Tuesday, December 13, 2011

a little more on power laws

I wanted to respond to several insightful comments on my recent post on power laws in finance. And, after that, pose a question on the economics/finance history of financial time series that I hope someone out there might be able to help me with.

First, comments:

ivansml said...
Why exactly is power-law distribution for asset returns inconsistent with EMH? It is trivial to write "standard" economic model where returns have fat tails, e.g. if we assume that stochastic process for dividends / firm profits has fat tails. That of course may not be very satisfactory explanation, but it still shows that EMH != normal distribution. In fact, Fama wrote about non-gaussian returns back in 1960's (and Mandelbrot before him), so the idea is not exactly new. The work you describe here is certainly useful and interesting, but pure patterns in data (or "stylized facts", as economists would call them) by themselves are not enough - we need some theory to make sense of them, and it would be interesting to hear more about contributions from econophysics in that area.
James Picerno said...
It's also worth pointing out that EMH, as I understand it, doesn't assume or dismiss that returns follow some specific distribution. Rather, EMH simply posits that prices reflect known information. For many years, analysts presumed that EMH implies a random distribution, but the empirical record says otherwise. But the random walk isn't a condition of EMH. Andrew Lo of MIT has discussed this point at length. The market may or may not be efficient, but it's not conditional on random price fluctuations. Separately, ivansmi makes a good point about models. You need a model to reject EMH. But that only brings you so far. Let's say we have a model of asset pricing that rejects EMH. Then the question is whether EMH or the model is wrong? That requires another model. In short, it's ultimately impossible to reject or accept EMH, unless of course you completely trust a given model. But that brings us back to square one. Welcome to economics.
I actually agree with these statements. Let me try to clarify. In my post I said, referring to the fat tails in returns and 1/t decay of volatility correlations, that  "None of these patterns can be explained by anything in the standard economic theories of markets (the EMH etc)." The key word is of course "explained."

The EMH has so much flexibility and is so loosely linked to real data that it is indeed consistent with these observations, as Ivansml (Mark) and James rightly point out. I think it is probably consistent with any conceivable time series of prices. But "being consistent with" isn't a very strong claim, especially if the consistency comes from making further subsidiary assumptions about how these fat tails might come from fluctuations in fundamental values. This seems like a "just so" story (even if the idea that fluctuations in fundamental values could have fat tails is not at all preposterous).

The point I wanted to make is that nothing (that I know of) in traditional economics/finance (i.e. coming out of the EMH paradigm) gives a natural and convincing explanation of these statistical regularities. Such an explanation would start from simple well accepted facts about the behaviour of individuals, firms, etc., market structures and so on, and then demonstrate how -- because of certain logical consequences following from these facts and their interactions -- we should actually expect to find just these kinds of power laws, with the same exponents, etc., and in many different markets. Reading such an explanation, you would say "Oh, now I see where it comes from and how it works!"

To illustrate some possibilities, one class of proposed explanations sees large market movements as having inherently collective origins, i.e. as reflecting large avalanches of trading behaviour coming out of the interactions of market participants. Early models in this class include the famous Santa Fe Institute Stock Market model developed in the mid 1990s. This nice historical summary by Blake LeBaron explores the motivations of this early agent-based model, the first of which was to include a focus on the interactions among market participants, and so go beyond the usual simplifying assumptions of standard theories which assume interactions can be ignored. As LeBaron notes, this work began in part...
... from a desire to understand the impact of agent interactions and group learning dynamics in a financial setting. While agent-based markets have many goals, I see their first scientific use as a tool for understanding the dynamics in relatively traditional economic models. It is these models for which economists often invoke the heroic assumption of convergence to rational expectations equilibrium where agents’ beliefs and behavior have converged to a self-consistent world view. Obviously, this would be a nice place to get to, but the dynamics of this journey are rarely spelled out. Given that financial markets appear to thrive on diverse opinions and behavior, a first level test of rational expectations from a heterogeneous learning perspective was always needed.   
I'm going to write posts on this kind of work soon looking in much more detail. This early model has been greatly extended and had many diverse offspring; a more recent review by LeBaron gives an updated view. In many such models one finds the natural emergence of power law distributions for returns, and also long-term correlations in volatility. These appear to be linked to various kinds of interactions between participants. Essentially, the market is an ecology of interacting trading strategies, and it has naturally rich dynamics as new strategies invade and old strategies, which had been successful, fall into disuse. The market never settles into an equilibrium, but has continuous ongoing fluctuations.

Now, these various models haven't yet explained anything, but they do pose potentially explanatory mechanisms, which need to be tested in detail. Just because these mechanisms CAN produce the right numbers doesn't mean this is really how it works in markets. Indeed, some physicists and economists working together have proposed a very different kind of explanation for the power law with exponent 3 for the (cumulative) distribution of returns which links it to the known power law distribution of the wealth of investors (and hence the size of the trades they can make). This model sees large movements as arising in the large actions of very wealthy market participants. However, this is more than merely attributing the effect to unknown fat tails in fundamentals, as would be the case with EMH based explanations. It starts with empirical observations of tail behaviour in several market quantities and argues that these together imply what we see for market returns.

There are more models and proposed explanations, and I hope to get into all this in some detail soon. But I hope this explains a little why I don't find the EMH based ideas very interesting. Being consistent with these statistical regularities is not as interesting as suggesting clear paths by which they arise.

Of course, I might make one other point too, and maybe this is, deep down, what I find most empty about the EMH paradigm. It essentially assumes away any dynamics in the market. Fundamentals get changed by external forces and the theory supposes that this great complex mass of heterogenous humanity which is the market responds instantaneously to find the new equilibrium which incorporates all information correctly. So, it treats the non-market part of the world -- the weather, politics, business, technology and so on -- as a rich thing with potentially complicated dynamics. Then it treats the market as a really simply dynamical thing which just gets driven in slave fashion by the outside. This to me seems perversely unnatural and impossible to take seriously. But it is indeed very difficult to rule out with hard data. The idea can always be contorted to remain consistent with observations.

Finally, another valuable comment:
David K. Waltz said...
In one of Taleeb's books, didn't he make mention that something cannot be proven true, only disproven? I think it was the whole swan thing - if you have an appropriate sample and count 100% white swans does not prove there are ONLY white swans, while a sample that has a black one proves that there are not ONLY white swans.
Again, I agree completely. This is a basic point about science. We don't ever prove a theory, only disprove it. And the best science works by trying to find data to disprove a hypothesis, not by trying to prove it.

I assume David is referring to my discussion of the empirical cubic power law for market returns. This is indeed a tentative stylized fact which seems to hold with appreciable accuracy in many markets, but there may well be markets in which it doesn't hold (or periods in which the exponent changes). Finding such deviations  would be very interesting as it might offer further clues as to the mechanism behind this phenomenon.

NOW, for the question I wanted to pose. I've been doing some research on the history of finance, and there's something I can't quite understand. Here's the problem:

1. Mandelbrot in the early 1960s showed that market returns had fat tails; he conjectured that they fit the so-called Stable Paretian (now called Stable Levy) distributions which have power law tails. These have the nice property (like the Gaussian) that the composition of the returns for longer intervals, built up from component Stable Paretian distributions, also has the same form. The market looks the same at different time scales.
2. However, Mandelbrot noted in that same paper a shortcoming of his proposal. You can't think of returns as being independent and identically distributed (i.i.d.) over different time intervals because the volatility clusters -- high volatility predicts more to follow, and vice versa. We don't just have an i.i.d. process.
3. Lots of people documented volatility clustering over the next few decades, and in the 1980s Robert Engle and others introduced ARCH/GARCH and all that -- simple time series models able to reproduce the realistic properties of financial times, including volatility clustering.
4. But today I found several papers from the 1990s (and later) still discussing the Stable Paretian distribution as a plausible model for financial time series.

My question is simply -- why was anyone even 20 years ago still writing about the Stable Paretian distribution when the reality of volatility clustering was so well known? My understanding is that this distribution was proposed as a way to save the i.i.d. property (by showing that such a process can still create market fluctuations having similar character on all time scales). But volatility clustering is enough on its own to rule out any i.i.d. process.

Of course, the Stable Paretian business has by now been completely ruled out by empirical work establishing the value of the exponent for returns, which is too large to be consistent with such distributions. I just can't see why it wasn't relegated to the history books long before.

The only possibility, it just dawns on me, is that people may have thought that some minor variation of the original Mandelbrot view might work best. That is, let the distribution over any interval be Stable Paretian, but let the parameters vary a little from one moment to the next. You give up the i.i.d. but might still get some kind of nice stability properties as short intervals get put together into longer ones. You could put Mandelbrot's distribution into ARCH/GARCH rather than the Gaussian. But this is only a guess. Does anyone know?

7 comments:

  1. I think I see two types of people who propose stable distribution models:

    *) Those who see the logic of stable distributions and ignore all the evidence that they don't fit. Very much like those who see the logic (or simplicity) of Gaussian models and ignore all the evidence that they don't fit.

    *) Those who want to have a model in which they can assume a distribution of returns for an arbitrary length of time. I certainly see the attraction, I'm not sure how much downside there is.

    The appropriateness of such models, as always, will depend heavily on the use to which they are put.

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  2. I believe that one reason for differences between economists and others might be due to different questions asked. For example, a macroeconomist probably wouldn't care much about short-term correlations in market data, yet he would be very interested in how market prices reflect fundamentals (and expectations of those fundamentals) over longer horizons (say, months or years). In that case, thinking about prices as function of fluctuating fundamentals makes more sense. On the other hand, a quant trying to compute daily VaR would be in quite a different position.

    Unfortunately, I don't know answer to your question about modeling return distribution. In fact when I think about it, I don't recall much (if any) discussion of this issue in my economics courses, except for discussing GARCH in time series econometrics course. I guess that confirms that not everyone considers this to be a central question.

    And just to clarify, I'm not Mark. My first name is (not surprisingly) Ivan, and I'm an economics grad student at non-top but mainstream department in Europe.

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  3. Mark,

    I think I remember hearing that Mandelbrodt's early work was not well received in the academic circles for quite some time.

    That might explain why later on people wrote as if they hadn't heard of him - in fact they hadn't?

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  4. There was a whole revival of Mandelbrodt's work following the rise of the wavelet transform and multi-fractal formalism (i.e. in studying intermittence in fully developed-turbulence) which eventually flowed to finance.

    See for instance this article:

    http://www.wilmott.com/pdfs/0503_bouchaud.pdf

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  5. Mandelbrodt's work is quite impressive so I guess it's worth to talk to about. This is a good read, worth my time spent in reading it! Thanks for sharing!

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  6. perhaps you may find something useful by browsing through this series of papers written by Mandelbrot himself and published in Quantitative Finance a few years ago (2001).

    Quantitative Finance volume 1 pages 113-123 and 124-30 are likely to be a useful introduction for the mathematically-inclined, whereas volume 1 pages 427-40 and 641-49 describe further developments based upon the applications of fractals in finance.

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