Chaos Theory and Its Applications to Economic Dynamics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 10429

Special Issue Editor


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Guest Editor
Facultad De Estudios Estadisticos, Universidad Complutense de Madrid, Avda. Puerta de Hierro 1, 28040 Madrid, Spain
Interests: deterministic chaos mathematics and its applications to economic dynamics; data mining in financial risk management and risk scoring; applied econometrics; data analysis in social studies, economics and finance; spatial econometrics

Special Issue Information

Dear Colleagues,

Chaos theory refers to the behaviour of certain deterministic nonlinear dynamical systems whose solutions, although globally stable, are locally unstable. These chaotic systems describe aperiodic, irregular, apparently random and erratic trajectories, i.e., deterministic complex dynamics.

Most economic time series exhibit this type of erratic and irregular cyclical behaviour. The way in which traditionally economists have dealt with these complex dynamics has been through linear deterministic models to which purely random shocks are added to explain irregularity. Chaos theory provides a new possibility to explain the irregularity and aperiodicity of economic phenomena without the need to appeal to purely stochastic behaviour.

In this Special Issue, we aim to present the recent developments in the Applications of Chaos Theory to Economics Dynamics in two main areas. On the one hand, in the development of theoretical models that, based on a rigorous economic foundation, allow the emergence of chaotic solutions.

The second main area of application is the empirical or statistical analysis of chaotic economic and financial time series. More specifically, the application of tools to detect chaotic behavior from time series, that is, to differentiate chaotic motions from purely random fluctuations even when the economic systems that generates the time series are unknown.

This Special Issue will accept high-quality papers containing original research results and review articles of exceptional merit in the following fields.

  • Chaos models in economics (business cycle, economic growth, income distribution, economic networks, imperfect competition markets)
  • Chaos theory in financial markets
  • Detection of Chaos in economic and financial time series (application of Lyapunov exponents, Fractal Dimension, test for nonlinearities, …)
  • Predictability and Chaos control in Economics

Prof. Dr. Lorenzo Escot
Guest Editor

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Keywords

  • chaos theory in economics
  • chaos theory in financial markets
  • nonlinear economic dynamics
  • complex economics dynamics
  • detection of chaotic behaviour in economics and financial markets
  • Lyapunov exponents
  • fractal dimension
  • nonlinear economic and financial time series analysis
  • chaos control in economics
  • forecasting economic and financial time series

Published Papers (5 papers)

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Research

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21 pages, 2600 KiB  
Article
A Study on the Nature of Complexity in the Spanish Electricity Market Using a Comprehensive Methodological Framework
by Lucía Inglada-Pérez and Sandra González y Gil
Mathematics 2024, 12(6), 893; https://doi.org/10.3390/math12060893 - 18 Mar 2024
Viewed by 513
Abstract
The existence of chaos is particularly relevant, as the identification of a chaotic behavior in a time series could lead to reliable short-term forecasting. This paper evaluates the existence of nonlinearity and chaos in the underlying process of the spot prices of the [...] Read more.
The existence of chaos is particularly relevant, as the identification of a chaotic behavior in a time series could lead to reliable short-term forecasting. This paper evaluates the existence of nonlinearity and chaos in the underlying process of the spot prices of the Spanish electricity market. To this end, we used daily data spanning from 1 January 2013, to 31 March 2021 and we applied a comprehensive framework that encompassed a wide range of techniques. Nonlinearity was analyzed using the BDS method, while the existence of a chaotic structure was studied through Lyapunov exponents, recurrence plots, and quantitative recurrence analysis. While nonlinearity was detected in the underlying process, conclusive evidence supporting chaos was not found. In addition, the generalized autoregressive conditional heteroscedastic (GARCH) model accounts for part of the nonlinear structure that is unveiled in the electricity market. These findings hold substantial value for electricity market forecasters, traders, producers, and market regulators. Full article
(This article belongs to the Special Issue Chaos Theory and Its Applications to Economic Dynamics)
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18 pages, 4992 KiB  
Article
Detecting Structural Changes in Time Series by Using the BDS Test Recursively: An Application to COVID-19 Effects on International Stock Markets
by Lorenzo Escot, Julio E. Sandubete and Łukasz Pietrych
Mathematics 2023, 11(23), 4843; https://doi.org/10.3390/math11234843 - 1 Dec 2023
Viewed by 1884
Abstract
Structural change tests aim to identify evidence of a structural break or change in the underlying generating process of a time series. The BDS test has its origins in chaos theory and seeks to test, using the correlation integral, the hypothesis that a [...] Read more.
Structural change tests aim to identify evidence of a structural break or change in the underlying generating process of a time series. The BDS test has its origins in chaos theory and seeks to test, using the correlation integral, the hypothesis that a time series is generated by an identically and independently distributed (IID) stochastic process over time. The BDS test is already widely used as a powerful tool for testing the hypothesis of white noise in the residuals of time series models. In this paper, we illustrate how the BDS test can be implemented also in a recursive manner to evaluate the hypothesis of structural change in a time series, taking advantage of its ability to test the IID hypothesis. We apply the BDS test repeatedly, starting with a sub-sample of the original time series and incrementally increasing the number of observations until it is applied to the full sample time series. A structural change in the unknown underlying generator model is detected when a change in the trend shown by this recursively computed BDS statistic is detected. The strength of this recursive BDS test lies in the fact that it does not require making any assumptions about the underlying time series generator model. We ilustrate the power and potential of this recursive BDS test through an application to real economic data. In this sense, we apply the test to assess the structural changes caused by the COVID-19 pandemic in international financial markets. Using daily data from the world’s top stock indices, we have detected strong and statistically significant evidence of two major structural changes during the period from June 2018 to June 2022. The first occurred in March 2020, coinciding with the onset of economic restrictions in the main Western countries as a result of the pandemic. The second occurred towards the end of August 2020, with the end of the main economic restrictions and the beginning of a new post-pandemic economic scenario. This methodology to test for structural changes in a time series is easy to implement and can detect changes in any system or process behind the time series even when this generating system is not known, and without the need to specify or estimate any a priori generating model. In this sense, the recursive BDS test could be incorporated as an initial preliminary step to any exercise of time series modeling. If a structural change is detected in a time series, rather than estimating a single predictive model for the full-sample time series, efforts should be made to estimate different predictive models, one for the time before and one for the time after the detected structural change. Full article
(This article belongs to the Special Issue Chaos Theory and Its Applications to Economic Dynamics)
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27 pages, 1407 KiB  
Article
Entropy-Based Tests for Complex Dependence in Economic and Financial Time Series with the R Package tseriesEntropy
by Simone Giannerini and Greta Goracci
Mathematics 2023, 11(3), 757; https://doi.org/10.3390/math11030757 - 2 Feb 2023
Cited by 2 | Viewed by 1415
Abstract
Testing for complex serial dependence in economic and financial time series is a crucial task that bears many practical implications. However, the linear paradigm remains pervasive among practitioners as the autocorrelation function, because, despite its known shortcomings, it is still one of the [...] Read more.
Testing for complex serial dependence in economic and financial time series is a crucial task that bears many practical implications. However, the linear paradigm remains pervasive among practitioners as the autocorrelation function, because, despite its known shortcomings, it is still one of the most used tools in time series analysis. We propose a solution to the problem, by introducing the R package tseriesEntropy, dedicated to testing for serial/cross dependence and nonlinear serial dependence in time series, based on the entropy metric Sρ. The package implements tests for both continuous and categorical data. The nonparametric tests, based on Sρ, rely on minimal assumptions and have also been shown to be powerful for small sample sizes. The measure can be used as a nonlinear auto/cross-dependence function, both as an exploratory tool, or as a diagnostic measure, if computed on the residuals from a fitted model. Different null hypotheses of either independence or linear dependence can be tested by means of resampling methods, backed up by a sound theoretical background. We showcase our methods on a panel of commodity price time series. The results hint at the presence of a complex dependence in the conditional mean, together with conditional heteroskedasticity, and indicate the need for an appropriate nonlinear specification. Full article
(This article belongs to the Special Issue Chaos Theory and Its Applications to Economic Dynamics)
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29 pages, 1388 KiB  
Article
Testing the Efficient Market Hypothesis and the Model-Data Paradox of Chaos on Top Currencies from the Foreign Exchange Market (FOREX)
by Julio E. Sandubete, León Beleña and Juan Carlos García-Villalobos
Mathematics 2023, 11(2), 286; https://doi.org/10.3390/math11020286 - 5 Jan 2023
Cited by 3 | Viewed by 2054
Abstract
In this paper, we analyse two interesting applications related to the dynamics of economic phenomena linked to the Efficient Market Hypothesis (EMH), informative surprises, and the Model-Data Paradox of Chaos in certain top currency pairs from the foreign exchange market (FOREX). On the [...] Read more.
In this paper, we analyse two interesting applications related to the dynamics of economic phenomena linked to the Efficient Market Hypothesis (EMH), informative surprises, and the Model-Data Paradox of Chaos in certain top currency pairs from the foreign exchange market (FOREX). On the one hand, we empirically show that the FOREX market reacts under the Efficient Market Hypothesis in some cases, creating a significant variation in a short period of time (15, 30, and 60 min) in the quotes of the main currencies from the most important economic regions in the West (the United States, Europe, and the United Kingdom). This variation would depend on the actual deviation of high-impact macroeconomic news reported by these markets in relation to trade balance, unemployment rate, Gross Domestic Product (GDP), retail sales, the Industrial Production Index (IPI), and the Consumer Price Index (CPI). On the other hand, by testing the Model-Data Paradox of Chaos, we empirically verify that if we consider all the information available in the financial markets of currencies (or at least, more desegregated data) instead of daily data, and we apply a robust chaotic behaviour detection method, we can find differences in relation to the detection of chaos on the same series but with different temporal frequencies. This allows us to confirm that behind these financial time series which show an apparently random irregular evolution, there would be a generating system which, although unknown in principle, would be deterministic (and nonlinear), and we could take advantage of that deterministic character to make predictions, even if only in the short term, understanding “short term” as the time it takes for the market to incorporate these informative surprises in the FOREX market analysed. Full article
(This article belongs to the Special Issue Chaos Theory and Its Applications to Economic Dynamics)
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Review

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20 pages, 1522 KiB  
Review
Overview and Perspectives of Chaos Theory and Its Applications in Economics
by Andrés Fernández-Díaz
Mathematics 2024, 12(1), 92; https://doi.org/10.3390/math12010092 - 27 Dec 2023
Cited by 3 | Viewed by 1976
Abstract
Starting from the contribution of such thinkers as the famous Giordano Bruno (1583) and the great mathematician and physicist Henri Poincaré (1889) and the surprising discovery of the meteorologist Edward Lorenz (1963), we consider the expansion of the mathematics of chaos in this [...] Read more.
Starting from the contribution of such thinkers as the famous Giordano Bruno (1583) and the great mathematician and physicist Henri Poincaré (1889) and the surprising discovery of the meteorologist Edward Lorenz (1963), we consider the expansion of the mathematics of chaos in this article, paying attention to topology, qualitative geometry, and Catastrophe Theory, on the one hand, and addressing the possibilities derived from the new Computer Science as Quantum Algorithms and the advances in Artificial Intelligence, on the other. We especially highlight the section on computing chaos, which we consider to be new calculation and analysis instruments, such as machine learning and its algorithm called reservoir computing, through which we can know the dynamics of a chaotic system. With past data, with equations like Karamoto–Sivashinsky, one can improve predictions of the system eight times further ahead than in previous methods. Integrating the machine learning approach and traditional model-based prediction, one could obtain accurate predictions twelve Lyapunov times. As we know, in the framework of chaos theory, it is habitually accepted that the idea of long-term prediction seems impossible because we live under a veil of uncertainty. But with technological advances, the landscape begins to change, both in chaos theory and in its applications, especially in the field of economics, to which we devote particular attention, carrying out as an example the analysis of the evolution of the Madrid Stock Exchange in the 2006–2013 crisis. Above all this, a reflection of a general nature is necessary to enlighten us on the possibility of opening a new horizon. Full article
(This article belongs to the Special Issue Chaos Theory and Its Applications to Economic Dynamics)
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