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Tails of the unexpected

"The interactions which generate non-normalities in children’s games repeat themselves in real world systems – natural, social, economic, financial. Where there is interaction, there is non-normality."
Andrew G Haldane and Benjamin Nelson
Bank of England, 8 June 2012

In a paper given at “The Credit Crisis Five Years On: Unpacking the Crisis” conference held at the University of Edinburgh Business School, Andrew Haldane – Executive Director for Financial Stability and member of the Financial Policy Committee – argues that economic and financial systems are prone to tail events, as demonstrated during the financial crisis, that are not captured by traditional macro-economic and risk-pricing models.  In doing so, he draws a number of important lessons for economic and financial policymakers.

Excerpts:

"For almost a century, the world of economics and finance has been dominated by randomness.  Much of modern economic theory describes behaviour by a random walk, whether financial behaviour such as asset prices (Cochrane (2001)) or economic behaviour such as consumption (Hall (1978)).  Much of modern econometric theory is likewise underpinned by the assumption of randomness in variables and estimated error terms (Hayashi (2000)).

But as Nassim Taleb reminded us, it is possible to be Fooled by Randomness (Taleb (2001)).  For Taleb, the origin of this mistake was the ubiquity in economics and finance of a particular way of describing the distribution of possible real world outcomes.  For non-nerds, this distribution is often called the bell-curve.  For nerds, it is the normal distribution.  For nerds who like to show-off, the distribution is Gaussian.  

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The interactions which generate non-normalities in children’s games repeat themselves in real world systems – natural, social, economic, financial.  Where there is interaction, there is non-normality.  But risks in real-world systems are no game.  They can wreak havoc, from earthquakes and power outages, to depressions and financial crises.  Failing to recognise those tail events – being fooled by randomness – risks catastrophic policy error.

So is economics and finance being fooled by randomness?  And if so, how did that happen?  That requires a little history.

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In the period since, the models of Markowitz and Arrow/Debreu, with embedded assumptions of normality, have dominated asset-pricing in economics and finance.   In economics, the Arrow/Debreu equilibrium model is the intellectual antecedent of today’s real business cycle models, the dominant macro-economic framework for the past 20 years (for example, Kiyotaki (2011)).  Typically, these models have Gaussian-distributed impulses powering a Quetelet-inspired representative agent.

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In the 1870s, the German statistician Wilhelm Lexis began to develop the first statistical tests for normality.  Strikingly, the only series Lexis could find which closely matched the Gaussian distribution was birth rates.  In 1929, E B Wilson and M M Hilferty re-examined using formal statistical techniques the datasets used by Peirce (Wilson and Hilferty (1929)).  The distribution of the original data was found to be incompatible with the normal model.  The natural world suddenly began to feel a little less normal.     

The period since has seen mounting evidence of non-normality in a variety of real-world settings.  In its place, natural and social scientists have often unearthed behaviour consistent with an alternative distribution the so-called power law distribution.

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So what explains these fat-tailed outcomes in natural and social systems?  One factor holds the key - interactions.  The central limit theorem is predicated on the assumption of independence of observations.  In complex systems, natural and social, that assumption is almost certain to be violated.  Systems are systems precisely because they are interdependent.  In a nutshell, that is why so few systems behave normally in the statistical sense. 

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Like de Moivre in the 18th century and Galton in the 19th, the economics profession has for much of the 20th century been bewitched by normality.  Real business cycle theory in economics and efficient markets theory  in finance bear the tell-tale signs of this intellectual infatuation.  So too does much of econometrics.  All three are rooted in a Frisch/Slutsky meets Arrow/Debreu framework, with normal impulses acting on linear propagation rules.  Predictably, this generates near-Gaussian outcomes for macro-economic variables.    

Over the past five years, the real world has behaved in ways which make a monkey of these theories.  In the face of shocks, some of them modest, the economic and financial world has often responded in highly irregular and discontinuous ways.  Tipping points and phase transitions have been the name of the game.  The disconnect between theory and reality has been stark.

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Taking a step forward may require economics and finance to first take a step back.  In 1921, Frank Knight drew an important distinction between risk on the one hand and uncertainty on the other (Knight (1921)).  Risk arises when the statistical distribution of the future can be calculated or is known.  Uncertainty arises when this distribution is incalculable, perhaps unknown.

Many of the biggest intellectual figures in 20th century economics took this distinction seriously.  Indeed, they placed uncertainty centre-stage in their policy prescriptions.  Keynes in the 1930s, Hayek in the 1950s and Friedman in the 1960s all emphasised the role of uncertainty, as distinct from risk, when it came to understanding economic systems.  Hayek criticised economics in general, and economic policymakers in particular, for labouring under a “pretence of knowledge” (Hayek (1974)).

Yet it is risk, rather than uncertainty, that has dominated the economics profession for much of the past 50 years.  By assuming future states of the world were drawn from known distributions, Arrow and Debreu enabled risk to be priced with statistical precision and uncertainty to be conveniently side-stepped.

Uncertainty was, quite literally, ruled out of the equation.  But if economic and financial systems operate on the border between order and disorder, ignoring uncertainty is deeply unrealistic.  

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Uncertainty profoundly alters the way systems behave.  Take asset pricing.  Under uncertainty rather than risk, asset prices are no longer defined by a single price.  Instead their equilibrium price is defined by a range (Epstein and Wang (1994)).  Prices systematically differ from their fundamental values.  If uncertainties escalate, they experience phase shifts."