Tuesday, August 12, 2014

investing volatility01 boring01 variance01 rentseeking power laws umich : not in case of boring thermodynamic equilibrium, i.e. boring ones, should have correlations which decay exponentially over space and time

http://vserver1.cscs.lsa.umich.edu/~crshalizi/notabene/power-laws.html

http://quantivity.wordpress.com/2012/01/03/physics-biology-peltzman-finance/#more-9021

variance hits like blk swan in groups
Notebooks

Power Law Distributions, 1/f Noise, Long-Memory Time Series

05 Sep 2013 11:26

Why do physicists care about power laws so much? I'm probably not the best person to speak on behalf of our tribal obsessions (there was a long debate among the faculty at my thesis defense as to whether "this stuff is really physics"), but I'll do my best. There are two parts to this: power-law decay of correlations, and power-law size distributions. The link is tenuous, at best, but they tend to get run together in our heads, so I'll treat them both here. The reason we care about power law correlations is that we're conditioned to think they're a sign of something interesting and complicated happening. The first step is to convince ourselves that in boring situations, we don't see power laws. This is fairly easy: there are pretty good and rather generic arguments which say that systems in thermodynamic equilibrium, i.e. boring ones, should have correlations which decay exponentially over space and time; the reciprocals of the decay rates are the correlation length and the correlation time, and say how big a typical fluctuation should be. This is roughly first-semester graduate statistical mechanics. (You can find those arguments in, say, volume one of Landau and Lifshitz's Statistical Physics.) Second semester graduate stat. mech. is where those arguments break down --- either for systems which are far from equilibrium (e.g., turbulent flows), or in equilibrium but very close to a critical point (e.g., the transition from a solid to liquid phase, or from a non-magnetic phase to a magnetized one). Phase transitions have fluctuations which decay like power laws, and many non-equilibrium systems do too. (Again, for phase transitions, Landau and Lifshitz has a good discussion.) If you're a statistical physicist, phase transitions and non-equilibrium processes define the terms "complex" and "interesting" --- especially phase transitions, since we've spent the last forty years or so developing a very successful theory of critical phenomena. Accordingly, whenever we see power law correlations, we assume there must be something complex and interesting going on to produce them. (If this sounds like the fallacy of affirming the consequent, that's because it is.) By a kind of transitivity, this makes power laws interesting in themselves. Since, as physicists, we're generally more comfortable working in the frequency domain than the time domain, we often transform the autocorrelation function into the Fourier spectrum. A power-law decay for the correlations as a function of time translates into a power-law decay of the spectrum as a function of frequency, so this is also called "1/f noise". Similarly for power-law distributions. A simple use of the Einstein fluctuation formula says that thermodynamic variables will have Gaussian distributions with the equilibrium value as their mean. (The usual version of this argument is not very precise.) We're also used to seeing exponential distributions, as the probabilities of microscopic states. Other distributions weird us out. Power-law distributions weird us out even more, because they seem to say there's no typical scale or size for the variable, whereas the exponential and the Gaussian cases both have natural scale parameters. There is a connection here with fractals, which also lack typical scales, but I don't feel up to going into that, and certainly a lot of the power laws physicists get excited about have no obvious connection to any kind of (approximate) fractal geometry. And there are lots of power law distributions in all kinds of data, especially social data --- that's why they're also called Pareto distributions, after the sociologist. Physicists have devoted quite a bit of time over the last two decades to seizing on what look like power-laws in various non-physical sets of data, and trying to explain them in terms we're familiar with, especially phase transitions. (Thus "self-organized criticality".) So badly are we infatuated that there is now a huge, rapidly growing literature devoted to "Tsallis statistics" or "non-extensive thermodynamics", which is a recipe for modifying normal statistical mechanics so that it produces power law distributions; and this, so far as I can see, is its only good feature. (I will not attempt, here, to support that sweeping negative verdict on the work of many people who have more credentials and experience than I do.) This has not been one of our more successful undertakings, though the basic motivation --- "let's see what we can do!" --- is one I'm certainly in sympathy with. There have been two problems with the efforts to explain all power laws using the things statistical physicists know. One is that (to mangle Kipling) there turn out to be nine and sixty ways of constructing power laws, and every single one of them is right, in that it does indeed produce a power law. Power laws turn out to result from a kind of central limit theorem for multiplicative growth processes, an observation which apparently dates back to Herbert Simon, and which has been rediscovered by a number of physicists (for instance, Sornette). Reed and Hughes have established an even more deflating explanation (see below). Now, just because these simple mechanisms exist, doesn't mean they explain any particular case, but it does mean that you can't legitimately argue "My favorite mechanism produces a power law; there is a power law here; it is very unlikely there would be a power law if my mechanism were not at work; therefore, it is reasonable to believe my mechanism is at work here." (Deborah Mayo would say that finding a power law does not constitute a severe test of your hypothesis.) You need to do "differential diagnosis", by identifying other, non-power-law consequences of your mechanism, which other possible explanations don't share. This, we hardly ever do. Similarly for 1/f noise. Many different kinds of stochastic process, with no connection to critical phenomena, have power-law correlations. Econometricians and time-series analysts have studied them for quite a while, under the general heading of "long-memory" processes. You can get them from things as simple as a superposition of Gaussian autoregressive processes. (We have begun to awaken to this fact, under the heading of "fractional Brownian motion".) The other problem with our efforts has been that a lot of the power-laws we've been trying to explain are not, in fact, power-laws. I should perhaps explain that statistical physicists are called that, not because we know a lot of statistics, but because we study the large-scaled, aggregated effects of the interactions of large numbers of particles, including, specifically, the effects which show up as fluctuations and noise. In doing this we learn, basically, nothing about drawing inferences from empirical data, beyond what we may remember about curve fitting and propagation of errors from our undergraduate lab courses. Some of us, naturally, do know a lot of statistics, and even teach it --- I might mention Josef Honerkamp's superb Stochastic Dynamical Systems. (Of course, that book is out of print and hardly ever cited...) If I had, oh, let's say fifty dollars for every time I've seen a slide (or a preprint) where one of us physicists makes a log-log plot of their data, and then reports as the exponent of a new power law the slope they got from doing a least-squares linear fit, I'd at least not grumble. If my colleagues had gone to statistics textbooks and looked up how to estimate the parameters of a Pareto distribution, I'd be a happier man. If any of them had actually tested the hypothesis that they had a power law against alternatives like stretched exponentials, or especially log-normals, I'd think the millennium was at hand. (If you want to know how to do these things, please read this paper, whose merits are entirely due to my co-authors.) The situation for 1/f noise is not so dire, but there have been and still are plenty of abuses, starting with the fact that simply taking the fast Fourier transform of the autocovariance function does not give you a reliable estimate of the power spectrum, particularly in the tails. (On that point, see, for instance, Honerkamp.)
See also: Chaos and Dynamical Systems; Complex Networks; Self-Organized Criticality; Time Series; Tsallis Statistics

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William J. Reed and Barry D. Hughes, "From Gene Families and Genera to Incomes and Internet File Sizes: Why Power Laws are so Common in Nature", Physical Review E 66 (2002): 067103 [This is, as I said, perhaps the most deflating possible explanation for power law size distributions. Imagine you have some set of piles, each of which grows, multiplicatively, at a constant rate. New piles are started at random times, with a constant probability per unit time. (This is a good model of my office.) Then, at any time, the age of the piles is exponentially distributed, and their size is an exponential function of their age; the two exponentials cancel and give you a power-law size distribution. The basic combination of exponential growth and random observation times turns out to work even if it's only the mean size of piles which grows exponentially.]
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R. Alexander Bentley, Paul Ormerod, Michael Batty, "An evolutionary model of long tailed distributions in the social sciences", arxiv:0903.2533 [This is a minor modification of the classical Yule/Simon mechanism for random growth, with the main advantage being that (with the right parameter tweaking) it allows for more turn-over of which values are most common. Unsurprisingly, this is done by adding extra parameters, and so the family of distributions is more flexible. But they use bad statistical procedures, and the finding that the estimated power law exponent grows as the amount of data held in the tail shrinks is simply explained: the tails aren't power laws.]

Mason Porter's Power Law Shop

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Cline, heavy-tailed noise, 1983 (?)
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Alexander Roitershtein, "One-dimensional linear recursions with Markov-dependent coefficients", math/0409335 = Annals of Applied Probability 17 (2007): 572--608 [To summarize the abstract, suppose S(n) = A(n) + B(n)*S(n-1), where A(n) and B(n) are Markov sequences. Then "the distribution tail of its stationary solution has a power law decay." This sounds like Simon's argument made more general.]
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Gennady Samorodnitsky and Murad S. Taqqu, Stable Non-Gaussian Random Processes
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Attilio L. Stella, Fulvio Baldovin, "Anomalous scaling due to correlations: Limit theorems and self-similar processes", arxiv:0909.0906
Stilian A Stoev, George Michailidis, "On the Estimation of the Heavy-Tail Exponent in Time Series using the Max-Spectrum", arxiv:1005.4329
Stilian A. Stoev and Murad S. Taqqu, "Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights", Methodology and Computing in Applied Probability 9 (2007): 55--87
Sarah Touati, Mark Naylor, and Ian G. Main, "Origin and Nonuniversality of the Earthquake Interevent Time Distribution", Physical Review Letters 102 (2009): 168501
Ciprian Tudor and Frederi Viens, "Variations and estimators for the selfsimilarity order through Malliavin calculus", arxiv:0709.3896
Caglar Tuncay, "A universal model for languages and cities, and their lifetimes", physics/0703144
Marta Tyran-Kaminska, "Convergence to Lévy stable processes under strong mixing conditions", arxiv:0907.1185
Sergio Venturini, Francesca Dominici, Giovanni Parmigiani, "Gamma shape mixtures for heavy-tailed distributions", Annals of Applied Statistics 2 (2008): 756--776 = arxiv:0807.4663
Yogesh Virkar, Aaron Clauset, "Power-law distributions in binned empirical data", arxiv:1208.3524
Rafal Weron
- "Estimating long range dependence: finite sample properties and confidence intervals," cond-mat/0103510
- "Measuring long-range dependence in electricity prices," cond-mat/0103621
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Notebooks

Burned all my notebooks
What good are notebooks
If they won't help me survive?

But a curiosity of my type remains after all the most agreeable of all vices --- sorry, I meant to say: the love of truth has its reward in heaven and even on earth. ---Nietzsche, Beyond Good and Evil, 45

They're, well, notebooks --- things I find amusing, outrageous, strange or otherwise noteworthy; notes towards works-in-glacial-progress; hemi-demi-semi-rants; things I want to learn more about; lists of references; quotations from the Talking Heads where appropriate. If you can help with any of these, I'd be grateful; if you can tell me of anything I can profitably prune, I'd be even more grateful.
There is a list of frequently asked questions (FAQ), along with answers, and a colophon, which explains more than anyone would want to know about how these pages are put together. If your question isn't answered in either place, feel free to write, though, sadly, I can't promise a timely reply.
---Cosma
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Physics, Biology, or Peltzman?

January 3, 2012

Quantivity is fortunate to be acquainted with numerous folks who have earned consistent returns over multiple decades without significant drawdown. Although they have varying trading strategies, there is a common theme which unifies them: top-down systematic focus on the sociology of market participants.
This focus is not behavioral finance, in search of anomalies driven by cognitive biases divergent from equilibrium (although majority do that too). Rather asking inferential sociological questions, such as: was the market “efficient”, in the Fama sense, during the post-war decades prior to 2000 because people expected it to be (blissfully ignoring a few hiccups); in contrast to how it is commonly understood and formalized, with reverse causality: market is assumed to be efficient, thus people understand it as such.
Similarly, have the past 15 years been “inefficient”, in the bubble and anomaly sense, because cultural faith among investors in such “efficiency” was lost; or, did they lose faith because the market became inefficient? Big difference.
In other words: is finance governed by physics, biology, or Peltzman?

The traditional answer of market hypothesis, provided by financial economics via microeconomic principles of equilibrium and efficiency: causality flows from market to investor. This explanation comes in two variants, known by their colloquial analogical fields:

Physics: market is governed by immutable mathematical principles and can be formalized into coherent predictive models, either in favor or contradiction of excess returns; exemplified by classic weak/strong EMH theory
Biology: market is governed by evolutionary principles ala Darwin, as exemplified by Lo’s 2004 AMH article: “Very existence of active liquid financial markets implies that profit opportunities must be present. As they are exploited, they disappear. But new opportunities are also constantly being created as certain species die out, as others are born, and as institutions and business conditions change.” (p. 24)

Yet, both these explanations suffer from implicitly begging the question: conjure “a market” with desired attributes and then derive conclusions. The physics perspective assumes immutability, conceivability, and mathematical expressiveness for its hypothesized market. While the biology perspective endows the hypothesized market with even more sophisticated Darwinian traits, presumably driven by underlying physical principles so inscrutable as to defy mathematical formalization.
An alternative explanation is to apply the self-fulfilling Peltzman effect to financial markets, and reverse causality: markets behave as they do because of investor sociology, rather than arising emergent from implicit cooperation of equilibrium-seeking rational microeconomic agents.
In other words: when investors believe the market is rational (irrespective of whether that belief is well-founded), then they embody Dunning-Kruger by ex ante faithfully dumping money into their 401K each month; in doing so collectively, the investment management industry undertakes its rent seeking activity resulting in the market possessing ex post “efficient” characteristics. Conversely, when investors believe the market is irrational, they either: go to cash, pursue uninformed non-collective trading, or both. Both of which result in anomalous market behavior, uncontrollable by the industry, either due to decreased liquidity or absence of predictable momentum.
If the market is indeed Peltzmanian, then the real question is how to best quantify and model primary and spillover effects resulting from investor sociology as they unfold ephemerally.

LiveJournal syndication (not by me)

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