Random matrix theory and the evolution of business cycle synchronisation 1886-2006

Paul Ormerod

Volterra Consulting, London, UK and Institute of Advance Study, University of Durham, UK

March 2008

pormerod@volterra.co.uk

2 Abstract

The standard literature on business cycle convergence relies upon the estimation of an empirical correlation matrix of time series data of macroeconomic aggregates in the various countries.

The major study by Bordo and Helbing (2003) analyses the business cycle in Western economies over the 1881-2001 period. They examine four distinct periods in economic history and conclude that there is a secular trend towards greater synchronisation for much of the 20th century, and that it takes place across these different regimes.

However due to the finite size of both the number of economies and the number of observations, a reliable determination of the correlation matrix may prove to be problematic. The structure of the correlation matrix may be dominated by noise rather than by true information. Random matrix theory was developed in physics to overcome this problem, and to enable true information in a matrix to be distinguished from noise.

Using a very similar data set to Bordo and Helbing, I use random matrix theory, and the associated technique of agglomerative hierarchical clustering, to examine the evolution of convergence of the business cycle between the capitalist economies over the long-run.

Contrary to the findings of Bordo and Helbing, it does not seem possible to speak of a ‘secular trend’ towards greater synchronisation over the period as a whole. During the pre-First World War period the international business cycle does not exist in any meaningful sense. The cross-country correlations of annual real GDP growth are indistinguishable from those which could be generated by a purely random matrix. The periods 1920-1938 and 1948-1972 do show a certain degree of synchronisation – very similar in both periods in fact – but it is very weak. In particular, the cycles of the major economies cannot be said to be synchronised during these periods. Such synchronisation as exists in the overall data set is due to meaningful co-movements in sub-groups.

So the degree of synchronisation has evolved fitfully, and it is only in the most recent period, 1973-2006, that we can speak of a strong level of synchronisation of business cycles between countries.

JEL classification: C69, E32, N10 Keywords: international business cycle; synchronisation; random matrix theory

3

Introduction

Bordo and Helbing (2003) examine the evolution of the synchronisation of the business cycle in 16 capitalist economies over the 1880 to 2001 period. They use data that covers four distinct eras with different international monetary regimes. The four eras are 1880- 1913 when much of the world adhered to the classical Gold Standard, the interwar period (1920-1938), the Bretton Woods regime of fixed but adjustable exchange rates (1948- 1972), and the modern period of managed floating among the major currency areas (1973 to 2001).

The authors conclude that ‘using three different methodologies that there is a secular trend towards increased synchronization for much of the twentieth century and that it occurs across diverse exchange rate regimes’.

These methodologies rely on empirical estimates of the correlation matrix of time series data of macroeconomic aggregates in the various countries. However due to the finite size of both the number of economies and the number of observations, a reliable determination of the correlation matrix may prove to be problematic. The structure of the correlation matrix may be dominated by noise rather than by true information. In other words, the apparent increase in sychronisation might be due to noise in the correlation matrix rather then to genuine differences in information. If this is the case, we cannot rely on apparent differences in values of correlation matrices calculated over different time periods.

Random matrix theory has been successfully applied by physicists to financial market data in order to overcome this problem (for example, Laloux et.al. (1999), Bouchaud and Potters (2000), Mantegna and Stanley (2000), Plerou et.al. (2000)). Ormerod and Mounfield (2002) apply the technique to recent quarterly real GDP growth data in the main EU economies.

4 This short paper investigates the application of the concepts of random matrix theory to the correlations between the annual growth rates of real GDP to a very similar set of economies over a very similar time period to that of Bordo and Helbing.

Section 2 discusses the data and methodology, and the results are set out in section 3.

2 Data and methodology

The annual real GDP data for 16 countries 1885-1994 is taken

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