Fat tails and Epochs
February 8, 2011 3 Comments
Different explanations of the crash of 2007/8
The term ‘fat tails’ has been much in evidence since the crash of 2007/8. Nicholas Taleb had been warning that statistical tails were fat and of its significance, and the term has since been taken up to explain the crash.
There has also been a revival in references to Keynes and some references (e.g. by Gordon Brown) to Whitehead’s notion of epochs. Both are often seen as alternatives to the previously fashionable ‘sophisticated mathematics’ slated in the Turner Review.
Implications of fat-tails
From a mathematical view these alternative explanations are quite different. If the crash was due to a ‘fat tail’ then the problem was that it’s probability had been under-estimated. A variant of the ‘fat-tail’ notion is the one-sided fat tail. Here sample statistics tend to seriously underestimate the probability of occurrence for quite long periods, followed by an erratic ‘correction’. Thus competitive pressure favours those who optimise in the short-run, but (unless bailed out) crashes weed them out, leaving those who had carried some ‘fat’. The solution to ‘fat tails’ is to fatten up one’s body. Similarly, in evolutionary systems we would expect to see systems that are not too optimised. If one only has a fat-tail then after the event the distribution is unchanged and – assuming that the event has been survived – the best thing to do is supposedly to update the estimated probability distribution and carry on as before. This is William James’ ‘pragmatic’ approach. Even if one is aware of fat tails the best strategy might be to optimise like everyone else (to stay in the game), cope with the crisis as best one can, and then learn what one can and carry on.
Implications of epochs
The Keynes, Whitehead and Smuts view is quite different. It incorporates a different view of evolution. It informed a different management style (not particularly well documented). It was based on a different view of logic and uncertainty. It had profound impact on Keynes’ view of economics.
All observations relate to some epoch. Empirical predictions are only really extrapolations, conditional on the epoch not changing. A given epoch has characteristic behaviours. The assumption that an epoch will endure entails assumptions about those behaviours. Within an epoch one typically has distributions, which may have fat tails. Some, but not all, fat-tail events may destabilise the epoch. Thus, from this view, fat-tail events are a problem. But external events can also destabilise an epoch, leading to a change in ‘the rules of the game’.
Thus match-fixing in cricket may lead to a new epoch in the way cricket is organised. But match-fixing in rugby could also lead to a change in the way cricket is organised. If the match-fixing had been rife, one may see a significant, enduring, shift in the statistics, rather than a ‘blip’.
The term ‘fat-tail’ seems to be being used to indicate any shocking event (some refer to Shackle). But the term ‘fat-tail’ implies that events were probabilistic, the problem being that low-probability is not the same as no probability. The term ‘epoch’ indicates that the rules have changed, for example an assumption was violated. Thus the terms might be used to complement each other rather than using ‘fat-tail’ as a portmanteau term. ‘Shock’ would seem a suitable less precise term.
A further problem is that in the epochs of interest the assumptions of probability theory do not necessarily hold (e.g., see Turner) so perhaps we need a different term. What are the important distinctions? Maybe we need to distinguish between:
- Surprises that do not change our view of what may happen in the future, only of what has happened.
- Surprises that change our view of what may happen, but which are not evidence of a fundamental change in the situation (just our view of it) and may not cause one.
- Surprises that are not evidence of a fundamental change having actually occurred already, but which may lead to one.
- Surprises that suggest a fundamental change in the actual situation.
Thus, for small investors, a changing view of the market does not change the market, but en mass may lead to a change. It seems inevitable that any terminology would need to address appropriate concepts of ‘causality’.