A new test for chaos and determinism based on symbolic dynamics

Abstract

a b s t r a c tWe propose a novel test to determine, given a time series, if the dynamics are generated by a deterministic (including low dimensional chaos), rather than a stochastic, process. In addition, we introduce a new nonparametric bootstrap test for independence which is consistent against a broad class of alternatives. The conditions under which the tests can be applied are very weak. The advantages of the presented methods are simplicity, invariance with respect to monotonic transformations and the applicability of the tests regardless of the discrete or continuous nature of the data generating process. We conduct several simulation studies to evaluate the performance of our tests on well-known dynamic processes. Finally, our tests are applied to several sets of financial returns that have been recently studied.

A Comprehensive Framework for Uncovering Non-Linearity and Chaos in Financial Markets: Empirical Evidence for Four Major Stock Market Indices

“…Matilla-Garcia and Ruiz Marin (2010) [6] They study the chaotic behavior of the daily series of the Dow Jones Industrial Average stock market index.…”

“…Many scholars have empirically studied the existence of nonlinear dynamics in financial series [ 4 ], meanwhile several theoretical models consistent with the presence of nonlinearity in asset prices have arisen [ 5 ]. The nonlinear approach can capture the characteristics of the financial series and their sudden fluctuations, and, therefore, plays an important role in economic modeling [ 6 , 7 ]. The initial interest in the application of nonlinear models has been extended to solve the question of whether the nonlinear dynamics have a stochastic or deterministic behavior.…”

“…Indeed, the former studies applied a methodology, such as the correlation dimension method, that is not sensitive to noise, an intrinsic feature of financial series. However, studies that have used new and more robust methods, such as the test proposed by Matilla-Garcia and Ruiz Marin (MGRM test) [ 6 ] and the Lyapunov test based on BenSaïda and Litimi [ 14 ], do not find evidence for chaos. In fact, to ensure that conclusions are sound and reliable, traditional tests should be applied to time series that consist of many observations and are noise-free.…”

“…The 0/1 test [46] that provides a value close to zero or one, indicating the latter value existence of chaos in the series. • MGRM test [6], which has the main advantage of not needing to build the attractor to assess the existence of chaos.…”

Permutation Entropy and Information Recovery in Nonlinear Dynamic Economic Time Series

“…Using symbolic dynamics, they predicted the daily change in the volatility of German and the British stock markets. Mensi et al (2014), Matilla-García and Marín (2010), Matilla-García (2007) and Stutzer (1980) constitute some other applications of the concept of symbolic dynamics in the economics literature. Ordinal patterns-permutations that arrange the values of a time series according to their order are used by Bandt (2005) and Groth (2005) as a robust method with respect to nonlinear perturbations, which describe the intrinsic patterns that are hidden in dynamics economic systems.…”