Synchronization of business cycles using random matrix theory

Here we show the synchronization of quarterly GDP growth in Germany, Japan, the UK, and the US over the 1973Q1 to 2019Q1 period. We use a window of 24 quarters, which approximates the length of the business cycle.

Note: the first observation is 1979Q1 which shows the degree of convergence over the previous 24 quarters, i.e. 1973Q1 until 1978Q4

Typically, the degree of synchronization of the cycle is reasonable but not dramatic. Notice however, the very considerable rise in synchronization displayed in the data when the period of the financial crisis of the late 2000s is included in the data window.

Random matrix theory (RMT) is an important tool in statistical physics. Perhaps its first application to economic data was to changes in asset prices [Laloux L, Cizeau P, Potters M, Bouchaud JP. Random matrix theory and financial correlations. International Journal of Theoretical and Applied Finance. 2000 Jul;3(03):391-7.]

Empirical correlation matrices are often dominated by noise. RMT reveals the true amount of information in a correlation matrix.  Standard statistical tests of significance in economics do not capture this.

One important finding was the existence of the “market mode”. In other words, the extent to which all assets in the set being analysed tend to move together.

This is measured by the size of the principal eigenvalue of the correlation matrix as a proportion of the trace of the eigenvalue decomposition.

Ormerod extended this principle to the analysis of GDP growth rates over time in the main economies, e.g., [Ormerod P, Mounfield C. The convergence of European business cycles 1978–2000. Physica A: Statistical Mechanics and its Applications. 2002 May 1;307(3-4):494-504] and [Ormerod P. Random Matrix Theory and Macro-Economic Time-Series: An Illustration Using the Evolution of Business Cycle Synchronisation, 1886-2006. Economics: The Open-Access, Open-Assessment E-Journal. 2008;2]