## Ergodicity – the biggest mistake ever made in economics

10 February, 2013 at 16:30 | Posted in Economics | 5 CommentsPaul Samuelson claimed that the “ergodic hypothesis” is essential for advancing economics from the realm of history to the realm of science.

But is it really tenable to assume – as Samuelson and most other neoclassical economists – that ergodicity is essential to economics?

The answer can only be – as I have argued here, here, here, here and 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.”

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In Taleb’s Fooled by Randomness he speaks of Ergodicity, “…namely that time will eliminate the annoying effects of randomness.” On the other hand an online definition/discussion of ergodicity states: “Why are election polls often inaccurate? Why is racism wrong? Why are your assumptions often mistaken? The answers to all these questions and to many others have a lot to do with the non-ergodicity of human ensembles.” Many scientists agree that ergodicity is one of the most important concepts in statistics.

G Charles-Cadogan has three research papers that look not only at ergodic behaviour, but also behavioural chaos. It turns out that human beings operate on a circadian cycle that affects their time of day decisions. He uses entropy analysis popularized by E. T. Jaynes to derive a harmonic probability weighting function. That function shows that when our probability cycles are broken, we tend to make different decisions even though we are faced with the same underlying stimuli. Also, our confidence levels can be represented by how we perceive gains and losses, and how we project on those domains.. Literature reviews and perhaps comparatively nontechnical aspects of the above can be examined by reading the introduction sections of the following papers:

A. http://papers.ssrn.com/abstract=2081376

B. http://papers.ssrn.com/abstract=2049425

C. http://papers.ssrn.com/abstract=1971954

Comment by Colin— 11 February, 2013 #

I think there is a mistake here. Brownian motion is ergodic.

Comment by Noah Smith— 11 February, 2013 #

Actually, no mistake at all, as e.g. Peters and Klein forcefully show in their recently published paper “Partial ensemble averages in geometric Brownian motion”

(http://arxiv.org/pdf/1209.4517v2.pdf)

Comment by Lars P Syll— 11 February, 2013 #

[...] The biggest mistake ever made in [...]

Pingback by Further reading | FT Alphaville | MG-PT— 11 February, 2013 #

You are absolutely right about ergodicity! And what you say abot the scientific method there.

Luckily progress has been made in maths since Samuelson’s time. (Some of which I was reading at the week-end.) If you check isomorphismes.tumblr.com on Tuesday I’ll post a picture of a germ (I’m going to use the term “wiggly number”).

I don’t understand the tail of your comments, after firmly agreeing with everything up to that:

What is the relationship between b.e. and risk? I agree with you that probably economists in general are not talking about risk in the most sensible way. We struggle with philosophy of probability and the meaning of “could”.

That blockquote is also confusing because, isn’t all of economics about behaviour? So which part is behavioural–I assume you mean some experiments without a larger theory tying them together?

Which is … ?

Comment by isomorphismes— 24 February, 2013 #