Abstract
Mineral commodity prices are influenced by economic, technological, psychological, and geopolitical factors. Stochastic approaches, and time series and econometric techniques have been used to represent the dynamics of mineral commodity markets and predict prices. However, these techniques cannot provide a comprehensive representation of market dynamics because they do not recognise the relationship between these factors over time, and they are unable to capture both the evolution and the cumulative effects of these factors on prices. Stability of motion and chaos theories can detect sensitivity to initial conditions, and therefore the evolutionary patterns allowing a proper understanding and representation of mineral commodity market dynamics. Most of the techniques used to assess chaos require a colossal amount of data, so the use of small data sets to assess chaos has been largely criticised. Nevertheless, by definition, the dynamics of a chaotic system remain at different scales owing to its self-organisation features that exhibit ordered patterns in the absence of codes or rules. Therefore, any deterministic chaotic behaviour of mineral commodity prices can be captured by using small data sets if a detailed qualitative and quantitative analysis are carried out. This paper examines the chaotic behaviour of annual copper prices between 1900 and 2015. To do so, we combine chaos theory, stability of motion and statistical techniques to reconstruct the long-term dynamics of copper prices. First, we examine the time dependency and the presence of a strange attractor by a visual analysis of the time series and phase space reconstruction based on Takens’ theorem and determine embedding parameters. Then we examine the dynamic characteristics of the system which assesses its complexity and regularity patterns to measure the system’s entropy. Finally, we calculate the largest Lyapunov exponent λ to assess the sensitivity to initial conditions and determine chaotic behaviour supported by a surrogate test. We find that annual copper prices have a chaotic behaviour embedded in a high-dimensional space and short time delay. The study suggests that copper prices exhibit only a single state of low prices, which fluctuate through transitional periods of high prices. It challenges the assertion that metal markets have fluctuated over four major super cycles and debate the adequacy of stochastic and econometric models for representing mineral commodity market behaviour.This study recommends that the use of chaotic behaviour improves our understanding of mineral commodity markets and narrows the data searching, processing and monitoring requirements for forecasting. Therefore, it improves the performance of traditional techniques for selecting key factors that influence the market dynamics, and may also be used to select the most suitable algorithm for forecasting prices.
Highlights
The impression that small causes may have a significant effect through time is known as “sensitivity dependence to initial conditions” (Lorenz, 1995) and has long been used to explain historical events and their effects through time
Takens’ theorem is one of the most well-known, and has been widely used in tests to recognise chaotic behaviour in time series. It asserts that the orbits of chaotic systems are attracted to one specific limited area of the phase space, so-called “strange attractors”, and that changes of their shape provide significant information hidden inside the dynamics of the system (Huke, 2006; Perc, 2006; Povinelli, 2001; Takens, 1981)
We prove that small datasets of 116 observations can be used to investigate the chaotic behaviour of mineral commodity prices by using the Lyapunov exponent λ method (Becks et al, 2005; Blank, 1991; Chen et al, 2016; Gaspard et al, 1998; Gottwald, 2009; Kodba et al, 2005; Navarro–Urrios et al, 2017; Panas, 2001; Panas and Ninni, 2000; Perc, 2006; Raffalt et al, 2017; Reynolds et al, 2016; Rosenstein et al, 1993; Savi, 2005; Showalter and Hamilton, 2015; Sivakumar, 2000; Wernecke et al, 2017; Zhong et al, 2017) using copper as a representative mineral
Summary
The impression that small causes may have a significant effect through time is known as “sensitivity dependence to initial conditions” (Lorenz, 1995) and has long been used to explain historical events and their effects through time. Sensitivity to changes in the initial conditions is the most important characteristic of chaotic behaviour in complex dynamic systems (Showalter and Hamilton, 2015) It is demonstrated by the fast exponential growth of the divergence between two near initial trajectories of the system across time known as strange attractors (Azar and Vaidyanathan, 2015, p 10; Becks et al, 2005; Guegan, 2009; Hegger et al, 1999; Kodba et al, 2005; Navarro–Urrios et al, 2017; Panas, 2001; Panas and Ninni, 2000; Reynolds et al, 2016; Rosenstein et al, 1993; Savi, 2005; Vlad et al, 2010; Wernecke et al, 2017; Yamamoto, 1999). The same methodology has been successfully applied to study the complexity of various dynamic systems, including electrocardiograms, human gait recording, and laser droplet generation (Krese et al, 2010; Perc, 2005a, 2005b)
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