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Energy Economics 22 2000 549]568
Are oil markets chaotic? A non-linear dynamic analysis
Epaminondas Panas
U
, Vassilia Ninni
Athens Uni
¨
ersity of Economics and Business, 76 Patission St., Athens 104 34, Greece
Abstract
The analysis of products’ price behaviour continues to be an important empirical issue. This study contributes to the current literature on price dynamics of products by examining
for the presence of chaos and non-linear dynamics in daily oil products for the Rotterdam and Mediterranean petroleum markets. Previous studies using only one invariant, such as
the correlation dimension may not effectively determine the chaotic structure of the underlying time series. To obtain better information on the time series structure, a
framework is developed, where both invariant and non-invariant quantities were also examined. In this paper various invariants for detecting a chaotic time series were analysed
along with the associated Brock’s theorem and Eckman]Ruelle condition, to return series for the prices of oil products. An additional non-invariant quantity, the BDS statistic, was
also examined. The correlation dimension, entropies and Lyapunov exponents show strong evidence of chaos in a number of oil products considered. Q 2000 Elsevier Science B.V. All
rights reserved.
Keywords: Petroleum products; Correlation dimension; Entropy; Lyapunov exponent; Non-linear
dynamic; Chaos
1. Introduction
Ž .
In the last 10 years since the publications of Frank and Stengos 1989 , who found evidence of non-linear structure in the rates of return of gold and silver,
economists have been engaged in an attempt to detect non-linear structure in Ž
. economic time series. Various authors, such as Frank and Stengos
1989 ,
U
Corresponding author. Ž
. E-mail address:
panasaveb.gr E. Panas . 0140-9883r00r - see front matter Q 2000 Elsevier Science B.V. All rights reserved.
Ž .
PII: S 0 1 4 0 - 9 8 8 3 0 0 0 0 0 4 9 - 9
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 550
Ž .
Ž .
Ž .
Scheinkman and LeBaron 1989a,b , Blank 1991 , Hsieh 1991 , DeCoster et al. Ž
. Ž
. Ž
. Ž
. 1992 , Yang and Brorsen 1993 , Fang et al. 1994 , Kohzadi and Boyd 1995 , have
found strong evidence of non-linear structure and, in particular, of chaotic struc- ture in the ‘behaviour of economic’ time series.
In particular, the empirical analysis of the economic time series in a non-linear framework is important for at least three principal reasons. First, the effort
devoted to study time series reflects the fact that non-linearities convey informa- tion about the structure of the series under study. Second, these non-linearities
provide insight into the nature of the process governing the structure of these time series. Third, in the absence of information about the structure of these time series
it is difficult to distinguish the stochastic from the chaotic process.
Ž .
Ž .
Various authors, e.g. DeCoster et al. 1992 , Yang and Brorsen 1993 , have examined for chaos in economic data based only on one invariant. More specifi-
cally, the correlation dimension technique has been used. This technique, however, can only give an unreliable indication of chaos if for example the Eckmann]Ruelle
Ž .
1992 condition is not satisfied. This non-reliability is increased given that, in almost all empirical applications
the choice of a time delay is arbitrary. More specifically, in dealing with chaotic behaviour the problem posed in time series analysis is the determination of the
following invariants: fractal dimension; entropies; and Lyapunov exponents. From 4
the knowledge of economic time series Z
, t s 1,2,...,n , we can produce a
t
Ž .
Ž .
reconstructed phase-space or embedding space by the method of Takens 1981 , 4
Z , Z , Z
,...,Z , having the same invariant structure as Z , t s
t tqt
tq 2 t
tqŽ my 1.t
t
4 1,2,...,n . In almost all empirical results the time delay, t, is often chosen arbitrarily.
The specification of the time delay is of central importance, since the choice of t influences the values of the invariants. For this reason, in this paper, we focus our
attention on how to choose t. In this fashion we will address the issue of the quality of reconstructed time series.
There are a variety of criteria or methods that can be used to examine the chaotic behaviour of time series. There are advantages and disadvantages to each
of these criteria and there is no obviously superior approach. These criteria are: Ž .
Ž . Ž .
Ž . i correlation dimension; ii entropy; iii maximal Lyapunov exponent; iv
Ž . Ž .
Eckmann]Ruelle condition; v Brocks or residual test theorem; and vi BDS statistic test.
The importance of these criteria in examining the chaotic structure lies not only in their usefulness in analysing the non-linear structure but also in their relevance
and potential utility to distinguish between stochastic behaviour and deterministic chaos. It is, therefore, essential that a careful investigation be made into the
validity of these criteria } towards which goal this paper is a small first step. An alternative approach to the analysis of economic time series is to use all the
available criteria. Such a procedure should be sufficiently general in order to minimise an incomplete or even ambiguous picture of the time series. Indeed, this
methodology is used in the present study to obtain information on the structure of our time series. Therefore, the purpose of this paper is to apply this methodology
in the search of chaotic structure in daily price data for each of eight products:
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 551
Ž .
NAPHTHA; MOGAS PREM. 0.15 GrL; JET FUELr KERO ; GASOIL 0.2 SULFUR; FO 3.5 SULFUR; FO 1.0 SULFUR; MOGAS REG UNL; MOGAS
PREM. UNL 95 for the Rotterdam and Mediterranean markets. This paper is organised as follows. Following this introductory section, we
present our empirical results in Section 2. Finally our main concluding remarks are presented in Section 3.
2. Empirical results
