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...hypothesis chaos is then investigated via the stability of the largest Lyapunov exponent. Evidence of chaos is found in futures returns. Global modelling techniques, like genetic algorithms, have been used in order to estimate potential motion equations. In addition, short term forecasts in futures price movements have been conducted with these estimated equations. The results show that although forecast errors are statistically smaller than those computed with other stochastic approaches, further research on these topics needs to be done.
1. INTRODUCTION
The behaviour of daily market energy futures can be profitable for chartists when predicting future trends as well as for analysts when explaining and determining market dynamics. Energy futures are clearly characterized by unpredictable and volatile price movements. As is well known, chaos theory shows that both characteristics are compatible with a nonlinear deterministic explanation of price movements, and not only with a pure random nonlinear approach. Despite the fact that fluctuations in prices might be attributed to some perfectly deterministic nonlinear feedback mechanism, only very short run predictions can be obtained, basically, because of the sensitivity to initial conditions that characterizes chaotic systems. If it is found that the data can be approximately described according to a nonlinear deterministic motion equation, it is worthwhile estimating it, since it might constitute a powerful forecasting tool.
Researchers in economics and finance have been interested in testing for nonlinear dependence and chaos for more than a decade now. Interest in nonlinear models has developed in parallel with an expansion in the knowledge of the properties of tools for nonlinear data analysis. Financial market data like stock market returns, exchange rate returns, natural gas futures and daily oil production have been studied, among others, by Scheinkman et al. (1989), Hsieh (1989), Chwee (1998) and Panas et al. (2000), respectively. The interest in looking for chaos on financial markets has been recently renewed (see Moshiri and Foroutan, 2006; Fernandez-Rodriguez et al. 2005; and Shintani and Linton, 2004).
The core of this approach is that the market consists of a large number of traders who are organized into dynamic, volatile, complex, and adaptive systems that are sensitive to environmental constraints, and that evolve according to their internal structures. Daily price fluctuations are the outcome of these systems. Much research in financial economics has relied on the theory of dynamical systems to analyze price movements. This theory deals with the behaviour of the evolution of a dynamical process over time. It is realistic to assume that the equations describing the underlying futures' dynamical processes are unknown. Additionally, the researcher only observes time series of prices or returns. Fortunately, time delay space reconstruction, due to Takens (1981), connects time series observation data and the underlying dynamical system. Several nonlinear techniques, based on such reconstructed space, have been developed to detect nonlinearities and chaos in observed data. Particularly, Chwee (1998) used the BDS statistic and the Lyapunov spectra to test for nonlinearity and chaos in natural gas futures. Evidence in favour of chaos was not found.
This paper analyzes the nature of three energy futures: natural gas, unleaded gasoline and light crude oil. The first goal of the paper is to examine the nonlinear and potential chaotic properties of these relevant energy futures. To this end several techniques are used: (1) the recent generalization of the well known BDS statistic; (2) the Kaplan statistic procedure; and (3) the stability of the largest Lyapunov exponent. The second goal of the paper is to estimate, if chaos is a potential source of nonlinearity, the motion equations driving energy futures returns. To this end genetic algorithms are used. A comparison, in terms of forecasting, with two other stochastic models is provided.
The paper is organized as follows. Firstly, in section 2, the data and their basic properties, as univariate time series, are described; secondly, the generalized BDS statistic tests and the Kaplan test are presented along with their respective results for the time series under study. The third section introduces the concept of the largest Lyapunov exponent, together with a recent test for the null hypothesis of chaos. Accordingly, tests for chaos are then conducted and reported. In section 4, the estimated equations, via genetic algorithms (GA), are presented. Conclusions are provided in section 5.
2. TESTING FOR NONLINEARITY
The data consist of the following daily futures at the New York Mercantile Exchange (NYMEX): Natural Gas from 04/03/1990 to 10/19/2005 (3892 observations), Unleaded Gasoline from 03/17/1992 to 01/31/06 (3499 observations) and Light Crude Oil from 04/03/1990 to 10/19/2005 (3892). We focus, however, on market returns from these three futures prices. Stationary data sets are required when testing for nonlinearity. Returns ([z.sub.t]) are defined as the difference of the logarithms of the future settlement prices [z.sup.i.sub.t] [equivalent to] 1n [P.sup.i.sub.t] I - ln [P.sup.i.sub.t-1], where i = Natural Gas, Unleaded Gas and Light Crude Oil.
Table 1 presents the descriptive statistics for the three returns. In all cases, there is excess kurtosis relative to the standard distribution. The distribution of all of them is negatively skewed (1). These observations lead us to suspect that energy futures returns are not normally distributed as is suggested by Jarque-Bera statistics.
Due to the fact that nonlinearity is a necessary (but not sufficient) condition for chaos, two tests for nonlinearity are conducted in this section: A generalized version of the well known BDS statistic which incorporates different time delays and so a fine search is guaranteed, and a direct test known as the Kaplan test.
2.1 Generalized BDS Test
The BDS test (Brock et al., 1996) is used to test the null of whiteness against the alternative of nonwhite linear and nonwhite nonlinear dependence. It is based on the estimation of the correlation integral, which was introduced in the context of dynamical systems by Grassberger and Procaccia (1983).
The basic idea behind state space reconstruction is that the past and the future of a time series both contain information about unobserved state variables that can be used to define a state at the present time. Reconstruction is done from a scalar time series and all relevant components (relative to the underlying dynamics) have to be extracted from it. Takens (1981) showed that this type of reconstruction yields a topologically equivalent attractor leaving the dynamic parameters invariant. The required reconstruction will embed the univariate observations into a multivariate phase space. To that end, information is encapsulated in the delay vector called the m-history.
The Grassberger and Procaccia correlation integral is based on Takens' 'time delay method', and it consists of the two following steps: (1) for established values of m (dimension) and [tau](delay time), to convert the scalar time series {[z.sub.1], [z.sub.2],..., [z.sub.T]} into a set of m-histories: [z.sub.i.sup.m[tau]], = {[z.sub.i], [z.sub.i+[tau]], [z.sub.i+2[tau]], ... ,[z.sub.i+(m-1)[tau]]}; (2) to compute the correlation function or integral which is estimated by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where n = T-(m-1)[tau] is the number of m-histories, with [tau]-delay time, that can be formed from T observations; and H is the Heaviside function so that H([z.sub.i.sup.m,[tau]], [z.sub.j.sup.m,[tau]]) takes the value 1 if both observations are within distance [epsilon] of each other, and otherwise. In words, (1) measures the fraction of the pairs of points [z.sub.i] that are within a distance of [epsilon] from each other. This distance is chosen relative to the standard deviation divided by the spread of the data.
It is known that the choice of time delay is crucial when estimating the correlation dimension (a measure based on the correlation integral), to the extent that an unfortunate time delay choice yields misleading results concerning the dimension of well known attractors. However, as Kantz and Schreiber (2004) indicate, the relevant mathematical framework for a proper choice of a time delay has not been convincingly studied.
Since the BDS test fixes [tau] = 1, it does not take into account all the potential power of Takens' 'time delay method' which implies a connection between geometric concepts (such as dimensions) and the analysis of time series. The 'time delay method' allows the reconstruction of phase space. By fixing [tau] = 1, m successive observations are stacked in producing the embedded phase space vectors. Real-world time series are, however, noisy and finite. These restrictions make the selection of time delay crucial: For very small [tau], the coordinates of each reconstructed state, z, do not significantly differ from one another and therefore the points are scattered along the diagonal. As a consequence, the dynamics in the space state, that take place in the coordinates of the reconstructed space, are almost linearly dependent (which is not the case for the real observable of a nonlinear system). On the other hand, a large delay time will cause the coordinates to disjoin by stretching and folding, so this will lead to vectors whose components are (seemingly) randomly distributed in the embedding space.
Recently, a new test called BDS-G (Matilla et al., 2004) has included time delay as a new parameter, and in this light, the well known BDS is interpreted as a particular case of the BDS-G test.
The BDS-G statistic:
BDS - G(m, T, [epsilon], [tau]) = [square...
NOTE: All illustrations and photos
have been removed from this article.
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