The ever-returning golden ratio

Bert de Groot

Bert de Groot, Professor of Governance and Strategic Investment Policy at Erasmus School of Economics, has recently seen his paper, titled Disentangling the enigma of multi-structured economic cycles - A new appearance of the golden ratio, getting published in Technological Forecasting & Social Change. In this paper, De Groot examines cycles in GDP growth and encounters the golden ratio.

The use of determining economic cycles

De Groot offers an empirical approach to chart a pattern in the lengths of subcycles in GDP growth. One might ask, what is the use of finding interrelationships between the length of economic subcycles? The answer is the following. If these relationships can be found, cycles and their fluctuations can more easily be detected. This can then be used to signal future changes in cyclical behaviour, giving us an extra instrument to combat economic and societal distress. This subject can rightfully be called enigmatic. For over hundred years, many researchers have attempted to unravel the complex reality of multi-structured cycles. For his econometric analysis, De Groot uses time series data of GDP growth regarding 25 OECD countries and Europe.

Tools for analysis

To detect cycles, Fourier analysis is a very useful tool and at the core of De Groot’s analysis. The idea of the analysis is that a time series (a series of observations of a variable over a given period of time) can be decomposed into cosine waves. However, there is a problem. The analysis is not useful when the data shows irregular trends (non-stationarity). This is the case with lots of data, including the data of this research. De Groot uses a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to account for this obstacle. This makes the data less irregular, still preserving the cyclic behaviour of the original data. After performing these two tests, De Groot uses an algorithm to figure out which cycles are prominent. Finally, a trend-cycle is estimated to describe the amplitudes and phases of the economic cycles.


The models confirm that the cycles that were found, strongly correlate with the time series data. Almost all the estimates turn out to be statistically significant at a level of 5%. The detected cycles from the sample can describe swings in GDP growth rates of up to 5 percentage points. The models can be useful to predict future turning points of an economy as well: by letting the model forecast a part of the data that is left out of the sample, but is readily available, it is possible to assess whether the model actually forecasts what happens (out-of sample estimates). “The results indicate that in each economy, between two and five cycles are present. Cycles with a length between 5–6 years and between 9–10 years appear most frequently. Finally, De Groot has made an interesting observation, which contradicts the current notion on subcycles: “A meta-analysis on the detected cycle lengths reveals that the ratio between the lengths of consecutive cycles often closely matches the golden ratio.”

Something to think about

Euclides ( 250 BC) already defined the golden ratio.  Leonardo da Vinci’s mathematics professor Luca Pacioli (1447-1517), wrote a book “Divina Proportione” (1509, in which mathematics, art and architecture were connected, Leonardo da Vinci made the illustrations. Five centuries later, we still enjoy the brilliance of many painters. Apart from their genius in painting, they shared a passion for applying the golden ratio in their paintings. The golden ratio these days is no longer a hidden secret, many architects have applied it in their appealing constructions. More examples can be found in various sciences from biology to astronomy. Why would the golden ratio show up in economic- and social systems?

More information

You can read the full paper, published in Technological Forecasting & Social Change, here.

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