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The analysis shows that the accuracy of results improves with the increase in the number of recording points and the length of the time series, data points being sufficient to identify deterministic dynamics. All rights reserved. Introduction Chaos is widely found in the fields of physics and other natural sciences, however, the existence of chaos in economic data is still an open question.
In addition, a new international journal, which is exclusively devoted to, and entitled, Studies in Non-linear Dynamics and Econometrics, has recently been founded which is testimony to the interest in this area. Some of the main problems, which pervade the area of economic time series evidence on chaos, are the effects of noise, trend, and more general structural change.
These problems are frequently compounded by the paucity of available data. Harrison et al. While these tests have revealed an abundance of previously unexplained non-linear structure and yielded a deeper understanding of the dynamics of many different economic time series, the case of deterministic chaos in these types of series is yet to be clarified.
The main problem we identify in this study is that of noise which degrades these measurement techniques. The use of conventional filtering methods such as low pass filtering using Fourier transforms, moving averages, etc. Testing approach Before economic data can be analysed for the existence of deterministic chaos, the twin problems of growing time trends and noise require consideration. The main contribution of this paper will be to the latter where new non-linear noise reduction NNR techniques will be applied to the data.
However, the following general methodology will be followed. Firstly, the log data will be adjusted to remove systematic calendar effects and trend effects by differencing, following [10]. Secondly, in order to reconstruct a chaotic attractor in phase space, two basic parameters, the embedding dimension m, and delay time h, must be correctly determined.
An efficient method to determine an acceptable minimum m from experimental time series is the so-called false nearest neighbour FNN recently developed using a geometrical construction. It monitors the behaviour of near neighbours under changes in the embedding dimension from m!
When the number of the false nearest neighbours arising through projection is zero in dimension m, the attractor has unfolded in this embedding dimension m.
This technique is robust to the noise and a correct region of the embedding dimension can be determined in the presence of noise, which is important for the type of data used here.
An estimate of the value of the delay time h is provided by the autocorrelation function ACF. Here we concentrate on the latter. For a chaotic attractor, D2 is a non-integer, the value of which determines whether the system is low- or high-dimensional. The use of this approach must, however, be applied with caution since it describes a kind of scaling of behaviour in the limit as the distance between points on the attractor approaches zero and therefore is sensitive to the presence of noise.
Moreover, noise can also prevent precise prediction. Here we use NNR algorithms based on finding and extracting the approximate trajectory, which is close to the original clean dynamics in reconstructed phase space from the observed time series.
The implementation of the algorithms involves three basic steps: i to reconstruct the underlying attractor from the observed series, ii to estimate the local dynamical behaviour choosing a class of models and fitting the parameters statistically, and iii to adjust the observations to make them consistent with the clean dynamics. The technique can reduce noise by about one order of magnitude.
If some standard techniques are employed to pre-process the data, such as band-pass filtering, filtered embedding and singular value decomposition, significantly larger amounts of noise can be reduced since the local dynamics are enhanced.
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