Randomness and ergodic theory in economics – what went wrong?

18 Feb, 2012 at 12:23 | Posted in Statistics & Econometrics, Theory of Science & Methodology | 6 Comments

Ergodicity is a difficult concept that many students of economics have problems with understanding. In the very instructive video below, Ole Peters – from the Department of Mathematics at the Imperial College of London – has made an admirably simplified and pedagogical exposition of what it means for probability structures of stationary processses and ensembles to be ergodic. Using a progression of simulated coin flips, his example shows the all-important difference between time averages and ensemble averages for this kind of processes:

To understand real world ”non-routine” decisions and unforeseeable changes in behaviour, ergodic probability distributions are of no avail. In a world full of genuine uncertainty – where real historical time rules the roost – the probabilities that ruled the past are not those that will rule the future.

When we cannot accept that the observations, along the time-series available to us, are independent … we have, in strict logic, no more than one observation, all of the separate items having to be taken together. For the analysis of that the probability calculus is useless; it does not apply … I am bold enough to conclude, from these considerations that the usefulness of ‘statistical’ or ‘stochastic’ methods in economics is a good deal less than is now conventionally supposed … We should always ask ourselves, before we apply them, whether they are appropriate to the problem in hand. Very often they are not … The probability calculus is no excuse for forgetfulness.

John Hicks, Causality in Economics, 1979:121

Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies.

Added I:Some of my readers have asked why the difference between ensemble and time averages is of importance. Well, basically, because when you assume the processes to be ergodic,ensemble and time averages are identical. Let me giva an example even simpler than the one Peters gives:

Assume we have a market with an asset priced at 100 €. Then imagine the price first goes up by 50% and then later falls by 50%. The ensemble average for this asset would be 100 €- because we here envision two parallel universes (markets) where the assetprice falls in one universe (market) with 50% to 50 €, and in another universe (market) it goes up with 50% to 150 €, giving an average of 100 € ((150+50)/2). The time average for this asset would be 75 € – because we here envision one universe (market) where the assetprice first rises by 50% to 150 €, and then falls by 50% to 75 € (0.5*150).

From the ensemble perspective nothing really, on average, happens. From the time perspective lots of things really, on average, happen.

Assuming ergodicity there would have been no difference at all.

Added II: Just in case you think this is just an academic quibble without repercussion to our real lives, let me quote from an article of Peters in the Santa Fe Institute Bulletin from 2009 – On Time and Risk – that makes it perfectly clear that the flaw in thinking about uncertainty in terms of “rational expectations” and ensemble averages has had real repercussions on the functioning of the financial system:

In an investment context, the difference between ensemble averages and time averages is often small. It becomes important, however, when risks increase, when correlation hinders diversification, when leverage pumps up fluctuations, when money is made cheap, when capital requirements are relaxed. If reward structures—such as bonuses that reward gains but don’t punish losses, and also certain commission schemes—provide incentives for excessive risk, problems arise. This is especially true if the only limits to risk-taking derive from utility functions that express risk preference, instead of the objective argument of time irreversibility. In other words, using the ensemble average without sufficiently restrictive utility functions will lead to excessive risk-taking and eventual collapse. Sound familiar?

Added III: Still having problems understanding the ergodicity concept? Let me cite one last example that hopefully will make the concept more accessible on an intuitive level:

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. So, what is it?

Suppose you are concerned with determining what the most visited parks in a city are. One idea is to take a momentary snapshot: to see how many people are this moment in park A, how many are in park B and so on. Another idea is to look at one individual (or few of them) and to follow him for a certain period of time, e.g. a year. Then, you observe how often the individual is going to park A, how often he is going to park B and so on.

Thus, you obtain two different results: one statistical analysis over the entire ensemble of people at a certain moment in time, and one statistical analysis for one person over a certain period of time. The first one may not be representative for a longer period of time, while the second one may not be representative for all the people. The idea is that an ensemble is ergodic if the two types of statistics give the same result. Many ensembles, like the human populations, are not ergodic.

1. Thanks, Lars, this is very interesting.

2. But repeating a statement which is wrong doesn’t make it true! I question, first, whether or not risk management actually disregard the “effect of time”. First, the bet suggested in your example would be rejected by anyone with a slight degree of risk aversion. In fact, even with a downside risk of 40% loss, the bet would be rejected by a person with coefficient of relative risk aversion of unity. Precisely because the “time-average growth rate is negative”. So how can one claim that these effects are ignored, when they are internalized in the most plain vanilla textbook example I could come up with? Second, most financial models (although I am no expert here) use brownian motions. As far as I recall, these are the continuous time variants of random walks, and not ergodic.

3. You’re still missing the main point: assuming ergodicity there would/should/could have been no difference at all between the two cases.
Re the rather different question of neoclassical utility theory and its neglect of “the arrow of time” – and concomitant inability to adequately cope with the irreversibility of time – I can’t but cite Ole again: “Today’s risk management often solely relies
on investors specifying their risk preferences, or, synonymously, their utility functions, without explicitly considering the effects of time. My bank asked me the other day what risk type I am, apparently expecting a reply like “I like a good
gamble,” or “I always wear my bicycle helmet.” When I replied with a statement regarding time and answered, truthfully, that I’m the type who likes to see his money grow fast, they thought I was joking.”

4. Just read Ole’s article and realized that he reasons in a similar way: “The time-average
growth rate for this game, just like the expected logarithmic utility, is negatively infinite”. The same goes with your example: The time-average growth rate for your example is negative. That’s why it’s not a good bet, and that’s also why I teach my students to calculate growth rates and evaluate percentage changes accordingly.

5. Really, Pontus, you have to come up with something better!
My example is a standard one (you can find an analogous example all ready in Georgescu-Roegen’s critique of Birkhoff’s version of the ergodic theorem in The Entropy Law and the Economic Process). So, the only thing here that may be characterized as “meaningless”, is, sorry to say, your own “argumentation”.

6. But if you would define percentage changes in terms of its rate of growth — which every economist would do — instead of your primary school version, your example is turned on its head. What is worse, Ole Peters talk becomes meaningless.

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