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
We use two major petroleum markets, namely those of Rotterdam and the Mediterranean. The sample consists of the daily prices of different oil products
from 4 January 1994 to 7 August 1998, resulting in 1161 observations. All prices were collected from OPEC.
Let p be the oil product price at date t then
t
Ž .
Ž . Z s
log p rp 1
t t
ty 1
is the daily log price relatives. Ž .
Thus, the transformed data given by Eq. 1 are rates of return. In this paper we will study the daily behaviour of oil product prices, using the scalar time series
Ž . 4
Z s log p rp
t t
ty 1
Chaos tests can be conducted on the original data directly without involving filters. However, in the approach taken here we will use both the original as well as
Ž . Ž
. the filtered data. The filters used in this paper are obtained by AR k -GARCH p,q
models. There are many different types of GARCH models used in finance. In this study
we determine the ‘best’ among the various types of GARCH models by minimising Akaike’s information criterion. Table 1 reports the results of fitting the ‘best’
GARCH model to each oil product return. The table reports t-statistics using robust standard errors in parentheses below each coefficient estimate, see Boller-
Ž .
slev and Wooldridge 1992 . It follows from Table 1 that: 1. Except for the Naphtha series in the Rotterdam market, the return series are
all positively skewed, that is, the distribution of these series is skewed to the right. The skewness results in Table 1 show that the distributions of all daily
returns are significantly skewed. The kurtosis coefficients are in all cases significantly leptokurtic. The combination of a significant asymmetry and
leptokurtosis indicates that oil product returns are non-normally distributed.
Ž .
By inspecting Table 1, we see that the BJ Bera]Jarque test rejects the null hypothesis of a normal distribution for both the Rotterdam and Mediterranean
markets. These results imply that returns on oil products are not normally distributed for both Rotterdam and Mediterranean markets, and according to
Ž .
Fang et al. 1994 , ‘the significant deviations from normality can be a symptom of non-linear dynamics’.
E. Panas,
V. Ninni
r Energy
Economics 22
2000 549
] 568
552
Table 1
a
Ž . Ž
. Parameter estimates for the AR K -GARCH p,q models
Ž .
Parameters NAPHTHA MOGAS PREM. 0.15 GrL
JET FUELr KERO GASOIL 0.2
SULFUR
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
I II
I II
I II
I II
Mean equation Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. f
0.00026 1.1 0.00395 2.6
y 0.00005 3.9
0.0003 2.2 0.00355 2.6
0.002574 2.9 0.00383 2.6
0.00207 2.34 Ž
. Ž
. f
0.32674 10.04 0.22305 6.8
1
Ž .
Ž .
Ž .
Ž .
Ž .
g y
0.28877 2.12 y
0.28239 2.51 y11.4113 2.85 y
0.28803 2.6 y
8.9985 2.31 l
0.5 0.5
2 0.5
2 Variance equation
a 0.000003
0.000028 0.000039
0.00004 0.00004
0.00006 0.000006
0.000007 Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. a
0.08082 2.1 0.0858 2.3
0.17853 3.0 0.0245 3.75
0.10797 3.3 0.00738 2.9
0.10500 2.9 0.08378 3.2
1
Ž .
Ž .
Ž .
Ž .
Ž .
Ž .
Ž .
Ž .
a 0.90524 19.03
0.89684 17.2 0.69802 9.2
0.51495 5.59 0.87744 30.9
0.90431 35.7 0.87542 32.2
0.89236 31.76
2
a q a 0.98606
0.98264 0.87655
0.53945 0.98541
0.91169 0.98042
0.97614
1 2
Ž .
Ž .
Q 25 and QS 25 of standardised residuals of the models Ž
. Q 25
5.85 31.4
32.0 30.1
32.1 12.97
30.76 25.4
Ž .
QS 25 7.23
22.6 22.6
13.8 28.6
21.1 18.3
25.6 Descriptive statistics of the oil products returns
U U
U U
U U
U U
SK y
0.641 0.993
0.184 0.769
0.520 0.394
0.233 0.274
U U
U U
U U
U U
EK 242.109
12.072 5.823
7.783 9.589
7.671 3.664
3.507 BJ
2835665.6 q 7240.64 q
1646.82 q 3044.75 q
4500.35 q 2876.63 q
659.93 q 609.49 q
E. Panas,
V. Ninni
r Energy
Economics 22
2000 549
] 568
553 Ž
. Table 1
Continued Parameters
FO 3.5 SULFUR FO 1.0 SULFUR
MOGAS REG UNL MOGAS PREM.
UNL 95
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
Ž I.
I II
I II
I II
I II
Mean equation Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. f
0.00002 2.2 0.00141 4.1
y 0.00018 1.88
0.01635 4.6 y
0.00064 2.1 NA
0.00016 3.1 NA
Ž .
Ž .
Ž .
Ž .
f 0.09815 2.9
0.32212 7.9 0.22501 6.74
NA 0.25214 7.8
NA
1
Ž .
g y
1.2609 4.3 l
0.5 NA
NA Variance equation
a 0.0008
0.0002 0.00001
0.00004 0.00003
NA 0.00003
NA Ž
. Ž
. Ž
. Ž
. Ž
. Ž
. a
0.06953 2.7 0.13096 2.9
0.12417 2.23 0.0386 2.2
0.17231 2.6 NA
0.14212 3.4 NA
1
Ž .
Ž .
Ž .
Ž .
Ž .
Ž .
a 0.91912 31.8
0.73999 11.2 0.81414 11.4
0.62295 7.9 0.70185 8.6
NA 0.72410 9.4
NA
2
a q a 0.98865
0.87095 0.93831
0.66155 0.87416
NA 0.86622
NA
1 2
Ž .
Ž .
Q 25 and QS 25 of standardised residuals of the models Ž
. Q 25
13.57 16.88
13.45 33.91
6.41 NA
10.6 NA
Ž .
QS 25 26.3
32.5 17.7
16.3 11.4
NA 24.4
NA Descriptive statistics of the oil products returns
U U
U U
U U
SK 0.180
0.122 0.582
0.709 0.306
NA 0.115
NA
U U
U U
U U
EK 6.353
5.96 13.732
12.021 5.607
NA 5.667
NA BJ
1958.71 q 1721.24
9187.51 q 7087.67
1538.95 q NA
1556.12 q NA
a
Notes. I, Rotterdam; II, Mediterranean; SK, skewness; EK, excess kurtosis; NA, non-available; BJ, Bera]Jarque; a , a and a need to be positive; The
1 2
sum of GARCH-parameters a and a must be less than one to guarantee a covariance stationary model; ‘
U
’ Denote skewness or excess kurtosis
1 2
significantly different from zero at the 5 level; ‘q’ Indicates that the null hypothesis of normal distribution is rejected at the 1 level of significance.
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 554
2. In the Rotterdam market six out of eight return series followed a GARCH Ž
. 1,1 process:
Ž . Z s f q f Z
q « 2
t 1
ty 1
t
Ž .
« s f ? s , f ; iid 0,1
t t
t t
2 2
2
Ž . s s a q a «
q a s 3
t 1 ty1
2 ty
1
3. In the Mediterranean oil market four out of six return series followed a Ž
. GARCH 1,1 } M model that has the following form
l
Ž . Z s f q g ? s q «
4
t t
t 2
2 2
Ž . s s a q a «
q a s 5
t 1 ty1
2 ty
1
4. The sum of coefficients a and a , in Table 1, defines the persistence of
1 2
variance. When that sum is close to one, as in the case of eight out of 14 return series, the persistence is high. The degree of persistence seems to change by a
lot between two oil products, namely FO 1.0 SULFUR and MOGAS PREM. 0.15 GrL across the two markets; and
5. If the GARCH model is specified correctly, the standardised residuals
2
Ž .
« r s
are a white noise process.
ˆ
t t
The use of the GARCH filters results in a significant reduction in the autocorre- Ž
. lation. The Q 25 statistics, for the standardised residuals from the GARCH model,
indicate the absence of autocorrelation. This is even more obvious if we observe Ž
. Ž
. the Q statistics for the squared standardised residuals, QS 25 . The QS 25 statistic,
based on squared residuals, provides an indirect test of conditional heteroskedas- Ž
. Ž
. ticity. The values of Q 25 and QS 25 statistics are shown in Table 1. The
standardised residuals from the GARCH models serve as the filtered data. We begin our analysis of the characteristics of the non-linear dynamics by
Ž . carefully constructing the phase space. The choice of the time delay t, see Eq. 6
in Appendix A, is important for properly constructing the phase space. In most applications the value of t is almost arbitrary. The value of t, however, determines
the quality of the constructed orbits in the phase space and consequently the values obtained for the dimensions and entropies. For the choice of the time delay,
Ž .
t , we used the autocorrelation function see Zeng et al., 1992 to ensure that the
components are independent. Ž
. Ž
. Ž
. Correlation dimensions s d , entropies s k
and Lyapunov exponents s l
2 2
are among the major experimental tools for studying the chaotic behaviour of our oil returns time series. For review of the methodology of these invariants see
Appendix A.
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 555
In our empirical analysis we used the following procedure: given a list of criteria, the stationary return series of oil products is characterised as being chaotic with
respect to that list if: 1. The Eckmann]Ruelle condition is satisfied. The most obvious restriction,
which the estimates of various dimensions have to satisfy, is the Ž
. Eckmann]Ruelle condition } see Eq. 11 in Appendix A.
Ž . Ž .
2. d
, K and Lyapunov exponents have a saturation level see Eqs. 8 , 12 and
2 2
Ž .
13 , respectively, in Appendix A. 3. Brock’s residual test theorem is valid.
Ž .
4. The BDS test } see Eq. 14 in Appendix A } can help distinguish between a random and a chaotic process.
Thus, in our identification of returns of oil products we adopted a procedure which combined the theoretical considerations discussed in General Methodology
} see Appendix A } with empirical observations, designed to extract the
maximum of information from our observed returns series. In the following tables we present our results. Table 2 contains the estimated
correlation dimensions of the raw and filtered data for the Rotterdam and Mediterranean markets. By calculating the correlation dimension, one is able to
decide whether a time series is generated by a dynamic process, with many or few degrees of freedom. In the case of many degrees of freedom, it is adequate to
model the scalar time series by a stochastic process. In addition, by the correlation dimension it is possible to distinguish between periodic and chaotic dynamics,
because a periodic dynamics results in an integer correlation dimension, whereas a chaotic results in a non-integer correlation dimension.
Starting with the correlation dimension, the first of the above criteria, we see from Table 2 that in five out of eight products of the Rotterdam market, namely
NAPHTHA, MOGAS PREM. 0.15 GrL, FO 3.5 SULFUR, FO 1.0 SULFUR and MOGAS REG UNL, the estimated correlation dimension is lower than the
embedding dimension and they satisfy the saturation condition. This can be an indication of a deterministic structure. Furthermore, the correlation dimension for
these five oil products is consistent with Brock’s theorem, the second in the list of the above criteria. The estimates from the filtered data are not very different from
the estimates from the raw data. In addition, the correlation dimension for the filtered data is slightly greater than that of raw data. Also, these five oil products
satisfy the Eckmann]Ruelle restriction, third in the list of the above criteria, as
Ž .
well. It seems, from Table 2 that in the case of JET FUELr KERO and MOGAS PREM. UNL 95. the correlation dimension is lower than the embedding dimension
and they satisfy Brock’s theorem but in these products it was observed that they do not satisfy the Eckmann]Ruelle restriction.
To summarise, so far five out of eight oil products of the Rotterdam oil market satisfy the first three criteria in our list. Namely, they satisfy correlation dimension,
Brock’s theorem and the Eckmann]Ruelle condition. If, in addition, they are
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 556
Table 2
a
Estimates of the correlation dimension for the Rotterdam and Mediterranean oil product markets Embedding
Oil products dimension
NAPHTHA MOGAS JET GASOIL FO
FO MOGAS MOGAS
PREM. FUELr 0.2
3.5 1.0
REG PREM.
Ž .
0.15 GrL KERO SULFUR SULFUR SULFUR UNL UNL 95
Rotterdam Original data
4 1.778
3.464 4.028
3.858 3.820
0.026 3.417
3.993 5
2.200 4.490
4.332 4.675
4.540 0.179
4.417 4.580
6 3.853
4.970 4.962
5.179 5.065
0.305 4.924
5.195 7
3.864 5.569
5.354 5.488
5.572 1.770
5.212 5.711
8 4.330
5.967 5.810
5.899 5.810
2.633 5.680
5.810 9
3.672 6.100
6.018 6.275
6.115 2.795
5.692 6.242
10 4.151
6.211 6.329
6.611 6.224
3.659 6.016
6.577 Filtered data
4 1.976
3.875 3.839
3.919 3.841
2.085 3.787
3.962 5
3.566 4.405
4.570 4.656
4.534 3.392
4.480 4.626
6 4.779
5.051 5.135
5.397 5.183
4.080 5.012
5.215 7
5.309 5.424
5.641 5.726
5.573 4.250
5.538 5.581
8 5.434
5.925 6.132
6.171 5.996
4.495 6.056
6.000 9
5.507 6.265
6.343 6.233
6.106 4.924
6.043 6.261
10 5.890
6.264 6.425
6.586 6.292
5.279 6.052
6.587 Mediterranean
Original data 4
3.255 0.606
3.858 3.695
0.239 0.282
NA NA
5 4.275
2.943 4.702
4.539 1.182
1.727 NA
NA 6
4.837 4.124
5.002 5.216
2.709 3.424
NA NA
7 5.337
4.532 5.591
5.630 3.665
3.470 NA
NA 8
5.636 4.979
5.659 5.920
4.005 4.278
NA NA
9 6.019
4.995 6.124
6.218 4.915
5.134 NA
NA 10
5.972 5.085
6.463 6.301
5.169 5.122
NA NA
Filtered data 4
3.531 0.763
3.905 3.816
1.152 2.787
NA NA
5 4.337
2.988 4.578
4.530 1.473
3.609 NA
NA 6
4.982 3.997
5.292 5.228
3.050 4.155
NA NA
7 5.553
4.473 5.770
5.676 3.704
4.491 NA
NA 8
5.980 4.805
5.992 5.945
4.531 5.088
NA NA
9 6.129
5.235 6.300
6.278 4.820
5.829 NA
NA 10
6.116 5.684
6.462 6.558
5.204 6.032
NA NA
a
NA, non-available.
found to satisfy the remaining two criteria in our list we will be in a position to argue that they are consistent with deterministic chaos. Regarding the other three
products of the Rotterdam market with strong scepticism, it can be said that they are not consistent with deterministic chaos.
The results shown in Table 2 in the case of the Mediterranean oil market can be
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 557
summarised as follows. On one hand the returns for NAPHTHA, MOGAS PREM. 0.15 GrL, FO 3.5 SULFUR and FO 1.0 SULFUR show the correlation
dimension to be lower than the embedding dimension, there is strong evidence for saturation of the estimated correlation dimension, the Brock’s theorem and the
Eckmann]Ruelle restriction are satisfied. On the other hand, the return series for
Ž .
GASOIL 0.2 SULFUR and JET FUELr KERO turn out to be more consistent with a stochastic process. The empirical results for GASOIL 0.2 SULFUR and
Ž .
JET FUELr KERO are very instructive. One might naively conclude from Brock’s theorem that the returns for both products demonstrate chaotic behaviour. This is
quite erroneous. Neither the Eckmann]Ruelle restriction nor the saturation crite- rion is satisfied.
We now turn to a discussion of entropy index, K . Recall that K entropy
2 2
measures the degree of chaos in a system. Table 3 represents the estimates of K
2
entropy, for original as well as filtered data. Three striking conclusions emerge Ž .
Ž .
from Table 3: i K approaches saturation level except for the JET FUELr KERO
2
Ž .
and GASOIL 0.2 SULFUR for the Mediterranean and for JET FUELr KERO , GASOIL 0.2 SULFUR, MOGAS REG UNL and MOGAS PREM. UNL 95 for
Ž . Rotterdam; ii
K -entropy estimates based on filtered data are greater than
2
Ž . K
-entropy estimates based on the original data; and iii the K -entropy results
2 2
also agree with the correlation dimension results. The correlation dimension Ž
. results Table 2 and the convergence of the K -entropy to a finite value for the oil
2
products NAPHTHA, MOGAS PREM. 0.15 GrL, FO 3.5 SULFUR, FO 1.0 SULFUR and MOGAS REG UN provide a strong indication of chaotic behaviour
for these products. One of the most important characteristics of chaos is sensitivity to initial
conditions. Lyapunov exponents measure the sensitivity of the system to changes in initial conditions, or determine the stability of the periodic orbits.
A positive value of the largest Lyapunov exponent is a characteristic of determin- istic chaotic behaviour and, in addition, provides a qualitative measure of the
predictability of the system. Table 4 represent the largest Lyapunov exponents obtained using original and filtered data for both markets. The results of the above
Ž . tables lead to the following conclusions: i in general, the largest Lyapunov
exponents estimates are all positive, but differ according to the embedding dimen- sion, the evolution, and the original and filtered time series data used. This
Ž . evidence strongly supports the presence of chaos in our returns series; ii If we
combine the results of correlation dimension, entropy, Eckmann]Ruelle condition, and Brock’s theorem with the Lyapunov exponent estimates, the arising evidence
Ž .
supports that all products except for JET FUELr KERO and GASOIL 0.2 Ž
. SULFUR for the Mediterranean and JET FUELr KERO , GASOIL 0.2 SUL-
FUR, MOGAS REG UNL and MOGAS PREM. UNL 95 for the Rotterdam are generated by a deterministic chaotic process. The results given in Table 4, there-
fore, complement those given in Tables 2 and 3, which showed that the chaotic behaviour for NAPHTHA, MOGAS PREM. 0.15 GrL, FO 3.5 SULFUR, FO
Ž . 1.0 SULFUR and MOGAS REG UNL can be maintained; iii Lyapunov
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 558
Table 3
a
K -entropy estimates for the oil products
2
Embedding Oil products
dimension NAPHTHA MOGAS JET
GASOIL FO FO
MOGAS MOGAS PREM.
FUELr 0.2 3.5
1.0 REG
PREM. Ž
. 0.15 GrL
KERO SULFUR SULFUR SULFUR UNL
UNL 95 Rotterdam
Original data 4
0.418 0.416
0.449 0.490
0.581 0.521
0.458 0.497
5 0.428
0.494 0.528
0.571 0.634
0.581 0.536
0.578 6
0.498 0.581
0.573 0.641
0.660 0.658
0.590 0.653
7 0.519
0.645 0.604
0.706 0.696
0.668 0.659
0.726 8
0.550 0.725
0.628 0.744
0.741 0.668
0.720 0.816
9 0.580
0.797 0.675
0.784 0.738
0.658 0.773
0.889 10
0.580 0.797
0.675 0.784
0.738 0.658
0.773 0.889
Filtered data 4
0.468 0.308
0.363 0.360
0.327 0.371
0.297 0.336
5 0.544
0.404 0.471
0.474 0.434
0.448 0.401
0.439 6
0.564 0.496
0.554 0.568
0.513 0.516
0.471 0.527
7 0.600
0.566 0.626
0.654 0.583
0.588 0.529
0.606 8
0.621 0.615
0.675 0.714
0.646 0.649
0.581 0.673
9 0.633
0.680 0.734
0.790 0.687
0.696 0.639
0.729 10
0.662 0.680
0.734 0.790
0.687 0.696
0.639 0.729
Mediterranean Original data
4 0.303
0.555 0.562
0.299 0.552
0.508 NA
NA 5
0.390 0.630
0.639 0.371
0.630 0.592
NA NA
6 0.461
0.710 0.708
0.449 0.681
0.661 NA
NA 7
0.510 0.748
0.773 0.508
0.698 0.667
NA NA
8 0.556
0.792 0.788
0.562 0.755
0.706 NA
NA 9
0.596 0.870
0.855 0.612
0.842 0.704
NA NA
10 0.596
0.870 0.855
0.612 0.842
0.704 NA
NA Filtered data
4 0.304
0.563 0.335
0.370 0.501
0.423 NA
NA 5
0.409 0.648
0.436 0.478
0.593 0.539
NA NA
6 0.496
0.732 0.530
0.576 0.659
0.635 NA
NA 7
0.562 0.759
0.602 0.638
0.678 0.708
NA NA
8 0.619
0.793 0.669
0.716 0.724
0.763 NA
NA 9
0.681 0.888
0.747 0.774
0.748 0.820
NA NA
10 0.681
0.888 0.759
0.774 0.748
0.820 NA
NA
a
NA, non-available.
exponents characterise the horizon of prediction of the underlying model. That is, predictions based on periods longer than the order of the inverse of the largest
Lyapunov exponent are not accurate. For example, the saturation value of the Lyapunov exponent in Table 4 in the case of FO 3.5 SULFUR is 0.066. Roughly,
Ž . this means that the predictability of the underlying model is 15 days; and iv For a
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 559
Table 4
a
Estimates of the largest Lyapunov exponent for the oil products Oil
Evolution or Embedding dimension
products separation
Original data Filtered data
periods 4
7 10
4 7
10 Rotterdam
NAPHTHA 3
0.072 0.036
0.018 0.161
0.121 0.053
6 0.036
0.022 0.010
0.108 0.071
0.032 10
0.037 0.019
0.013 0.067
0.047 0.007
MOGAS PREM. 0.15 GrL 3
0.341 0.119
0.056 0.356
0.114 0.083
6 0.199
0.071 0.042
0.229 0.085
0.072 10
0.130 0.049
0.036 0.139
0.066 0.015
Ž .
JET FUELr KERO 3
0.341 0.08
0.031 0.384
0.131 0.105
6 0.187
0.062 0.026
0.217 0.108
0.090 10
0.117 0.051
0.021 0.137
0.071 0.079
GASOIL 0.2 SULFUR 3
0.345 0.107
0.057 0.396
0.214 0.140
6 0.206
0.001 0.051
0.227 0.163
0.138 10
0.120 0.051
0.033 0.132
0.120 0.104
FO 3.5 SULFUR 3
0.317 0.075
0.045 0.355
0.102 0.065
6 0.187
0.051 0.028
0.213 0.069
0.042 10
0.113 0.040
0.024 0.127
0.052 0.029
FO 1.0 SULFUR 3
0.287 0.135
0.068 0.352
0.140 0.074
6 0.196
0.093 0.043
0.253 0.101
0.045 10
0.133 0.058
0.028 0.139
0.053 0.042
MOGAS REG UNL 3
0.369 0.103
0.067 0.364
0.159 0.122
6 0.220
0.086 0.052
0.219 0.126
0.096 10
0.123 0.061
0.047 0.129
0.028 0.084
MOGAS PREM. UNL 95 3
0.346 0.111
0.058 0.351
0.124 0.101
6 0.214
0.080 0.045
0.218 0.101
0.074 10
0.128 0.053
0.032 0.141
0.065 0.067
Mediterranean NAPHTHA
3 0.328
0.102 0.044
0.369 0.129
0.111 6
0.201 0.072
0.025 0.223
0.106 0.083
10 0.118
0.045 0.022
0.123 0.067
0.082 MOGAS PREM. 0.15 GrL
3 0.344
0.107 0.062
0.414 0.150
0.113 6
0.204 0.102
0.051 0.242
0.132 0.095
10 0.143
0.062 0.038
0.138 0.089
0.082 Ž
. JET FUELr KERO
3 0.333
0.082 0.034
0.366 0.112
0.072 6
0.205 0.057
0.025 0.217
0.089 0.063
10 0.122
0.043 0.024
0.131 0.059
0.059 GASOIL 0.2 SULFUR
3 0.339
0.105 0.050
0.379 0.248
0.152 6
0.212 0.089
0.055 0.240
0.190 0.149
10 0.122
0.058 0.039
0.154 0.116
0.116 FO 3.5 SULFUR
3 0.341
0.132 0.067
0.488 0.189
0.129 6
0.203 0.106
0.058 0.257
0.166 0.122
10 0.134
0.064 0.063
0.161 0.101
0.107 FO 1.0 SULFUR
3 0.339
0.113 0.056
0.384 0.133
0.095 6
0.217 0.079
0.045 0.225
0.109 0.078
10 0.132
0.051 0.032
0.156 0.070
0.074
E. Panas, V. Ninni r Energy Economics 22 2000 549]568 560
Ž .
Table 4 Continued
Oil Evolution or
Embedding dimension products
separation Original data
Filtered data periods
4 7
10 4
7 10
MOGAS REG UNL 3
NA NA
NA NA
NA NA
6 NA
NA NA
NA NA
NA 10
NA NA
NA NA
NA NA
MOGAS PREM. UNL 95 3
NA NA
NA NA
NA NA
6 NA
NA NA
NA NA
NA 10
NA NA
NA NA
NA NA
a
NA, non-available.
given oil product the values of the Lyapunov exponents decrease as the evolution length increases.
Ž Before concluding this analysis, let us consider the BDS statistical test results in
order to economise on space these results are available upon request from the .
authors . They indicate that the time series returns are chaotic and not a stochastic process. We cannot rely upon BDS results, however, as our evidence indicates that
the returns are non-linear processes with asymmetric distributions in all oil products. Therefore, the test is not appropriate in our case to provide a definite
Ž .
conclusion, see Hsieh 1991 .
3. Conclusions
