(The third edition corrects minor typos and adds 3 chapters synthesized from published papers plus an appendix on maximum entropy distributions.) The monograph investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible. Switching from thin tailed to fat tailed distributions requires more than "changing the color of the dress". Traditional asymptotics deal mainly with either n=1 or $n=\infty$, and the real world is in between, under of the "laws of the medium numbers" --which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence. A few examples: + The sample mean is rarely in line with the population mean, with effect on "naive empiricism", but can be sometimes be estimated via parametric methods. + The "empirical distribution" is rarely empirical. + Parameter uncertainty has compounding effects on statistical metrics. + Dimension reduction (principal components) fails. + Inequality estimators (GINI or quantile contributions) are not additive and produce wrong results. + Many "biases" found in psychology become entirely rational under more sophisticated probability distributions + Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions. This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles.
The Uncertainty Reading Group (Chapter 13) 1 Papers relied upon here are [52,54,55,56,57,113,124,147,177,194,270,273,274,275,277,278,279,280,289,293,294,295,296] C O N T E N T S Nontechnical chapters are indicated with a star *; Discussion chapters are indicated with a †; adaptation from published ("peer-reviewed") papers with a ‡. While chapters are indexed by Arabic numerals, expository and very brief mini-chapters (half way between appendices and full chapters) use letters such as A, B, etc.
The less you understand the world, the easier it is to make a decision.
Figure 1.1: The problem is not awareness of “fat tails”, but the lack of understanding of their consequences. Saying “it is fat tailed” implies much more than changing the name of the distribution, but a general overhaul of the statistical tools and types of decisions made. Credit Stefan Gasic.
The main idea behind the Incerto project is that while there is a lot of uncertainty and opacity about the world, and an incompleteness of information and understanding, there is little, if any, uncertainty about what actions should be taken based on such an incompleteness, in any given situation.
T his book consists in 1) published papers and 2) (uncensored) commentary, about classes of statistical distributions that deliver extreme events, and how we should deal with them for both statistical inference and decision making. Most “standard” statistics come from theorems designed for thin tails: they need to be adapted preasymptotically to fat tails, which is not trivial -or abandoned altogether. So many times this author has been told of course we know this or the beastly portmanteau nothing new about fat tails by a professor or practitioner who just produced an analysis using “variance”, “GARCH”, “kurtosis” , “Sharpe ratio”, or “value at risk”, or produced some “statistical significance” that is clearly not significant.
More generally, this book draws on the author’s multi-volume series, Incerto [272] and associated technical research program, which is about how to live in the real world, a world with a structure of uncertainty that is too complicated for us.
The Incerto tries to connect five different fields related to tail probabilities and extremes: mathematics, philosophy, social science, contract theory, decision theory, and the real world. If you wonder why contract theory, the answer is: option theory is based on the notion of contingent and probabilistic contracts designed to modify and share classes of exposures in the tails of the distribution; in a way option theory is mathematical contract theory. Decision theory is not about understanding the world, but getting out of trouble and ensuring survival. This point is the subject of the next volume of the Technical Incerto, with the temporary working title Convexity, Risk, and Fragility. a word on terminology “Thick tails” is often used in academic contexts. For us, here, it maps to much “higher kurtosis than the Gaussian” -to conform to the finance practitioner’s lingo. As to “Fat Tails”, we prefer to reserve it both extreme thick tails or membership in the power law class (which we show in Chapter 8 cannot be disentangled). For many it is meant to be a narrower definition, limited to “power laws” or “regular variations” -but we prefer to call “power laws” “power laws” (when we are quite certain about the process), so what we call “fat tails” may sometimes be more technically “extremely thick tails” for many. To avoid ambiguity, we stay away from designations such as “heavy tails” or “long tails”.
The next two chapters will clarify.
In addition to coauthors mentioned earlier, the author is indebted to Zhuo Xi, Jean-Philippe Bouchaud, Robert Frey, Spyros Makridakis, Mark Spitznagel, Brandon Yarkin, Raphael Douady, Peter Carr, Marco Avellaneda, Didier Sornette, Paul Embrechts, Bruno Dupire, Jamil Baz, Damir Delic, Yaneer Bar-Yam, Diego Zviovich, Ole Peters, Chitpuneet Mann, Harry Crane -and of course endless, really endless discussions with the great Benoit Mandelbrot. Social media volunteer editors such as Antonio Catalano, Maxime Biette, Caio Vinchi, Jason Thorell, Marco Alves, and Petri Helo cleared many typos. I am most thankful to Edmond Shami who found many such errors. Some of the papers that turned into chapters have been presented at conferences; the author thanks Lauren de Haan, Bert Zwart, and others for comments on extreme value related problems. More specific acknowledgements will be made within individual chapters. As usual, the author would like to express his gratitude to the staff at Naya restaurant in NY.
prologue * , † T his author presented the present book and the main points at the monthly Bloomberg Quant Conference in New York in Sep-tember 2018. After the lecture, a prominent mathematical finance professor came to see me. “This is very typical Taleb”, he said. “You show what’s wrong but don’t offer too many substitutes”.
Clearly, in business or in anything subjected
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