Friday, 6 January 2012

Why don't more economists see the potential of mathematics

A research student, working in econometrics has e-mailed me with the comment
I am a little confused why many economists do not see the potential of mathematics.
The discipline of econometrics was introduced in the 1940’s with the key monograph being Trygve Håvelmo’s The Probability Approach in Econometrics. Håvelmo’s motivation for writing the paper is eloquently stated in the preface
The method of econometric research aims, essentially, at a conjunction of economic theory and actual measurements, using the theory and technique of statistical inference as a bridge pier. But the bridge itself was never completely built. So far, the common procedure has been, ﬁrst to construct an economic theory involving exact functional relationships, then to compare this theory with some actual measurements, and, ﬁnally, “to judge” whether the correspondence is “good” or “bad”. Tools of statistical inference have been introduced, in some degree, to support such judgements, e.g., the calculation of a few standard errors and multiple-correlation coeﬃcients. The application of such simple “statistics” has been considered legitimate, while, at the same time, the adoption of deﬁnite probability models has been deemed a crime in economic research, a violation of the very nature of economic data. That is to say, it has been considered legitimate to use some of the tools developed in statistical theory without accepting the very foundation upon which statistical theory is built. For no tool developed in the theory of statistics has any meaning– except, perhaps, for descriptive purposes –without being referred to some stochastic scheme.

The reluctance among economists to accept probability models as a basis for economic research has, it seems, been founded upon a very narrow concept of probability and random variables. Probability schemes, it is held, apply only to such phenomena as lottery drawings, or, at best, to those series of observations where each observation may be considered as an independent drawing from one and the same “population”. From this point of view it has been argued, e.g., that most economic time series do not conform well to any probability model, “because the successive observations are not independent”. But it is not necessary that the observations should be independent and that they should all follow the same one–dimensional probability law. It is suﬃcient to assume that the whole set of, say $n$, observations may be considered as one observation of $n$ variables (or a “sample point”) following an $n$-dimensional joint probability law, the “existence” of which may be purely hypothetical. Then, one can test hypotheses regarding this joint probability law, and draw inference as to its possible form, by means of one sample point (in $n$ dimensions). Modern statistical theory has made considerable progress in solving such problems of statistical inference.

In fact, if we consider actual economic research–even that carried on by people who oppose the use of probability schemes–we ﬁnd that it rests, ultimately, upon some, perhaps very vague, notion of probability and random variables. For whenever we apply a theory to facts we do not–and we do not expect to–obtain exact agreement. Certain discrepancies are classiﬁed as “admissible”, others as “practically impossible” under the assumptions of the theory. And the principle of such classiﬁcation is itself a theoretical scheme, namely one in which the vague expressions “practically impossible” or “almost certain” are replaced by “the probability is near to zero”, or “the probability is near to one”.
This is nothing but a convenient way of expressing opinions about real phenomena. But the probability concept has the advantage that it is “analytic”, we can derive new statements from it by the rules of logic.
Håvelmo’s argument can be split into four key points. If economics is to be regarded as ‘scientiﬁc’, it needs to take probability theory seriously. He then notes that economists have taken a naive approach to probability, and possibly mathematics in general, and introduces the Lagrangian idea of representing $n$ points in one dimensional space by one point in $n$-dimensional space. Finally he makes Poincaré’s point that probability is a convenient solution, it makes the scientist’s life easier, and ﬁnally he makes Feller’s point that it enables the creation of new knowledge, new statements.

Håvelmo then goes on to tackle the issue that goes back as far as Cicero, at least, “there is no foreknowledge of things that happen by chance” by making the critical observation, nature looks stable because we look at it in a particular way
“In the natural sciences we have stable laws”, means not much more and not much less than this: The natural sciences have chosen very fruitful ways of looking on physical reality.
Håvelmo is saying that if economists look at the world in a diﬀerent way, if the right analytical tools are available to them, they may be able to identify stable laws.

At about the same time, Oskar Morgenstern was working with John von Nueumann on The Theory of Games and Economic Behavior, a “big book because they wrote it twice, once in symbols for mathematicians and once in prose for economists”. Morgenstern begins the book by describing the landscape. On the second page he, makes the case for using mathematics in economics, just as Håvelmo had, but with a more comprehensive argument. Morgenstern reviews the case as to why mathematics is inappropriate to economics, no doubt with von Neumann at his shoulder,
The arguments often heard that because of the human element, of psychological factors etc., or because there is – allegedly – no measurement of important factors, mathematics will ﬁnd no application [in economics] [von Neumann and Morgenstern 1967 p 3]
However, Morgenstern points out that Aristotle had the same opinion of the use of mathematics in physics
Almost all these objections have been made, or might have been made, many centuries ago in ﬁelds ﬁelds where mathematics is now the chief instrument of analysis.
While measurement may appear diﬃcult in economics, measurement appeared diﬃcult before the time of Albert the Great, again before Newton ﬁxed time and space, when objects were either ‘hot’ or ‘cold’ or before the idea of potential energy being released into kinetic energy emerged.
The reason why mathematics has not been more successful in economics must, consequently, be found elsewhere. The lack of real success is largely due to a combination of unfavourable circumstances, some of which can be removed gradually. To begin with economic problems were not formulated clearly and are often stated in such vague terms as to make mathematical treatment a priori appear hopeless because it is quite uncertain what the problems really are. There is no point in using exact methods where there is no clarity in the concepts and the issues to which they are to be applied. Consequently the initial task is to clarify the knowledge of the matter by further careful description. But even in those parts of economics where the descriptive problem has been handled more satisfactorily, mathematical tools have seldom been used appropriately. They were either inadequately handled, as in the attempts to determine a general economic equilibrium …, or they led to mere translations from a literary form of expression into symbols, without any subsequent mathematical analysis. [von Neumann and Morgenstern1967, p 4]
Morgenstern makes the critical observation, that the ‘correct’ use of mathematics in science leads to the creation of new mathematics
The decisive phase of the application of mathematics to physics – Newton’s creation of a rational discipline of mechanics – brought about, and can hardly be separated from, the discovery of [calculus]. (There are several other examples, but none stronger than this.)
The importance of social phenomena, the wealth and multiplicity of their manifestations, and the complexity of their structure, are at least equal to those in physics. It is therefore expected – or feared – that the mathematical discoveries of a stature comparable to that of calculus will be needed in order to produce decisive success in this ﬁeld. [von Neumann and Morgenstern 1967 p 5]
In 1989 Håvelmo was awarded the Nobel Prize in Economics “for his clariﬁcation of the probability theory foundations of econometrics and his analyses of simultaneous economic structures”. In his speech, the economist Håvelmo reﬂected on the impact of his work,
To some extent my conclusions [are] in a way negative. I [draw] attention to the – in itself sad – result that the new and, as we had thought, more satisfactory methods of measuring interrelations in economic life had caused some concern among those who had tried the new methods in practical work. It was found that the economic theories which we had inherited and believed in, were in fact less stringent than one could have been led to think by previous more rudimentary methods of measurement. To my mind this conclusion is not in itself totally negative. If the improved methods could be believed to show the truth, it is certainly better to know it. Also for practical economic policy it is useful to know this, because it may be possible to take preventive measures to reduce uncertainty. I also mentioned another thing that perhaps could be blamed for results that were not as good as one might have hoped for, namely economic theory in itself. The basis of econometrics, the economic theories that we had been led to believe in by our forefathers, were perhaps not good enough. It is quite obvious that if the theories we build to simulate actual economic life are not suﬃciently realistic, that is, if the data we get to work on in practice are not produced the way that economic theories suggest, then it is rather meaningless to confront actual observations with relations that describe something else. [ Prize Lecture Lecture to the memory of Alfred Nobel  ]
Håvelmo’s aim in the 1940s, along with that of John von Neuman, had been to improve economic methodology, the consequence was, in Håvelmo’s case, was that it highlighted deﬁciencies in economic theory. The question is, what happened in economics in the forty years between Håvelmo’s paper on econometrics and his Nobel Prize in 1989 to lead to such a negative reflection on the development of economics. I shall come back to this in my next post.

References

J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Wiley, 3rd edition, 1967.

1. Wow, a truly in-depth blog!

My critique of applying probability to macro-economics is something like the Schrodinger's Cat thought experiment - It goes something like this: atomic particles are represented by a probability space untill observed. But a "diabolical machine" can be constructed to kill/save a cat inside a box based on the behavior of the one atom (thru a geiger counter). Then the before the inside of the box is examined, the cat life itself has become aan unresolved probability distribution of both alive and dead. Clearly this result is absurd as it is designed to be.

Now let's jump to a different consideration I'm much more familiar with as an actuary: predicting deaths in an insurance population. There is a large margin of error in predicitng an INDIVIDUAL's expected lifetime: eg could only say with 50% centainty that a 45 year old will die between 70 and 80; a large interval with low confidence. But within a POPULATION of say 10,000 45 year olds, we can get a very accurate prediction of deaths thru CLT.

Now this latter example is a situation where prob abilistic methods become useful when the outcome is result is an aggregation of individual fortunes that are i.i.d. However, my contention is that this is not an accurate desciption of the modern economy, where animal spirits, and rational expectations play a heavy influence on the outcome. I think the modern economy is more like the Cat: really can't be expressed by PDF, it is either alive or dead in reality, there is no "random-sampling" of our one reality

2. I think some economists have come to appreciate the futility of mathematics in many human-sensitive economic applications. In such applications, historically, economists have tried to approximate economic behaviour-a largely social discipline-to the mathematical natural sciences, not without the resulting disastrous consequences. In recent decades, this obvious failure has proliferated the development of disciplines such behavioural finance and behavioural economics - which have continued to highlight the divergence between human behaviour and mathematical models.

3. Tim, An interesting paper.

In the late 90s the new government was concerned about the somewhat narrow and dogmatic views of economists and financiers, and the implications for financial stability. I was asked to get involved because of my interest in Keynes' work in his Treatise (mostly) on probability. There seemed to be three camps. The first regarded the dominant theories as correct, so that there was no need for any mathematics, only computation (Black Scholes ...); the second thought that mathematics was only about computation and prediction, and hence couldn't tackle the key issues; the third seemed very sensible but disheartened. I can't say that I got anywhere, but there do seem to be widespread misunderstandings about mathematics and probability. I suspect that this is because most of the mathematics used in finance and economics is of a type which isn't very useful if your concern is potential crashes.

Most people seemed to blame the short-term focus of those with the money, who thought that they would be able to pull out and retire to the Cotswolds in time. But Haavelmo’s points to a deeper problem. Like most, he has completely ignored Keynes’ work on probability, stability and representation. My own view is that mathematics is extremely powerful, so we should all put some effort into making sure that we have the appropriate mathematics before we start relying on it. Following the crash of 2007/8, Haavelmo’s arguments on probability, science, prediction and stability, although perhaps adequate myths in the short-term, surely seem lame in the extreme?

Good question.

4. Regarding differential topology, there has been a vein of economists working on this for some time, with Mas-Colell's book from 1985 an early effort. This has often also been associated with people more concerned with analyzing dynamics.

5. more math in econ?! there's way too much of it to my taste. take a look at any advanced micro textbook.