Lars Syll has recently published an excellent post on the dilemma of probability theory when applied to the social sciences in general and economics in particular. Syll argues that in order to apply probability theory — which is deeply embedded not simply in mainstream economic models but also in econometric techniques — we must first be sure that the underlying system being studied conforms to certain presuppositions of probability theory.

Syll illustrates this nicely by comparing the economy with a roulette wheel — something that, if ‘fairly balanced’, will actually yield outcomes to which probability theory can be applied.

Probability is a relational element. It always must come with a specification of the model from which it is calculated. And then to be of any empirical scientific value it has to be

shownto coincide with (or at least converge to) real data generating processes or structures – something seldom or never done!And this is the basic problem with economic data. If you have a fair roulette-wheel, you can arguably specify probabilities and probability density distributions. But how do you conceive of the analogous nomological machines for prices, gross domestic product, income distribution etc? Only by a leap of faith. And that does not suffice. You have to come up with some really good arguments if you want to persuade people into believing in the existence of socio-economic structures that generate data with characteristics conceivable as stochastic events portrayed by probabilistic density distributions!

I think that Syll could have buttressed his piece by providing some empirical evidence in this regard. One aspect of probability theory that data must conform to in order to be properly studied using techniques based on probability theory is known as the Central Limit Theorem.

The Central Limit Theorem basically states the conditions needed for a variable to be ‘normally distributed’. What this means is that the variable can be predicted using the well-known ‘bell-curves’ or ‘Gaussian functions’. If a variable is normally distributed we can use standard probabilistic techniques to analyse it. If it is not then we cannot.

So, what about economic variables? Are they normally distributed? Short answer: no, they are not.

The most well-studied of these is, of course, the stock market. After the crash it is often said that financial markets have ‘fat tails’ and that this implies that they are not predictable using probability theory. It means that events that in a normally distributed curve would be almost impossibly rare actually occur with some frequency in these markets. The best illustration of this is to plot the actual stock market against a normally distributed curve to see how far they diverge. Here is such a graph taken from the book *Fractal Market Analysis: Applying Chaos Theory to Investment and Economics*:

As we can see the actual returns of the Dow Jones diverge substantially from the normal distribution curve. The implications for this, of course, are enormous and I will allow the author of the book to explain them himself.

What does this mean? The risk of a large event’s occurring is much higher than the normal distribution implies. The normal distribution says that the probability of a greater-than-three standard deviation event’s occurring is 0.5%, or 5 in 1,000. Yet the above shows us that the actual probability is 2.4%, or 24 in 1,000. Thus, the probability of a large event is almost five times greater than the normal distribution implies. As we measure still larger events the gap between theory and reality becomes even more pronounced. The probability of a four standard deviation event is actually 1% instead of 0.01%, or 100 times greater. (p26)

But surely this is just the ever-skittish stock market, right? Nope! We find similar properties when studying many economic variables. For example, in a paper entitled *Are Output Growth-Rate Distributions Fat-Tailed? Some Evidence from OECD Countries* the authors find that GDP growth rates and other economic variables show very similar properties to stock market data. They write,

The foregoing evidence brings strong support to the claim that fat tails are an extremely robust stylized fact characterizing the time series of aggregate output in most industrialized economies… As mentioned, fat tails have been indeed discovered to be the case not only for cross-sections of countries, but also for plants, firms and industries in many countries. In other words, the general hint coming from this stream of literature is in favor of an increasingly “non-Gaussian” economics and econometrics. (p17)

So, what do these authors say in response? Well, the response is usually some vague allusion to some new statistical or mathematical approach. This seems to me to be wholly unfounded. What the authors are actually encountering in the data is not simply a failing of normal distribution curve but instead actual uncertainty or non-ergodicity.

This could probably be shown if different time periods were taken and the probability distributions calculated out. For example, in the Dow Jones data plotted above we could break the data down into, say, five year periods and plot each one. It would quickly become obvious that the curve for, say, 1927-1932 would look vastly different from that of 1955-1960.

What this implies is that events like the 1929 stock market crash are not simply ‘outliers’ that sit neatly off the normal distribution curve. Rather they are properly uncertain events. They cannot be tamed with better models and they cannot be fitted with better statistical equipment. They are simply uncertain — at least when viewed from the point-of-view of a single time-series.

Some, like Taleb, have taken this to mean that predictions are impossible. I entirely disagree with this point-of-view. Many of the uncertain events we see in economics can indeed be predicted. But they can only be predicted by looking at the data within a given historical context or constellation. They cannot be predicted using models or probability estimates or anything of the sort. Rather economists must learn how to read data properly; how to not be sucked in by silly trends; and above all how to appreciate that they are not dealing with material that they can just feed into a computer and expect a neat outcome.

Phil, I like your beginning and end (they pander to my prejudices) but I have a quibble about the middle:

* I agree with you and Taleb that economic processes aren’t normally distributed, and so one shouldn’t rely on mathematics that assumes that they are. (Garbage in, garbage out.)

* I agree with you and Lars (and Keynes) that classical probability theory isn’t up to economics beyond the short-run.

* But: even classical probability theory can cope with non-normal distributions, so if that were the only problem economists could simply go back to the standard probability texts and start again.

*So: I agree with Lars that economist should (as some are doing) investigate broader concepts of uncertainty, such as Keynes’.

*But: I don’t think that Keynes (or Good or Russell) gives us answers on a plate: only food for thought. (Hence my blog.)

Well, I began with normal distributions because it is standard fare. In the 4th paragraph from the end I clearly say that some new statistical approach will not work because the problems are more deeply rooted.

An interesting post. The normal distribution came to dominate because it was easier to compute. Just as quantitative economists stick to the maths and statistics they can manipulate in Excel or a stats package.. so does finance stick to prcing models they can compute in milliseconds.

I’m not surprised, Neil. I don’t think that using different distributions would make that much of a difference really. If I were using this sort of analysis it strikes me that weighing up the costs and benefits the normal distribution would be the more sensible option.

Phil, I agree that simply moving a few statistical deck-chairs wont help, but – to quibble – some statisticians are aware of the issues that you raise and trying to do something about it (e.g. Deborah Mayo). Perhaps they will succeed and just maybe the result will be called statistics. But perhaps not as you know it.

I also think that Keynes’ Treatise relevant to these issues, and some consider it ‘statistical’ in parts.

I just don’t see how they can get around the issue of non-ergodicity. As I say in the piece this implies that the distributions for any given time-series will look entirely different for any given time period. How do you make projections knowing this to be the case?

Firstly, pedantically, current techniques are perfectly sound for making extrapolations or ‘projections’. Thus from 2005 there was nothing much wrong with the Bank of England projections as such, it was just that we were approaching a critical instability, beyond which they were meaningless. So the problem was in the ‘meaning’ of those projections.

Conventional practice is to provide a ‘central projection’ together with some probabilistic ‘error bars’. I argue at http://djmarsay.wordpress.com/debates/which-type-of-mathematics-in-finance/ that this is pretty useless, and describe some of the support that was given to UK and EU governments through to 2008, to aid their thinking. I fondly imagine that things could have turned out much worse if decision-makers had relied on ‘predictions’.

Interesting paper, Dave. But I’m afraid that the economy is a bit like a vertical rod with innumerable mice moving up and down it, some running, some walking and some producing baby mice that behave in new ways. The notion of equilibrium simply doesn’t apply to the economy at all. The economic system behaves like the evolutionary system with unpredictable mutations taking place constantly. The difference is that what in evolution takes many years, in economies takes maybe a few days or weeks.

Phil, your comment doesn’t seem to have reply button – but that wont stop me!

As you probably know, evolutionary theories are not without their debates, similar to those in economics, and we lack clear, reliable models. Your mice and rod analogy is exactly how many in economics and similar fields describe their problems, and just the kind of thing that the pioneers of the mathematics had in mind. If they are correct then there is a very direct way in which the classical notion of an equilibrium applies to economics: it is false. But on the other hand the UK economy does seem to have been pretty ‘stable’ recently, and does seem to have gone through periods where its behaviour seems to have been more or less stable, and it does seem helpful to think in these terms (although you may have a better language?)

The theory I drew upon in my paper was fostered and developed by people with an interest in both economics and ecology. Unpredictable mutations take place constantly, but many ecologies ‘evolve’ in cycles rather like business cycles most of the time. But they also have critical instabilities.

It is quite true that economies can change much quicker than ecologies, but generally they don’t, and a ‘great moderation’ can sucker us into thinking that we have reached an equilibrium. But what would an actual equilibrium be like, in either economics or ecologies?

What I was keen to point out in my paper is that whereas economics has drawn upon a type of mathematics that cant address its key questions, other types are available. Similarly, where mathematicians regard the questions that politicians routinely ask as meaningless or worse, it can still provide insights that can be acted upon. As a minimum, the fact that Gordon Brown knew that the edifice of mainstream economics was ill-founded meant that he was able to maintain his equilibrium when ‘the system’ came crashing down.

“and above all how to appreciate that they are not dealing with material that they can just feed into a computer and expect a neat outcome.” Like Adam Curtis was implying in his latest trilogy?