“We were all Gaussians now”

This post focuses on a joint paper written in 2012 by Andrew Haldane and Benjamin Nelson titled “Tails of the unexpected”. The topic is the **normal distribution** which is obviously a bit technical but the paper is still readable even if you are not deeply versed in statistics and financial modelling. The condensed quote below captures the central idea I took away from the paper.

“For almost a century, the world of economics and finance has been dominated by randomness … 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.”

**The idea that the normal distribution should be used with care, and sometimes not at all, when seeking to analyse economic and financial systems is not news. **The paper’s discussion of why this is so is useful if you have not considered the issues before but probably does not offer much new insight if you have.

What I found most interesting was the **back story behind the development of the normal distribution. **In particular, the factors that Haldane and Nelson believe **help explain why it came to be so widely used and misused. ****Reading the history reminds us of what a cool idea it must have been when it was first discovered and developed.**

“By simply taking repeat samplings, the workings of an uncertain and mysterious world could seemingly be uncovered”.“To scientists seeking to explain the world, the attraction of the normal curve was obvious. It provided a statistical map of a physical world which otherwise appeared un-navigable. It suggested regularities in random real-world data. Moreover, these patterns could be fully described by two simple metrics – mean and variance. A statistical window on the world had been opened.”

**the term “normal” began to be independently applied to this statistical distribution.**They argue that adopting

**this label helped embed the idea that the “normal distribution” was the “usual” outcome that one should expect to observe.**

“In the 18th century, normality had been formalised. In the 19th century, it was socialised.”“Up until the late 19th century, no statistical tests of normality had been developed.

Having become an article of faith, it was deemed inappropriate to question the faith.

As Hacking put it, “thanks to superstition, laziness, equivocation, befuddlement with tables of numbers, dreams of social control, and propaganda from utilitarians, the law of large numbers became a synthetic a priori truth. We were all Gaussians now.”

Notwithstanding its widespread use today, in Haldane and Nelson’s account, **economics and finance were not early adopters of the statistical approach to analysis but eventually become enthusiastic converts. **The influence of physics on the analytical approaches employed in economics is widely recognised and Haldane cites the rise of probability based quantum physics over old school deterministic Newtonian physics as one of the factors that prompted economists to embrace probability and the normal distribution as a key tool.

” … in the early part of the 20th century, physics was in the throes of its own intellectual revolution. The emergence of quantum physics suggested that even simple systems had an irreducible random element. In physical systems, Classical determinism was steadily replaced by statistical laws. The natural world was suddenly ruled by randomness.”“Economics followed in these footsteps, shifting from models of Classical determinism to statistical laws.”“Whether by accident or design, finance theorists and practitioners had by the end of the 20th century evolved into fully paid-up members of the Gaussian sect.”

### Assessing the Evidence

**normality is not a good statistical description of real-world behaviour. In its place, natural and social scientists have often unearthed behaviour consistent with**an alternative distribution,

**the so-called power law distribution.**

“In consequence, Laplace’s central limit theorem may not apply to power law-distributed variables. There can be no “regression to the mean” if the mean is ill-defined and the variance unbounded. Indeed, means and variances may then tell us rather little about the statistical future. As a window on the world, they are broken”

**This section of the paper**probably does not introduce anything new to people who have spent any time looking at financial models. It

**does however beg some interesting questions. For example, to what extent bank loan losses are better described by a power law and, if so, what does this mean for the measures of expected loss that are employed in banking and prudential capital requirements**; i.e. how should banks and regulators respond if “…the means and variances … tell us rather little about the statistical future”? This is particularly relevant as banks transition to Expected Loss accounting for loan losses.

### Explaining Fat Tails

- Non- Linear dynamics
- Self organised criticality
- Preferential attachment
- Highly optimised tolerance

**competition which is typically assumed by economists to be a wholly benign force. This is generally true but Haldane and Nelson note the capacity for competition to contribute to self-organised criticality.**

### Where Next?

**Knight’s distinction between risk and uncertainty is a good foundation for developing better ways of dealing with a world that does not conform to the normal distribution**and note the distinguishied company that have also chosen to emphasise the importance of uncertainty and the limitations of risk.

“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.”

- agent based modelling
- simple rather than complex
- don’t aim to smooth out all volatility

#### Agent based modelling

Haldane and Nelson note that …

In response to the crisis, there has been a groundswell of recent interest in modelling economic and financial systems as complex, adaptive networks. For many years, work on agent-based modelling and complex systems has been a niche part of the economics and finance profession. The crisis has given these models a new lease of life in helping explain the discontinuities evident over recent years (for example, Kirman (2011), Haldane and May (2011))

- The “representative agents” conforming to simple economic laws are replaced by more complex interactions among a larger range of agents
- The single, stationary equilibrium gives way to Lorenz-like multiple, non-stationary equilibria.
- Linear deterministic models are usurped by non linear tipping points and phase shifts

#### Simple can be more robust than complex

The reason less can be more is that complex rules are less robust to mistakes in specification. They are inherently fragile. Harry Markowitz’s mean-variance optimal portfolio model has informed millions of investment decisions over the past 50 years – but not, interestingly, his own. In retirement, Markowitz instead used a much simpler equally-weighted asset approach. This, Markowitz believed, was a more robust way of navigating the fat-tailed uncertainties of investment returns (Benartzi and Thaler (2001)).

Themainstay of regulation for the past 30 years has been more complex estimates of banks’ capital ratios. These are prone to problems of highly-optimised tolerance. In part reflecting that, regulators will in future require banks to abide by a far simpler backstop measure of the leverage ratio. Like Markowitz’s retirement portfolio, this equally-weights the assets in a bank’s portfolio. Like that portfolio, it too will hopefully be more robust to fat-tailed uncertainties.

A second type of simple, yet robust, regulatory rule is to impose structural safeguards on worst-case outcomes. Technically, this goes by the name of a “minimax” strategy (Hansen and Sargent (2011)). The firebreaks introduced into some physical systems can be thought to be playing just this role. They provide a fail-safe against the risk of critical states emerging in complex systems, either in a self-organised manner or because of man-made intervention. These firebreak-type approaches are beginning to find their way into the language and practice of regulation.

Finally, in an uncertain world, fine-tuned policy responses can sometimes come at a potentially considerable cost. Complex intervention rules may simply add to existing uncertainties in the system. This is in many ways an old Hayekian lesson about the pretence of knowledge, combined with an old Friedman lesson about the avoidance of policy harm. It has relevance to the (complex, fine-tuned) regulatory environment which has emerged over the past few years.

#### Don’t aim to eliminate volatility

Systems which are adapted to volatility will tend to be stronger than systems that are sheltered from it, or in the words of Haldane and Nelson …

“And the argument can be taken one step further. Attempts to fine-tune risk control may add to the probability of fat-tailed catastrophes. Constraining small bumps in the road may make a system, in particular a social system, more prone to systemic collapse. Why? Because if instead of being released in small bursts pressures are constrained and accumulate beneath the surface, they risk an eventual volcanic eruption.”

I am a big fan of this idea. Nassim Taleb makes a similar argument in his book “Antifragile” as does Greg Ip in “Foolproof”. It also reflects Nietzsche’s somewhat more poetic dictum “that which does not kills us makes us stronger”.

### In conclusion

If you have read this far then thank you. I hope you found it useful and interesting. If you want to delve deeper then you can find my more detailed summary and comments on the paper here. If you think I have any of the above wrong then please let me know.

## 2 thoughts on “The rise of the normal distribution”