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## Non-normal normality

from Lars Syll

Asset price distributions are of great practical significance for portfolio managers. Standard finance theory assumes that asset price changes follow a normal distribution—the well-known bell curve. That this assumption is roughly accurate most of the time allows analysts to use very robust probability statistics. For example, for a sample that follows a normal distribution, you can identify the population average and characterize the likelihood of variance from that average.

However, much of nature—including the man-made stock market—is not normal. Many natural systems have two defining characteristics: an ever-larger number of smaller pieces and similar-looking pieces across the different size scales. For example, a tree has a large trunk and a number of ever-smaller branches, and the small branches resemble the big branches. These systems are fractal. Unlike a normal distribution, no average value adequately characterizes a fractal system. Fractal systems follow a power law.

Using the statistics of normal distributions to characterize a fractal system like financial markets is potentially very hazardous. Yet theoreticians and practitioners do it daily. The distinction between the two systems boils down to probabilities and payoffs. Fractal systems have few, very large observations that fall outside the normal distribution. The classic example is the crash of 1987. The probability (assuming a normal distribution) of the market’s 20%-plus plunge in one day was so infinitesimally low it was practically zero. And still the losses were a staggering \$2 trillion-plus.

1. August 2, 2019 at 1:43 am

Black swan, long tail, power law, Lévy (stable) distribution: these are different names of the same phenomena.

• August 2, 2019 at 3:19 pm

This distribution was studied mainly between two World Wars by French mathematician Paul Pierre Lévy, who develped almost alone the theory of stable stochastic processes and discovered Levy distribution (or more precisely stated, the explicit formular of the Laplace tranform of stable distributions).

One of his students in Ecole Polytechniques was Benoit Madelbrot.

2. August 2, 2019 at 7:19 am

TEST link: A 2011 blog post applying stable distributions, an early component of the fractal view, to S&P 500 price data.

3. August 2, 2019 at 7:44 am

I wrote about one early component of the fractal view–stable statistical distributions–at the Multiplier Effect blog in this 2011 post at that site, which is the house blog of the Levy Institute, my employer at the time. Also, I maintain links to some resources on stable distributions on the web at the Assorted Links page at my website, Greg Hannsgen’s Economics Blog.

Stable distributions, like Gaussian distributions, can be derived from limiting laws. In the non-Guassian case, stable distributions are fat-tailed and potentially asymmetrical. (They are important along with the power laws mentioned in this interesting RWER post!)

Also, I did some academic work on fat-tailed stable distributions in the residuals of orthodox-style structural vector autoregression, finding that they fit the residuals better than normal distributions and that they often fit better than other kinds of distributions such as Student’s t in horseraces among nonnested distributions.
These scholarly papers can be found among other places at my REPEC page at https://econpapers.repec.org/RAS/pha363.htm and at my profile at researchgate.net.

If the macroeconomic variables in the types of VARs I studied are distributed in this way, it could be because they are very much part of the same system that encompasses financial price variables with fat-tailed (and often asymmetric) distributions.

Historians of thought Phil Mirowski and Esther-Mirjam Sent have told much of the story of the development and eventual suppression of the fractal view within mainstream finance and economics. B. Mandelbrot’s fractals became the basis for a physics of “rough” phenomena such as coastlines, for example.

4. August 2, 2019 at 9:28 am

Black swan, long tail, power law distribution, Lévy (stable) distribution are different names of the same phenomena. Pareto law and Zipf law are special cases of the above distributions.

5. August 2, 2019 at 10:43 am

Nice intro…where’s the rest of the article?

6. August 2, 2019 at 2:54 pm

“However, much of nature—including the man-made stock market—is not normal. Many natural systems have two defining characteristics: an ever-larger number of smaller pieces and similar-looking pieces across the different size scales. For example, a tree has a large trunk and a number of ever-smaller branches, and the small branches resemble the big branches. These systems are fractal. Unlike a normal distribution, no average value adequately characterizes a fractal system. Fractal systems follow a power law”

This is a post that all would-be economists should read and study. In my research in finance, i found only self-similarity and power Law distributions. And often I reflect on the number of misleading conclusions derived from the assumption of normal distributions; just think of the so-called Efficient Market Hypothesis!

7. August 2, 2019 at 3:10 pm

Hardly Random – Donald pulls the market up or down with his tweets. I would bet he has traders that are informed of his tweets before he sends them.

You could argue the the GOP bought options on the market before the tax cut. Easy money.

8. August 3, 2019 at 4:37 am

As I have explained in the top two of my posts, the same non-normal distribution is studied by various different names, but we should know (1) why power law distributions were not recognized earlier and (2) why they are in a sense universal. These are highly mathematical problems which require physicist instinct. No ontology and philosophy could not detect even the existence of such phenomena. How can we study power law distributions and Lévy distribution without the aid of mathematics and physics?

As Dominique says, “This is a post that all would-be economists should read and study. ” But the contents tolled in the post have been studied since the seminal work Econophysics, 2000, by Mantegna and Stanley. The problem is why there has been a high barrier between those who work to study economic phenomena in unconventional new methods and those who are not satisfied by the mainstream economics.

• August 3, 2019 at 11:03 pm

Two typos:
(1 ) “physicist instinct” should be read “physicists’s intuition”.
(2) “tolled” should be corrected as “told”

9. August 7, 2019 at 10:11 am

The 20th century saw a universal experiment in impersonal society. Humans were everywhere organized by remote abstractions – states, capitalist markets, science. For most people it was impossible to make a meaningful connection with these anonymous institutions and this was reflected in intellectual disciplines whose structures of thought had no room for human society in them. Economics tops the list of those disciplines. Feeling disempowered, ordinary people everywhere sought self-expression where they could – in domestic life and informal economic practices. The three most important components of this impersonal modern economic life – people, machines and money – have never been properly considered by modern economics, or most other academic disciplines devoted to their study. These are not first and foremost mathematical problems. They are cultural problems created by literally dozens if not hundreds of events and agents in creating what I’ve labeled a “universal experiment in impersonal society” for much of the 20th century. The failure of financial markets to behave as predicted by economists and others is not unusual for human creations. In fact, few societal structures ever behave as expected. Funny thing, however, people keep assuming they will or must behave as predicted. I agree with Yoshinori that we should try to find out why power law distributions were not recognized earlier and why they seem common in financial markets. These are only the mathematical part of our work, however. More importantly, we need to look at the structure, functioning, regulation, and agents involved in these markets “on the ground,” not as depicted in textbooks or even government laws or policy to figure out why the “theory” says the markets operate and have results of a specific form but in actual operation do not have these results. And then consider what kinds of design is possible, feasible with what likely results. This is an iterative process. It cannot end. This kind of work is what anthropologists, sociologists, and supposedly economists bring to the table.

10. August 7, 2019 at 11:52 am

Ken Zimmerman>>
These are only the mathematical part of our work, however. More importantly, we need to look at the structure, functioning, regulation, and agents involved in these markets “on the ground,” not as depicted in textbooks or even government laws or policy to figure out why the “theory” says the markets operate and have results of a specific form but in actual operation do not have these results.

What econophysics found was very different from what textbooks described. Without that “only mathematical part”, we could not know how different way a stock market behaves. According to Mantegna and Stanley (2000), stock price fluctuation second by second (which is called tick data) can be characterized as truncated Lévy flight. Truncation is supposed to be caused by limit up/down rule. If there are no such rules, it is predicted that stock price fluctuation is more perfectly near to Lévy distribution that has no finite standard variation. Can you imagine such probability distribution? But this is what Paul Lévy was studying in 1930’s. Non-mathematical study can arrive to discover such phenomena. This was in 1930’s beyond physicists’ imagination but statistic physicists came to know that we can observe such statistics in many human phenomena (not particularly in physical phenomena). We know now even the reason why most phenomena in nature are described by normal distribution and why there are many human phenomena that are far from normal distributions. This is the power of mathematics and statistical physics.

Why Ken Zimmerman wants to excluded mathematics and physics? They can contribute to the thinner understanding of a phenomenon as well as other modes of investigations. Mathematics and physics have contributed to the understanding of ” structure, functioning”, and even “regulation” if we study the object seriously, in some cases far better than other methods.

• August 8, 2019 at 2:42 am

Yoshinori, I do not want to exclude mathematics and physics. I could not exclude them, even if I wanted to. Humans create them and embed them in many parts of their cultures. It would be like excluding English, French, or another major language. And humans have found both useful, more or less. But the future of mathematics is now largely controlled by experts (mathematicians, logicians, etc.). Same with physics. It’s this structure of control to which I object. The larger community now has little input into either. But it was the whole community that created each, originally. Mathematics may be the “language of the universe,” as suggested by some physicists today, or the language of God as suggest by Newton, but when we use mathematics and physics it’s not the universe or God that’s speaking. It’s humans. I want to ensure that isn’t lost in the mysteries of cosmology or the “truths” of the universe.

• August 8, 2019 at 3:37 am

Ken >> I do not want to exclude mathematics and physics.

OK! Agreed! It’s done.

11. August 8, 2019 at 5:29 am

Ken Zimmerman’s post at 2:42 am, August 8, 2019, presents an important point that becomes often cause of misunderstandings in considering economics research strategy. It is the relation between complexity and mathematics. People often argue that it is no good to try to use mathematics because economy is very complex and exceeds the possibility of mathematical analyses.

Let me cite two sentences from Ken:
>> Mathematics may be the “language of the universe,” as suggested by some physicists today, or the language of God as suggest by Newton, but when we use mathematics and physics it’s not the universe or God that’s speaking. It’s humans.

This is an idea that is often pointed both by mathematicians and physicists. Simply stated, this was true before the arrival of complexity science and thought. In 1948 Warren Weaver wrote a seminal paper: Science and Complexity, American Scientist 36(4): 536-544. (Wikipedia does not mention this paper and its influences, but you can find the paper on various sites.) He distinguished three phases of science development: (1) science of simplicity, (2) science of unorganized complexity and (3) science of organized complexity. Weaver claimed that third stage of science is now becoming possible mainly by appearance of computers. Weaver included economics in the science of organized complexity.

Now, how is mathematics is related to science of organized complexity? If mathematics is the sapience of God, complexity does not matter, because God knows everything and there is nothing that exceed his capability. For humans, complexity is one of most important facts when we want to think of science research strategy. The most difficult things in economics (as in many other sciences) is to overcome the gap of Difficulty and our Competence. The idea of CD gap was first advanced by Ronald Heiner in his paper: “The Origin of Predictable Behavior,” American Economic Review, September 1983, 73, 560-95. In Heiner’s case, CD gap was the gap for animals and human agents in routine works. But CD gap concept is also useful when we consider research strategy for economics. Economy is apparently a complex system and economists’ capability is limited. How to overcome this big gap? This must be the main subject matter to make a break-through in economics.

Computer simulation is one of possible methods. It has may merits and also demerits. Please see my paper:
A Guided Tour of the Backside of Agent-Based Simulation. Chapter 1, pp.3-50,in Kita, Taniguchi, and Nakajima (Eds.) Realistic Simulation of Financial Markets / Analyzing Market Behavior by the Third Mode of Science, Springer Japan, 2016.

As I have argued in this paper, conceptual work is very important and the starting point.
Collection of various facts precedes it. Computer simulation may open a new era for scientific research method. I have called it the third mode of scientific research. However, not speaking of laughable dictum “garbage in, garbage out,” simulation shows a possibility but it is quite difficult to get a law-like result. So the mathematical reasoning is necessary. If once we can overcome the apparent CD gap, we can see far more clearly what is happening in the seemingly complex economics processes. Of course, this is not an easy way. Risk that we fail in this attempt is always bigger than the possibility of success. But, once we find a suitable structure and the mathematics by which to analyze it, we can understand deeply what is happening under the seemingly complicated phenomena. Mathematics has its role, an important role that can be replaced by no other means. Therefore, we may roughly state that mathematics can be a tool which clarify the complexity.

Just this kind of story happened for us, as it is briefly told in the Preface of our new book:
Microfoundations of Evolutionary Economics, Springer Japan, 2019.
https://www.researchgate.net/publication/334508762_Microfoundations_of_Evolutionary_Economics

After my first attempt failed, Taniguchi succeeded to prove by computer simulation that input-output relations of firms’ quantity adjustment process converge for most of the cases if adjustment is based on average of past sales. But this knowledge was ambiguous because we cannot know how this is general and when the process diverges. It was Morioka’s ingenious work in mathematics that overcome this inconvenience and we get a good theorem. Of course, there are still several points to make clear, but we think a breakthrough was achieved.

If Ken send me an e-mail (y@shiozawa.net), I can send you a PDF copy of the new book.

• August 8, 2019 at 12:55 pm

Yoshinori, most networks are complex. Complexity is any network that is non-linear, non-periodic. The simplest series of measurements can become complex within a short time span. For example, measurements of GDP or unemployment (if economists believed in unemployment). Humans created the measurements. Made all the choices. So, complexity is the result of humans’ work. So, when choosing research strategies, it’s best to assume that whatever network (series of interrelated numbers) one examines that the network is complex.

What is a computer? Think about it this way. Imagine a squirrel that can calculate any algorithm in seconds. Thousands of times faster than any human. Has no connection to or understanding of the algorithm. Just calculates them. That’s a computer. They say AI will have both connection and understanding. Personally, I don’t agree. Why is there what you call a Difficulty and our Competence gap? Simply put, it’s the result of the creative process in economics. Humans create economies, but not alone. They do it via interactions with other humans and with the nonhuman (e.g. geography, disease, food, geology). There are millions of interactions over thousands of years. Leading to institutional forms for some of the resulting creations, from markets to money to economic science. Complexity all around. The creations grew beyond total human control hundreds of years ago. Makes changing these creations difficult. They answer the basic questions that face humans. How to make their creations durable and precise enough to protect the species. Computer modeling can help but it can assure neither of these goals. Just don’t forget that modeling is based on the assumptions humans put into them, all run at high speed. So, humans figuring out what’s been created becomes ever more difficult with these models.

As to your book, tell me what you assume is a micro foundation and I might take a look at it. However, I’d rather search archives or do ethnographic research on how people create their cultures, including economics. And the mathematics and philosophy humans create to explain these creations.

• August 8, 2019 at 4:56 pm

Ken, I have once contemplated and reflected on almost all problems you have listed above.

Chapter one of our new book, which has the same title as the book itself, is based on assumptions that (1) human beings are short sighted, (2) their rational capability is extremely bounded, and (3) their capability of action is limited and can change very small part of their environment. I have argued long on the ubiquity of intractable problems. Humans are acting in the complex environment. This is my presumption of formulating economic behaviors.

Chapter 2 of our book is titled “A Large Economic System with Minimally Rational Agents.” This presents a vision on how market economy works, which is totally different from Arrow and Debreu or any other neoclasssical economics vision that assumes price adjustment.

• August 8, 2019 at 5:45 pm

I contended that “I have once contemplated and reflected on almost all problems you have listed above.” You can check by yourself if you download papers that I have made public:

-Economic theory and the complexity of capitalism
https://www.researchgate.net/publication/236119923_Economic_theory_and_the_complexity_of_capitalism

-Evolutionray economics in the 21s century: a manifesto
https://www.researchgate.net/publication/233943458_Evolutionary_Economics_in_the_21st_Century_A_Manifesto

-The nature of kowledge in a complex world
https://www.researchgate.net/publication/236115982_The_Nature_of_Knowledge_in_a_Complex_World

• August 9, 2019 at 3:00 am

Yoshinori, all interesting. Just two questions. How did you arrive at the presumptions about humans and human actions? Individuals define the scope of their actions through their cultures. They must since they’re the only ones who can. That includes defining rational and not rational, the scope of their actions, and what their culture can and cannot achieve. That these sometimes, even often fail is shown in history. Also shown is that this work is done together with the nonhuman. It’s not up to you or anyone else, expert or not to re-define any of this as you please. Part of this work of these many humans and nonhumans is the creation on institutions that carry out culture in material form. Again, not your job to redefine any of these. Arrow and Debreu make the same errors. They want to reshape economic life in terms of a model they favor. The institution in which their lives are embedded, economic science is part of an aspect of western cultures involving elites who see it as their role in society to educate the ignorant masses about how economies ought to be organized and ought to perform. I suggest you distance yourself from this institution. Unfortunately, however economic science is not the only such institution. Large segments of science and the political institution share the same objectives. A couple of books that don’t pursue such an objective are worth a read. Economic Lives: How Culture Shapes the Economy by Viviana Zelizer, Stone Age Economics by Marshall Sahlins.

12. August 9, 2019 at 8:38 am

To know limits of three kind of capability is obvious. You can confirm by yourself by everyday life.

(a) Myopic sight
Do you know all prices of products that are sold in the market?
Do you know how much your competitors or your clients are planning to produce?
Do you know how much your product will sell tomorrow or the next week?
Even if we do not know none of them, a firm can make a plan how much it produces today, based on the past average of sales of your product. The details are given in Chapter 2 and Chapter 3 of our book.

(b) Bounded rationality
This is the question that is most argued since Herbert A. Simon presented this concept. You may think this is abstract but it is in fact a distilled knowledge of our lives.

In numbers of my papers including Chapter 1, and papers I have listed above, I have given a mathematical reasons why we cannot optimize many problems. This is the question widely argued under the title of intractability. Core of my argument is the ubiquity of intractable problems when they are formulated as optimization (maximization or minimization).

(c) Limited range to change the environment intentionally
Everybody feels this limits in almost all our life. Principle of Alibi assumes that we cannot be at place A and at place B at the same time when A and B are distanced.

You must know that in many cases we are payed by hours (when employed). This is because what we can do is limited and cannot complete all works even if we intended it.

There is no necessity to appeal to Sahlins, culture or institutions. I am questioning how these entities (culture, institutions and behavioral routines) arrive inevitably. There is a deeper level of conditions which forces us to depend or draw on culture and institutions.

This is the essence of my argument. CD gap necessitates that we behave according to routines. Have you ever imagined how a routine behavior is structured? Please read Chapter 1 or other papers. Shortly stated, we behave by a series of CD-transformation (reaction from Cognitive meaning to Directive meaning) when we face a big CD-gap (gap between our Capacity/Capability and the Difficulty of problems). I am sorry of this confusing terminology, but I adopted these terms from those who (I believe) first created these concepts: CD-gap from Ronald Heiner and CD-transformation from sociologist Tamito Yoshida.

Second point is to understand why a set of CD-transformations or routines are effective. It depends on the environment. However, in the case of economy, this environment itself is a product of human behaviors. Therefore, we need to see all these relations from evolutionary viewpoint. There are many micro-macro loops (I have explained this concept in my papers).
A conclusion that we can derive from the existence of micro-macro loops is that both methodological individualism and holism are both defective. They look economy from a single direction bottom-up and top-down.

• August 9, 2019 at 2:52 pm

Yoshinori, it’s not that simple. On knowledge we can conclude is that no one knows everything, but everyone knows some things. The focus of our research is figuring out at what point(s) on this continuum are the people on whom our research is centered. That’s what real social scientists do regularly.

Whether a person is rational depends on which of the multiple definitions of rationality one favors. Economic, scientific, logical, and mathematical rationality have no minimum or maximum since they are based on axioms generally believed to transcend experience. Pragmatic, planning, and democratic rationality place limits on people since they are believed to be based in human history. For the first four if one meets the axiomatic requirements then one must be considered rational. For the latter three, it’s questionable whether any decision about rationality can be made apart from situations limited in time and space.

You write, “There is no necessity to appeal to Sahlins, culture or institutions.” But immediately contradict yourself writing, “CD gap necessitates that we behave according to routines.” Routines are drawn from a culture. They tell us what is right and wrong in the big moral sense. But they also tell us how to get to the public restroom, or as they used to say about Hume, get to the local Pub. Culture fills in all the gaps. Including the one you write about. If culture fails in filling in the gaps, and that failure is not quickly corrected, culture begins to collapse, to be replaced by another. But you are correct that culture is situational. It’s created in a time and place, can perform well so long as the time and place remain generally stable. Culture then must be adapted. The central aspects of a culture usually don’t change in these adaptations but many of the operational parts do change. When the central aspects of the culture fail, the culture is done. It’s time for a replacement.

The concepts you use are irrelevant, since they must obscure our ability to reveal the work of the people making the economy. That’s the opposite of the goal of a social science, even economics, if it were a social science.

13. August 9, 2019 at 3:45 pm

Ken,
You mean that I am doing what what real social scientists do regularly. Thank you are right.

For other Ken’s arguments, it is sufficient to say that you have skipped reading this sentence:

Y >> There is a deeper level of conditions which forces us to depend or draw on culture and institutions.

I am considering from this layer. Culture and institutions are supported by this layer.

Do not interpret what I told you by your knowledge alone. Let us stop word association game. Why are you so afraid of reading our book or my articles? If you read them, you can deploy more concrete and productive argument. If you have no intentions to do so, I have to stop here. No one can make a horse drink water when it is not thirsty. It seems your brain is so filled. Or, perhaps you are too busy arguing this and that points with many other persons and have no time to reflect on our question seriously.

• August 10, 2019 at 2:04 am

Yoshinori, if you are not assuming before hand what people know or don’t know, whether they are rational or not, or just simply arational, or assuming before hand that some underlying force or level of existence explains their actions, then yes you are doing the work done by every social scientist. No before hand assumptions. Let the culture people create shine through.

But you don’t seem to be doing this. You write of “…a deeper level of conditions which forces us to depend or draw on culture and institutions. I am considering from this layer. Culture and institutions are supported by this layer.” This is gobbledygook. Humans live in and are created via their experiences. That includes the “mind,” and whatever adjuncts we add to mind. This is verifiable in history and anthropology. All this makes me not want to read your book. I only have a certain amount of time to do things like read books. I am prudent about how I use that time. Reading your book seems imprudent to me.

• August 13, 2019 at 11:41 am

Historical circumstances are different from cultural ones. Japan faced two (historical) events, Perry’s ships in Yokohama and what that meant to their cultural survival, and then, Japan’s total defeat in 1945, and how, once withdrawn from empire, how they could keep, without resources, from starving to death. Japanese proved flexible in adapting the basic management structures and above all the methods of management to the call of events, but did so while keeping the general attitudinal patterns inherited from the past, which proved to be situationally masterful. (Locke, The Collapse of the American Management Mystique, p.116-17)

Americans historical uniqueness was rooted in world hegemony in 1945, and an overweening confidence in the American way. Culture enters the picture when trying to cope with the historical event of decline — a stubborn refusal of Americans to accept culture’s consequences. Economic Science did not create the wealth of nations, and it can’t save the US. Or let’s say the battle goes on.

• August 14, 2019 at 12:07 am

Robert, all circumstances are historical. They occur at a place and time. Other circumstances arise from other places and times. Culture is created and changed via these experiences. And, in turn changes future circumstances, etc. Culture is historical. And history is cultural. To paraphrase your words, when events present challenges, culture either deals with them effectively, and survives. Or, fails to deal with them effectively and either begins a slow decline or with more severe tests collapses almost immediately.

Puritan preacher John Winthrop (1588–1649), the first governor of the Massachusetts Bay Colony, is credited with first identifying America as “A Shining City Upon a Hill.” Many American Presidents have noted those words as a source of inspiration, including John F. Kennedy and Ronald Reagan. To show the irony in this adulation, Winthrop did not represent a tradition of either democracy or religious tolerance. He hated democracy with a passion. The state he created did not hesitate to execute people like the Quakers and even brought to the “new” world the very popular tradition of medieval Europe, the trial and execution of witches. But George Washington admired the statement as well. A secular man devoted to the new nation’s future prosperity and power. American exceptionalism was part of American culture long before even the US became a nation.

• August 14, 2019 at 7:58 am

Re Arthur Miller’s Death of a Salesman and The Crucible; this exceptionalism has a lot of warts on it.

• August 14, 2019 at 8:43 am

Robert, it’s worse than warts. Several Presidents have used it to start wars.

14. August 11, 2019 at 4:48 am

I only answer for the rest of readers who have patiently followed our debate up to now. As for deeper level or layer than culture and institutions, I have I believe amply argued in my Chapter 1 of the book. The draft of this chapter with the title Microfoundations of Evolutionary Economics (the same as the book itself) has been uploaded since May 2016 in ResearchGate:
https://www.researchgate.net/publication/301766363_Microfoundations_of_Evolutionary_Economics

Anybody can download and read it. Although there are may revised expressions, the main argument is the same as the published chapter. Readers can judge by themselves.