Abstract
The traditional financial econometric studies presume the underlying data generating processes (DGP) of the time series observations to be linear and stochastic. These assumptions were taken face value for a long time; however, recent advances in dynamical systems theory and algorithms have enabled researchers to observe complicated dynamics of time series data, and test for validity of these assumptions. These developments include theory of time delay embedding and state space reconstruction of the dynamical system from a scalar time series, methods in detecting chaotic dynamics by computation of invariants such as Lyapunov exponents and correlation dimension, surrogate data analysis as well as the other methods of testing for nonlinearity, and mutual prediction as a method of testing for synchronization of oscillating systems. In this chapter, we will discuss the methods, and review the empirical results of the studies the authors of this chapter have undertaken over the last decade and half. Given the methodological and computational advances of the recent decades, the authors of this chapter have explored the possibility of detecting nonlinear, deterministic dynamics in the data generating processes of the financial time series that were examined. We have conjectured that the presence of nonlinear deterministic dynamics may have been blurred by strong noise in the time series, which could give the appearance of the randomness of the series. Accordingly, by using methods of nonlinear dynamics, we have aimed to tackle a set of lingering problems that the traditional linear, stochastic time series approaches to financial econometrics were unable to address successfully. We believe our methods have successfully addressed some, if not all, such lingering issues. We present our methods and empirical results of many of our studies in this chapter.
We are grateful to an anonymous referee for helpful comments on an earlier draft of this chapter.
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Notes
- 1.
As it will become clear in the discussion of surrogate analysis below, nonlinearity is not a property of a series; it is the absence of the property of linearity that is often detected. However, it is more straightforward, even though less accurate, to speak of presence of nonlinearity in a series throughout this chapter.
- 2.
For details, see an excellent introductory book by Hilborn (1994).
- 3.
The method was suggested by Liangyue Cao.
- 4.
We use the term correlation sum because we are dealing with discrete time series. In cases that deal with continuous time series, the term correlation integral is used.
- 5.
Note that we have unfolded the time series into d-dimensional space.
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Soofi, A.S., Galka, A., Li, Z., Zhang, Y., Hui, X. (2014). Applications of Methods and Algorithms of Nonlinear Dynamics in Economics and Finance. In: Faggini, M., Parziale, A. (eds) Complexity in Economics: Cutting Edge Research. New Economic Windows. Springer, Cham. https://doi.org/10.1007/978-3-319-05185-7_1
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