Stationary non-ergodicity (wonkish)

18 April, 2015 at 10:24 | Posted in Economics | 18 Comments

Let’s say we have a stationary process. That does not guarantee that it is also ergodic. The long-run time average of a single output function of the stationary process may not converge to the expectation of the corresponding variables — and so the long-run time average may not equal the probabilistic (expectational) average. cointossingSay we have two coins, where coin A has a probability of 1/2 of coming up heads, and coin B has a probability of 1/4 of coming up heads. We pick either of these coins with a probability of 1/2 and then toss the chosen coin over and over again. Now let H1, H2, … be either one or zero as the coin comes up heads or tales. This process is obviously stationary, but the time averages — [H1 + … + Hn]/n — converges to 1/2 if coin A is chosen, and 1/4 if coin B is chosen. Both these time averages have a probability of 1/2 and so their expectational average is 1/2 x 1/2 + 1/2 x 1/4 = 3/8, which obviously is not equal to 1/2 or 1/4. The time averages depend on which coin you happen to choose, while the probabilistic (expectational) average is calculated for the whole “system” consisting of both coin A and coin B.

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  1. Professor Syll,

    thanks for a very clear explanation. Could you give an example of some variable that is not ergodic and how the assumption of an economic model that it is leads to wrong conclusions?

    I am an interested student!

  2. Mathijs,

    Do you want a non-ergodic realist and relevant example instead of a toy model exploration? That’s not going to happen!

    • I do not understand what you mean, Pontus. Professor Syll seems to think that non-ergodicity is of paramount importance to understanding economics. I would like to know which processes he feels are non-ergodic and how economists model them as ergodic and how that leads them astray. It seems like a fair question.

      • Mathijs,

        Your question is highly relevant, and my comment was sarcastic. I wrote it because it is a helluva lot easier to show mickey-mouse example pointing out abstract situations where non-ergodicity is a real problem, and an entirely different to show where it really matters and what can be a useful way of analyzing such situations. Unfortunately, Professor Syll never addresses the latter point, but only the former (and then accuses “mainstream” economists for mainly dealing with mickey-mouse examples with no real world counterpart).

      • ” I would like to know which processes he feels are non-ergodic and how economists model them as ergodic and how that leads them astray.”

        I can’t speak for Prof Syll of course, but it seems there is a fairly obvious example.

        The Efficient Markets Hypothesis (and more generally any rational expectations based theory) posits an ergodic system governing investment decisions.
        https://larspsyll.wordpress.com/2014/12/27/rational-expectations-a-panel-discussion-with-robert-lucas/

        On the other hand as Keynes elucidated, processes around investment decisions are highly likely to be governed by a non-ergodic process.
        https://larspsyll.wordpress.com/2012/10/20/keynes-vs-bayes-on-information-and-uncertainty-wonkish/

        Surprising that Pontus can’t describe this example, considering how much time he spends posting on here. One would have thought he was familiar with the contents of this blog.

      • Nic,

        I do not think the EMH posits an ergodic system. In its basic form it simply states that asset prices contain all publicly relevant information, and that you cannot beat the market on the basis of that information. I believe there is ample empirical support for this statement.

        The Muth debate contains no statements that contradicts what I wrote above.

      • Thanks for the suggestion, Nic. However, like pontus I also do not see how the EMH posits an ergodic system. In fact, suppose a price followed precisely the the non-ergodic stochastic process that professor Syll described above, with the price equal to 3/8 in period 0. That would be a martingale process and thus be consistent with the EMH.
        (I want to thank you for giving the example. By discussing it we immediately become more concrete about what professor Syll might actually mean and why other people might not agree with him. Thus we have actual progress. I am happy to stand corrected if the above argument is wrong; that would also be progress🙂 )
        I would like to point out that, even if the EMH or generally RE theories are by definition ergodic, I do not think professor Syll demonstrated the point very clearly. I searched through his blog (with the search function in the upper right corner, which is probably not very sophisticated) for posts containing both the phrase “efficient market hypothesis” and the word “ergodic” (I think it picks up partial matches as well). I found about a dozen posts containing both, but none where he actually linked the EMH to ergodicity. Again I am happy to stand corrected and find an actual explicit discussion.
        I want to repeat myself a bit: I think there is a lot wrong with mainstream economics. But the only way to make things better is by applying a higher standard to ourselves. That means, amongst many other things, being as concrete and constructive as possible in our criticism. I have to say that I do not think professor Syll lives up to that standard. He seems to prefer facile words to hard work. I have asked him several questions over the last year (initially enthusiastically, later skeptically) pushing him to be more concrete and constructive. I have only received one answer, if memory serves, and that was entirely unsatisfactory. If professor Syll was genuinely interested in furthering understanding, he would have an answer ready for the question I asked here and he would gladly give it; even if he mistrusted my intentions or my heterodox credentials. After all, these discussions are more for the audience than the participants. Even if professor Syll thinks that I am some mainstream economist troll (which I am not), his intended readership would benefit from an actual answer.

      • Sorry, I just realized that the example I constructed is not a martingale, but it is easy to construct a martingale from it. I.e. let the value of the stochastic process be always equal to the expected value. So if we pick the first coin, rather than flipping it, we get outcome half deterministically forever, otherwise 1/4 forever. (I do not claim to be infallible!)

      • Well on this I think that Pontus is simply incorrect. The EMH, like any other rational expectations, theory relies implicitly (if not explicitly) on the ergodic axiom.

        According to various versions of the EMH investors are able to use their experience of the past to form accurate estimates of future investment returns (making the market unbeatable on average, at least using only public information). The ability to do this must clearly depend on past experience being indicative of future returns, in other words the investment system must exhibit statistical ergodicity.

        https://larspsyll.wordpress.com/2012/02/17/rational-expectations-and-ergodocity/

        Paul Davidson explains this well here,

        http://www.scielo.br/pdf/rep/v29n4/01.pdf

        Paul Samuelson said one must assume that the economy is ergodic or economics can’t be a science (which is simply extremely poor reasoning),

        https://larspsyll.wordpress.com/2012/08/07/paul-samuelson-and-the-ergodic-hypothesis/

        So if we want to make a statistical model of the economy we probably do need to assume some kind of ergodicity, and use prior statistics to estimate future outcomes. The resulting forecast will then be strictly limited in its skill, (and we should understand its limitations) as there is no reason to believe that the statistics collected for the past are going to be indicative for the future. But when you reach the level of assuming that markets follow ergodic processes, then you are no longer involved in advancing science, because that it patently impossible.

      • I’m confused now. The second link that you provide actually says that the Bachelier model and the geometric brownian motion are non-ergodic. And they are special cases of the EMH. So the EMH does not imply ergodicity. Also, my example (which is basically professor Syll’s example) still stands.
        Unfortunately I have no idea what Samueson said about ergodicity and why he said it; as far as I can tell there is no link to his comment. But if it was that ergodicity is necessary, then he is wrong and apparently many economist would disagree with him, if only implicitly, by using the EMH.
        There seem to be a misunderstanding about what ergodicity is. It is not the same as using the past to predict the future. For example, in professor Syll’s example the past throws are a reasonable indicator of the future. So that is not what ergodicity is about.
        Now, using the past to predict the future, in some way or another, is something that economists do. And it is in fact something that is simply unavoidable, although there are better and worse ways of doing it. This is just the Humean problem of induction. There is no fully satisfactory resolution, but what we usually do is reduce theorems to other things that we understand better. At any rate, if we don’t use induction at all, we cannot say anything. That would be unscientific, even if philosophically more precise. But at any rate, this has nothing to do with ergodicity.

      • Again, the EHM is a broader idea than what Nic makes it out to be. It is related to current and given public information (some stronger forms also relate it to private information, but most people feel that that’s a stretch).

        It is, however, certainly true that economist believe that there is much to learn from the past in order to deal with what the future may throw us. That is, history does — plus minus noise — tend to repeat itself.

      • “The second link that you provide actually says that the Bachelier model and the geometric brownian motion are non-ergodic. And they are special cases of the EMH. So the EMH does not imply ergodicity.”

        This discussion has nothing to do with the pattern of market prices, and everybody agrees that this is non-ergodic.

        “There seem to be a misunderstanding about what ergodicity is. It is not the same as using the past to predict the future. For example, in professor Syll’s example the past throws are a reasonable indicator of the future. So that is not what ergodicity is about.”

        The skill in using prior data to make predictions about the future is precisely what this is about. Prof. Syll’s example shows this well enough, before starting you know everything about the model, including the probability distribution of each coin but these statistics (at this moment there are no past throws, just knowledge of the model which is ‘perfect’ in the sense that it provides the same information as an infinite series of prior samples) *are not* a reasonable indicator of the future (expectation 3/8, not 1/2 or 1/4) because you are uncertain which coin will be chosen. This shows that no number of ‘past’ samples will ever be enough information, you need to know which coin will be used.

        “Now, using the past to predict the future, in some way or another, is something that economists do. And it is in fact something that is simply unavoidable, although there are better and worse ways of doing it. This is just the Humean problem of induction.”

        As I pointed out above, nobody has any problem with using statistics and induction to make estimates about the future. Making a forecast through a model is a reasonable example of doing this. But in making such a forecast would anybody actually be confident to say that their forecast is *definitely* going to be accurate? I don’t think that anybody would be so conceited to say so, and it would certainly not be borne out in practice. Yet this is the statement that the EMH makes about markets in some sense (e.g based on public information), it claims that markets can do the impossible and *accurately* forecast the future. We know (or should know) that this is not true due to the basic non-ergodic nature of the problem (which makes it impossible to make this accurate forecast based on prior experience). When your theory is based on the impossible happening, its simply incorrect. In other words, the market will get it ‘wrong’ from time to time, just as your forecast model will be ‘wrong’ from time to time, and this certainly needs to be kept in mind.

        EMH proponents will often argue at this point that the empirical challenges are based on markets where non-public information is traded on, or that random shocks have impacted the outcome in the short run, or that some market regulations are effecting this. Despite this there is serious issue built into the theory (as described above), it is literally impossible for the theory to be true based on our best scientific understanding of the world.

        @Pontus,
        “That is, history does — plus minus noise — tend to repeat itself.”

        That’s a pretty strong statement! I don’t think there is in any way sufficient evidence to support it.

      • Here is the original quote from Paul Samuelson,

        ‘Finally, there was an even more interesting third assumption implicit and explicit in the classical mind. It was a belief in unique long-run equilibrium independent of initial conditions. I shall call it the “ergodic hypothesis” by analogy to the use of this term in statistical mechanics.’

        which I took from ‘A History of Post Keynesian Economics Since 1936’, by J.E King. As you can see its not taken out of context in any way by either Davidson or Peters.

        Samuelson’s implication that the ergodic nature of the world would make certain kinds of progress in economics difficult may well be true. But this actually makes it un-scientific to simply create a theory which ignores this fact. If you can’t make general statements about the world then it is scientific to simply not make those kinds of statements and limit your statements to those which can be supported.

      • “[EHM] claims that markets can do the impossible and *accurately* forecast the future.”

        A common misconception, in particular amongst the heterodox, but no, the EHM does not rely on any such claim. It relies on the claim that you cannot forecast future stock prices more accurately than the 100,000 other investors trying to do the same with the same pieces of information. This is why Summers called it ketchup economics. The EHM is about informational efficiency, nothing else.

  3. Ah, I misunderstood. To be honest, I have also grown a bit disillusioned with professor Syll and the other heterodox that I have read (not extensively to be sure). There is obviously a lot wrong with mainstream economics, but neither the orthodox nor the heterodox seem all that interested in actually setting things right.

  4. I have the feeling that the argument is getting heated, which usually doesn’t help with the objectivity. Let me point out, as a way of hopefully appeasing, that my criticism of profssor Syll does not extend to Nic. Even if you haven’t convinced me yet, I think this is a proper dicussion. I also appreciate you pulling up the Samuelson quote.
    Having said that, the discussion started with my asking for an example of a theorem that was not ergodic, where the reality it sought to describe was.
    As I understand the EMH, it says that prices should behave as a martingale, i.e. there is no better forecast for the future than the current price. Many martingales are non-ergodic. Thus the EMH does not imply or assume ergodicity. I don’t see how it is an answer to my question.
    Perhaps you have a different understanding of the EMH. (I do not think the EMH is so well or clearly defined, so it could well be). I’m not entirely sure, but it seems you might think that either the EMH means that all publicly available information about future dividends (or cash flow or something) are reflected in the price, or even all private information, or even that it just perfectly captures all the future dividends. The last interpretation would definitely be at odds with the EMH. Even Eugene Fama, who is certifiably crazy, would not claim that. In my personal opinion, the EMH also does not imply the other two things. That is not generally agreed upon, but at any rate, even if the EMH implied that, that is not an ergodic theorem. For instance, the public info might be that dividends will be one or zero, to be determined by the throw of one of two coins, one with probability of heads 1/4, the other… Well, you see where I am going.
    I personally do not really believe in the EMH. I think it is pretty hard to beat the market and I expect that active manager have no added value. But if you know a firm has struck oil before word gets out, I think you stand to make quick buck. But I don’t think the problems with EMH have anything to do with ergodicity. If only because it does not assume or imply ergodicity, even in its zanier incarnations.

    I must admit that I do not really understand the point that you are trying to make about professor Syll’s example here:

    “The skill in using prior data to make predictions about the future is precisely what this is about. Prof. Syll’s example shows this well enough, before starting you know everything about the model, including the probability distribution of each coin but these statistics (at this moment there are no past throws, just knowledge of the model which is ‘perfect’ in the sense that it provides the same information as an infinite series of prior samples) *are not* a reasonable indicator of the future (expectation 3/8, not 1/2 or 1/4) because you are uncertain which coin will be chosen. This shows that no number of ‘past’ samples will ever be enough information, you need to know which coin will be used.”

    Since this is just a toy example, economists don’t actually have models about this artificial world. Clearly if the world behaved like this, then the best possible model would be non-ergodic. Are you saying that economists would not come up with such a model? Or that if the real world is like that, there cannot be a model? I’m really asking, I do not know exactly what your point is here.

    Finally, I think it is immaterial to our discussion, but I do not think the phrase “Samuelson said that we should accept the ergodic hypothesis because if a system is not ergodic you cannot treat it scientifically.” (from the beginning of the Ole Peters quote) is exactly a fair representation of the Samuelson quote you gave. His words do not exactly read like a ringing endorsement of the ergodicity hypothesis to me. But I don’t care about what Samuelson thought deep down, so the point is moot to me.

    • Sorry, I didn’t quite leave the reply in the appropriate spot. I’m new to the whole blogging thing.

    • At this moment I am just going to stick to explaining the example,

      “Since this is just a toy example, economists don’t actually have models about this artificial world.”
      We actually know the model, we know its exact specification. If we didn’t we could estimate its specification by running multiple games in an ensemble (in parallel), and recording the frequencies of various events. Running sufficient samples we might then derive a model of the example given. This expected value of this model, or a sample, would have an expected value of 3/8.

      “Clearly if the world behaved like this, then the best possible model would be non-ergodic. Are you saying that economists would not come up with such a model? Or that if the real world is like that, there cannot be a model?”
      I am saying something like the second, the best possible model of this, will not tell you the expected value of playing this game from the point of view of the player, which is either 1/4 or 1/2 depending on which coin is selected. The time average is how a player actually fares in this game, while the ensemble average is how the model is constructed.

      Given you were going to engage in this game, but your main risk was going bankrupt while playing. If you estimate your success based on both coins then you would observe an expected value of 3/8 with a larger variance. This is the actual effect of the game on an ensemble of players playing in parallel. You would set your risk profile higher in such a change than if you knew that the expected value was going to be 1/4 (I am imagining lower expected value makes you go bankrupt faster). On the other hand if you know that playing involves a risk of 1/4 expected value you might reasonably set a lower risk profile to try to avoid bankruptcy in case you end up in the 1/4 expected value scenario, even though this reduces your return in you end up in the 1/2 expected value scenario.

      There is no probability distribution (in the EMH this is the probability distribution of expected stock prices or the changes in them) for which the ensemble average and time average of the probability distribution match.


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