbling a few lines of equations. Of course it is not. It’s a model of growth, after all, so it can’t be ergodic, but you can actually make this completely formal and do the math, and not even the expectation value of the growth rate is equal to the time average of the growth rate. At the end of the day, what’s more important in finance than growth rates?
So Samuelson’s comment makes little sense. A hypothesis is about something we don’t know, but in the case of finance models this is something we do know. There’s no reason to hypothesize—the system is not ergodic. It’s like hypothesizing that 3 times 4 is 0 because it makes the mathematics simpler. But I can calculate that the product is 12. Of course, a formalism that’s based on the 3-times-4 hypothesis will run into trouble sooner or later. In economics, that happens with the ergodic hypothesis when we think about risk, or financial stability. Or inequality, as we’re just working out at the moment.
The reason this is so important is quite simple, and stems from a basic question: what does risk mean if the notion of time is not irreversible? The only reason risk exists is that we cannot go back and make decisions over again. Economics got very confused about the point of dealing with risk, and had to resort to introducing psychology and human behavior and all sorts of things. I don’t mean to say that we don’t need behavioral economics. What I mean is that there are lots of questions in economics that we can only answer behaviorally at the moment, but at the same time we have a perfectly formal natural physical analytic answer that’s very intuitive and sensible and that comes straight out of recognizing the non-ergodicity of the situation.
To be blunter, I’m pointing out that economics is internally inconsistent. I accept all the models that economists have developed. I could critique them, but I’m not worried about that. I didn’t make them up, the economists did. But when the economists treat the models as if they were ergodic, that’s when someone has to say “stop, that’s enough.”