# Models and model-makers

Nonlinear Dynamics and Economics
May 23, 1997

DESPITE have recently spent 15 months at Imperial College on sabbatical from my bank immersed in high-level mathematics, I was dazzled by the level of mathematical and statistical knowledge of all the authors of this high-quality and coherent collection of conference papers. But something does ring strange. The book's rigour in mathematical modelling is not matched in some of the papers by an equal rigour in trying to understand the events underlying the data or the process by which such data comes to exist.

For instance, Paolo Guarda and Mark Salmon speak of having identified a particular week from certain observations and then "checked past issues of The Economist magazine for relevant news at that date". That sounds like a team of expert engineers testing their models of metal fatigue in aeroplane components against data for actual aircraft failures obtained by scanning Business Week.

As an economics student I was once used as "slave labour" by an eminent professor. He gave me a method of analysing ten years of economic data and history in Nazi Germany. Hundreds of other students were asked to analyse other countries and periods. After several years' work, the product was a model so complex that it was mathematically unworkable. We students, however, gained a good sense of the relationships between economic data and political decisions and events. The balance has now tilted the other way. We have impressive, workable models, we understand increasingly well how they operate, we may even fit them to real data and calibrate them well enough to make them efficient for short periods - but our knowledge of the real world has not progressed in sync.

The book is divided into five sections: instability in economic theory; nonlinearity in financial markets; tests for nonlinearity and chaos; frequency domain methods and nonlinear business cycles. I shall consider five of the 18 contributions.

Alfredo Medio and Giorgio Negroni look at the possibility of chaotic dynamics in two-period OLG (overlapping generation) models with production. They conclude that introducing production into OLG models enormously increases the potential complexity of their dynamics; that periodic, aperiodic and chaotic behaviours occur in many possible combinations; and that the routes to chaos are several. They stress that parameters such as the elasticity of the utility functions, productivity coefficients and the elasticity of substitutions between factors generate bifurcations leading to cycles and chaos; and they point out that, by introducing production, OLG models no longer require an inverse relation between saving and interest rate to produce cycles and chaos. They try to disabuse us from believing that chaos is just a single property of a dynamical system. They maintain - and I am all for it - that complexity includes a large set of different classes of dynamical behaviour.

Gerlad Silverberg and Doris Lehnert deal instead with the generalisation of a dynamic Schumpeterian model, based on the Goodwin growth cycle, to allow for a continuing stochastic stream of capital-embodied innovations. They tell us that the model is "equivalent to a large-dimensional Lotka-Volterra system with stochastically perturbed coefficients". And they conclude that the new approach yields "robust dividends", according to which "it does not seem necessary for innovations to arrive in clusters" and for which "the slow tapering-off of the autocorrelation function ...is...an emerging property of the model". The entire chapter is well-written, stimulating and full of provocative observations.

Guarda and Salmon set themselves the task of detecting nonlinearity in foreign exchange data. They chose raw data, natural logarithms of raw data, the log differences and the standardised residuals from a generalised autoregressive conditional heteroskedasticity-GARCH (1,1)-model fitted to the log differences, and ran the following tests: Keenan, Ramsey RESET, Tsay, McLeod-Li, BDS, ARCH LM and Neural Network Test. They used full-sample and recursive analysis.

They came up with a two-page list of significant indications of nonlinearity and conclude that nonlinearity is much more pronounced at the higher frequency of data, eg weekly rather than monthly. Most traders would agree. More importantly, they conclude that "conditional inference may then provide a more relevant statistical framework for the detection of nonlinearity than the unconditional approach currently adopted".

In the tenth paper, Thomas Taylor addresses "the question of how one can distinguish chaotic time series from stochastic time series" and concludes that this is the wrong question to ask. Furthermore he deals with "the determination of nonlinear structures in time series and their use for the purpose of prediction", incorporating the tool of the embedding method. Finally, he reports on techniques that use embedding methods "to treat environments in which noise affects the observation of the chaotic signals".

I empathise with Taylor's statement that "a consequence of the exponential growth of the error of prediction is that the purely deterministic observation time series have qualitatively the same kind of difficulty with respect to prediction that a purely non-deterministic observation time series has, when viewed with limited accuracy or on a long time-scale. Thus, in practice, the distinctions between these ideal considerations of determinism and nondeterminism are blurred."

Steve Satchell and Allan Timmermann look for "evidence of nonlinear components in daily returns ... across 12 national stock-market indexes and a world index over the period 1980-92". They conclude that their results "support the broad conclusion that nonparametric predictions may be worthwhile to undertake". However, they maintain that "the transaction costs are important factors in determining the stochastic properties of the daily returns series in the markets under consideration and probably are sufficiently large to rule out economic profits from trading on the basis of non-parametric forecasts." So transaction costs, vilified for making markets imperfect, are responsible for the efficient market hypothesis? What a lovely paradox!

I recommend the challenge of reading this book to all those with a professional and/or personal interest in economics and finance. Despite the four or five-year delay between first appearance of the papers and publication in a single volume, this book is likely to last.

Rudi Bogni is a member of the group executive board, Swiss Bank Corporation.

## Nonlinear Dynamics and Economics: Proceedings of the Tenth International Symposium in Economic Theory and Econometrics

Editor - William A. Barnett, Alan P. Kirman and Mark Salmon
ISBN - 0 521 47141 9
Publisher - Cambridge University Press
Price - £40.00
Pages - 406