Discussion Paper

No. 2008-11 | April 04, 2008
Random Matrix Theory and the Evolution of Business Cycle Synchronisation, 1886–2006


Bordo and Helbing (2003) examine 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. Most of the analytical techniques used in the business cycle convergence literature rely upon the estimation of an empirical 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. Random matrix theory was developed in physics to overcome this problem, and to enable true information in a matrix to be distinguished from noise. It has been successfully applied in the analysis of financial data.
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. The results confirm that there is a very clear degree of synchronisation of the business cycle across countries during the 1973-2006 period. In contrast, during the pre-First World War period it is not possible to speak of an international business cycle 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.
Contrary to the findings of Bordo and Helbing, it does not seem possible to speak of a ‘secular trend’ towards greater synchronisation over the 1886-2006 period as a whole. The periods 1920-1938 and 1948-1972 do show a certain degree of synchronisation – very similar in both periods in fact – but it is 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. More detailed analysis of the evolution of synchronisation of the 6 major economies since 1948 suggests it can vary considerably over relatively short periods of time. During the 1990s, for example, the degree of synchronisation of the cycle was similar to that of the 1950s, and lower than it was in the 1970s and 1980s following the oil shocks.

JEL Classification:

C69, E32, N10



Cite As

Paul Ormerod (2008). Random Matrix Theory and the Evolution of Business Cycle Synchronisation, 1886–2006. Economics Discussion Papers, No 2008-11, Kiel Institute for the World Economy. http://www.economics-ejournal.org/economics/discussionpapers/2008-11

Comments and Questions

Yury Maksimenko - Admiration and a useful suggestion
April 20, 2008 - 14:57

Dear Paul Ormerod,
I highly appreciate your investigation of interaction of business cycles as a step towards better understanding the world economy. It's a pity that Russia was not included in the study, apparently for the lack of data. In this connection and view of your possible future efforts I ...[more]

... dare to suggest you a useful fragment of empirical facts about Russia. Though GDP doesn't figure out there explicitly it can be assesed indirectly.
Please see my home page http://nonmon.hotmail.ru and link to Empirical evidence.
Thank you

Christian Dreger - Comment
April 29, 2008 - 18:24

See pdf document

paul ormerod - response to Dreger
May 07, 2008 - 11:17

I am afraid these comments are not terribly helpful

1. ‘most of these results are already known, especially for the 1973-2006 period’. I address the 1886-2006 period, here these results are not really known. There is a good paper by Bordo and Helbing which I cite. My ...[more]

... paper partly confirms these results, but with important differences. The results are therefore new.
2. More generally, Christian Dreger’s comments do not seem to show an awareness of how much more powerful the results based on random matrix theory are compared to conventional ones which rely purely on the correlation matrix. The ECB paper cited by Dreger is an example of the latter.

3. Random matrix theory informs us about the true degree of information of the correlation matrix, regardless of the underlying statistical distribution of the data. They are therefore to be preferred to conventional results. If results based on the correlation matrix show increased convergence but those based on random matrix theory do not, we should rely on the latter and not the former

4. It was not the intention of the paper to explain why cycles converge or otherwise, but to establish what has actually happened to business cycles in the developed economies. This it does.

Paolo Pin - Referee Report
May 13, 2008 - 16:16

see attached file

paul ormerod - reply to referee 1
May 16, 2008 - 12:27

Comments on referee 1

These are helpful comments.

First, I think the differences between Bordo and Helbing and my own results are perhaps rather more substantial than I indicate. The lack of any trend to convergence pre-1973 in the RMT results is a clear differentiator. I ...[more]

... think I ought to have made this clearer. However, given that I am using techniques with which few economists are familiar, I perhaps laid too much emphasis on the similarities with results obtained using more conventional techniques in order to reassure economists that the RMT technique is not just ‘weird’ but sensible and powerful..

Second, I think the suggestion to use the paper as an illustration of the application of RMT to macro-data with this data set as an illustration is a good one. It is straightforward to do this.

However, it is precisely the fact that the RMT results can be compared to results obtained using more standard techniques which makes it important to publish in an economics journal rather than a physics journal. These are powerful techniques which economists should know about.

I have published articles on RMT and macro-data in physics journals e.g. ‘RMT and the Failure of Economic Forecasting’ Physica A, 2000 and ‘The Convergence of European Business Cycles 1980-2004’, Acta Physica Polonica B, 2005. But, as I say, economists need to be aware of the techniques as well.

Finally, econometric analysis relies upon the correlation matrix, both amongst the explanatory factors and between these and the dependent variable. RMT can be used to explain why time-series econometric relationships are in general both unstable over time and unsatisfactory in the sense that they have resolved few if any controversies. For example, it is 70 years since Keynes introduced the concept of the consumption function, yet time series econometric analysis of macro-data does not yet offer an answer to its empirical formulation. If the correlation matrix itself contains relatively little true information, then time-series macro-econometric will not tell us much of any use.

Anonymous - Contemplorary business cycle theory is weak
May 23, 2008 - 18:55

Contemporary to business cycle empirics, of which this paper is an example, is strong on statistical methodology, but weak both on economic theory and on looking for strong regularities in the data. Only GDP is looked at and any deviation from trend is considered to be an instance of ...[more]

... a ‘business cycle’ movement. The purpose of this comment is to bring to the attention of researchers in this area the existence of an approach that has obtained much stronger results both with respect to theory and empirical regularities.
I am referring to what I have called the ‘classical theory of business cycles’ which had evolved roughly over the period 1850-1950. The approach faded from the economic mainstream for reasons that I have argued were ideological rather than scientific. Specifically the rise of monetarism and the associated belief that the economy was intrinsically stable and only disturbed by exogenous shocks, particularly from the monetary sector. The classical theory was developed further over several decades at my institute SEMECON at the University of Munich. References are given at the end of this note. The main findings are outlined in the Survey Hillinger () from which I copied the following abstract:

The paper reports the principal findings of a long term research project on the description and explanation of business cycles. The research strongly confirmed the older view that business cycles have large systematic components that take the form of investment cycles. These quasi-periodic movements can be represented as low order, stochastic, dynamic processes with complex eigenvalues. Specifically, there is a fixed investment cycle of about 8 years and an inventory cycle of about 4 years. Maximum entropy spectral analysis was employed for the description of the cycles and continuous time econometrics for the explanatory models. The central explanatory mechanism is the second order accelerator, which incorporates adjustment costs both in relation to the capital stock and the rate of investment. By means of parametric resonance it was possible to show, both theoretically and empirically, how cycles aggregate from the micro to the macro level. The same mathematical tool was also used to explain the international convergence of cycles. I argue that the theory of investment cycles was abandoned for ideological, not for evidential reasons. Methodological issues are also discussed.

The abstract is taken from:
Hillinger, Claude (2005), Evidence and Ideology in Macroeconomics: The Case of Investment Cycles. Available at SSRN: http://ssrn.com/abstract=814527
Further references to the SEMECON research as well as to investment cycles generally can be found there.

Anonymous - Referee Report
June 03, 2008 - 10:24

see attached file

paul ormerod - author reply
June 03, 2008 - 12:05

these are simply points about drafting e.g. if the referee feels the abstract is too long, parts of it can be put in the introduction, which he/she feels is too weak. This would solve this particular point, the abstract deals with the referee's points made about the introduction.
Quite ...[more]

... frankly, I think these are all rather trivial points. They have some value, but not a great deal.

It is very hard to see how a recommendation of rejection could be given on this basis. The paper does advance knowledge, and makes economists aware of a powerful addition to their analytical tool-kit