Chaos in oil prices? Evidence from futures markets

Energy Economics 23 Ž2001. 405᎐425 Chaos in oil prices? Evidence from futures markets Bahram Adrangi a , Arjun Chatrath a,U , Kanwalroop Kathy Dhandaa , Kambiz Raffiee b a School of Business, Uni¨ ersity of Portland, 5000 N. Willamette Bl¨ d., Portland, OR 97203, USA b Department of Economics, College of Business, Uni¨ ersity of Ne¨ ada, Reno, NV 98511, USA Abstract We test for the presence of low-dimensional chaotic structure in crude oil, heating oil, and unleaded gasoline futures prices from the early 1980s. Evidence on chaos will have important implications for regulators and short-term trading strategies. While we find strong evidence of non-linear dependencies, the evidence is not consistent with chaos. Our test results indicate that ARCH-type processes, with controls for seasonal variation in prices, generally explain the non-linearities in the data. We also demonstrate that employing seasonally adjusted price series contributes to obtaining robust results via the existing tests for chaotic structure. Maximum likelihood methodologies, that are robust to the non-linear dynamics, lend support for Samuelson’s hypothesis on contract-maturity effects in futures price-changes. However, the tests for chaos are not found to be sensitive to the maturity effects in the futures contracts. The results are robust to controls for the oil shocks of 1986 and 1991. 䊚 2001 Elsevier Science B.V. All rights reserved. JEL classification: Q40 Keywords: Chaos; Crude oil futures; Energy 1. Introduction It has been well documented that non-linear relationships that are deterministic can yield highly complex time paths that will pass most standard tests of randomness Žsee Brock, 1986 for a survey.. Such random-looking but deterministic series U Corresponding author. School of Business Administration, University of Portland, Portland, OR 97203, USA. Tel.: q1-503-283-7465; fax: q1-503-978-8041. E-mail address: chatrath@up.edu ŽA. Chatrath.. 0140-9883r01r$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 4 0 - 9 8 8 3 Ž 0 0 . 0 0 0 7 9 - 7 406 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 have been termed ‘chaotic’ in the literature Že.g. Devaney, 1986; Guckenheimer and Holmes, 1986.. Chaotic dynamics are necessarily non-linear and may be able to explain a richer array of time series behavior. For example, linear models may not properly capture sudden movements and wide fluctuation in stock prices, exchange rates, or in other financial and economic series, while chaotic models may be quite suitable in explaining these behaviors. Furthermore, modeling non-linear processes may be considered less restrictive than linear structural systems as the former are not dependent on the knowledge of the specific underlying structures. Direct applications of chaos to economic theory has been initiated only in the last 20 years Že.g. Stutzer, 1980; Benhabib and Day, 1981, 1982., with researchers such as Brock and Sayers Ž1988. employing relatively new techniques to test the null of chaos in a number of macroeconomic series Žsuch as the US unemployment rate..1 The evidence of chaos in economic time series such as GNP and unemployment has thus far been weak Že.g. Brock and Sayers, 1988.. On the other hand, the few studies on the structure of commodity prices, employing a range of statistical tests, have generally found evidence consistent with low dimension chaos. Lichtenberg and Ujihara Ž1988. apply a non-linear cobweb model to US crude oil prices; Frank and Stengos Ž1989. estimate the correlation dimension and Kolmogorov entropy for gold and silver spot prices; Blank Ž1991. estimate the Lyapunov exponent for soybean futures; DeCoster et al. Ž1992. apply correlation dimension to daily sugar, silver, copper and coffee futures prices; Yang and Brorsen Ž1993. employ correlation dimension and the Brock, Dechert, and Scheinkman ŽBDS. test on several futures markets.2 Why is the evidence of chaos stronger for commodity prices than for economic time series? Baumol and Benhabib Ž1989. suggest that disaggregated variables Žsuch as commodity prices or production levels., that are inherently subject to resource constraints will make far better candidates for chaotic structure. There may be other reasons for this disparity. Prior studies on the structure of commodity prices suffer from a mixture of short data spans and fairly coarse tests for chaos. Several studies have also failed to control for seasonal variations in commodity prices. To what extent have these factors contributed to the evidence of chaos in commodity prices? In this paper, we provide evidence on the structure of commodity prices while addressing such questions. We examine the non-linear dynamics and their explanations for three important energy futures contracts: crude oil, heating oil, and unleaded gasoline, from the early 1980s. There is a surprising lack of evidence on 1 See Baumol and Benhabib Ž1989. for a more complete review on the application of chaos to economic theory. 2 Other papers document evidence of non-linearity for various financial and economic time series. For instance, see Hsieh Ž1989., and Aczel and Josephy Ž1991. for evidence on exchange rates, Scheinkman and LeBaron Ž1989., Hsieh Ž1991. for stock returns, Mayfield and Mizrach Ž1992. for S & P index prices, and Chwee Ž1998. for natural gas futures. However, the findings in these papers are not always consistent with chaos. Hsieh Ž1993. shows that conditional variance effects that are satisfactorily captured by ARCH-type models explain non-linearities in several currency futures contracts. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 407 possible non-linear dynamics in energy prices.3 These commodities play an obviously critical role in the world economy, and are necessarily subject to pricing pressures arising from world demand and supply conditions.4 Futures trading volume and prices in these commodities are known to be fairly seasonal, with prices generally rising in spring and early fall, in anticipation of increased demand. However, given that many disparate domestic and international factors affect the prices of these contracts, and that quantifiable information on such factors is generally incomplete, the accurate structural modelling of such commodities could be considered impossible.5 The evidence of chaos will offer some scope for modelling price behavior by simply employing the time series of prices. Testing for chaotic structure in commodity prices should be considered a meaningful exercise for other reasons. It has been speculated that technical analysis may succeed in forecasting short-term price behavior of various financial series because these series may be non-linear andror chaotic Žsee for example, Bohan, 1981; Brush, 1986; Pruitt and White, 1988; Pruitt and White, 1989; Clyde and Osler, 1997.. Several researchers show that technical analysis produces superior outcome relative to linear models in predicting price behavior of many financial instruments and economic time series Žsee LaBaron, 1991; Brock et al., 1992; Taylor, 1994; Blume et al., 1994; Chang and Osler, 1995; Kumar, 1992, among others.. Clyde and Osler Ž1997. conclude that it is worth investigating chaotic behavior because, unlike random processes, chaotic series are more conducive to technical analysis. Therefore, it would be informative to analyze the behavior of various financial data in order to determine the source of non-linearities, if they exist. If the non-linearity stems from chaos, then perhaps technical analysis could be applicable in the short-run for prediction purposes. However, chaos would also imply that while prices are deterministic, long-range prediction based on ‘technicals’ or statistical forecasting techniques becomes treacherous, as the slightest errors in function formulation will multiply exponentially. More is said on the matter in the next section. Our paper is distinguishable from previous studies on chaos in commodity futures markets in that: Ži. relatively long price histories are examined;6 Žii. unlike 3 Exceptions are Lichtenberg and Ujihara Ž1988. and Chwee Ž1998.. The US is the second largest producer of crude after Saudi Arabia, with approximately 6.6 million barrels a day. All but a handful of states are oil producers. 5 For instance, the supply and marketing of gasoline and distillate fuels can be as predictable as a major integrated refiner shipping its refinery output to a distribution terminal. Or, the chain can be long and varied, involving a combination of integrated or independent refiners, importers, resellers, and retailers. There may be similar unknowns at the production stage of these commodities. These may range from the variations in OPECs production quotas, to the agreements between producers and refiners on ‘posted prices’ set by the refiners. 6 Lichtenberg and Ujihara Ž1988. employ yearly data for crude from the mid-1960s to the mid-1980s; Chwee Ž1998. employs daily data for natural gas futures from 1990 to 1996. Yang and Brorsen Ž1993. examine the non-linear dynamics in daily futures prices for various commodity futures over the 1979᎐1988 interval. Blank Ž1991. examines only 2 years of data for soybean futures Žthe November 1986 contract.. DeCoster et al. Ž1992. cover an interval more comparable to ours, from October 1972 to March 1989, for the silver, copper, sugar, and coffee futures. 4 408 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 prior papers, the data are subject to adjustments for seasonalities and maturity effects that may otherwise have led to an erroneous conclusion of deterministic structure; Žiii. a wider range of ARCH-type models are considered as explanations to the non-linearities; and Živ. alternate statistical techniques are employed to test the null of chaotic structure. Like most prior studies we present strong evidence that commodity futures prices exhibit non-linear dependencies. Unlike these earlier studies, however, we find evidence that is clearly inconsistent with chaotic structure. The explanations for these inconsistencies may be attributed to differences in data size and methodology. We also make a case that employing seasonally adjusted price series may be important in obtaining robust results via the existing tests for chaotic structure. While we find notable maturity effects in the futures contracts, these effects were not found to be important to the chaos tests. We identify some commonly known ARCH-type processes that satisfactorily explain the non-linearities in the data. The GARCH model of Bollerslev Ž1986. and Exponential GARCH model of Nelson Ž1991. are found to generally perform the best in accounting for the non-linear dynamics in the commodities analyzed. The next section motivates the tests for chaos and further discusses the implications of chaotic structure in commodity prices. Simulated chaotic data are employed to highlight some important properties of chaos. Section 3 describes the procedures that this paper employs to test the null of chaos. Section 4 presents the test results for the three energy futures. Section 5 closes with a summary of the results. 2. Chaos: concepts and implications for commodity markets As the concepts of chaos are well developed in the literature, our descriptions are brief relative to some papers that we reference here. There are several definitions of chaos in use. A definition similar to the following is commonly found in the literature Že.g. Devaney, 1986; Brock, 1986; Deneckere and Pelikan, 1986; Brock and Dechert, 1988; Brock and Sayers, 1988; Brock et al., 1993.: the series a t has a chaotic explanation if there exists a system Ž h,F, x 0 . where a t s hŽ x t ., x tq1 s F Ž x t ., x 0 is the initial condition at t s 0, and where h maps the n-dimensional phase space, R n, to R1, and F maps R n to R n. It is also required that all trajectories, x t , lie on an attractor, A, and nearby trajectories diverge so that the system never reaches an equilibrium or even exactly repeats its path. For chaotic time series, if one knows Ž h,F . and could measure x t without error, one could forecast x tqi and, thus, a tqi perfectly. In that respect, chaos Žor strangeness . is the opposite of the process that is instantaneously unpredictable. With respect to the divergence property and attractor, A: in order that F generates random-looking behavior Žwhich is deterministic., nearby trajectories must diverge exponentially; moreover, in order that F generates deterministic behavior, locally diverging trajectories must eventually fold back on themselves. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 409 The attractor A may be thought of as a subset of the phase space towards which sufficiently close trajectories are asymptotically attracted Že.g. Brock and Sayers, 1988.. Chaotic time paths will have the following properties that should be of special interest to commodity market observers:7 Ži. the universality of certain routes Žsuch as the period folding over of trajectories. that are independent of the details of the map; ii. time paths that are extremely sensitive to microscopic changes in the initial conditions; this property is often termed sensitive dependence upon initial condition or SDIC;8 and Žiii. time series that appear stochastic even though they are generated by deterministic systems; i.e. the empirical spectrum and empirical autocovariance functions of chaotic series are the same as those generated by random variables, implying that chaotic series will not be identified as such by most standard techniques Žsuch as spectral analysis or autocovariance functions.. The presence of chaos will hamper the success of technical traders and long-range forecasting models. Both basic forecasting devices ᎏ extrapolation and estimation of structural forecasting models ᎏ become highly questionable in chaotic systems Žalso see Baumol and Benhabib, 1989.. Similarly, if a price series is chaotic, it is fair to say that regulators must have some knowledge of F,h to effect meaningful and more-than-transitory changes in the price patterns. Then too, it is not obvious that regulators will succeed in promoting their agenda. Without highly accurate information of F and h, and the current state x 0 , chaos would imply that regulators cannot extrapolate past behavior to assess future movements. In effect, they would only be guessing as to the need for regulation. In other words, the sensible technical analyst and policy maker ought to be indifferent to whether or not the non-linear structure is chaotic, unless of course, she had detailed knowledge of the underlying chaotic structure. It should be noted, however, that chaotic systems may provide some advantage to forecastingrtechnical analysis in the very short-run Žsay a few days when dealing with chaotic daily data.. As indicated earlier, a deterministic chaotic system is, in some respects, polar to an instantaneously unpredictable system. For instance, Clyde and Osler Ž1997. simulate a chaotic series and demonstrate that the heads-over-shoulder trading rule will be more consistent at generating profits Žrelative to random trading. when applied to a known chaotic system. However, the results in Clyde and Osler also indicate that this consistency declines dramatically, so that the frequency of ‘hits’ employing the trading rule is not distinguishable from that of a random strategy after just a few trading periods Ždays..9 7 See Brock et al. Ž1993. for a more complete description of the properties. This property follows from the requirement that local trajectories must diverge; if they were to converge, the system would be stable to disturbance, and non-chaotic. Often, researchers illustrate the property of sensitivity of initial conditions by demonstrating that, in a chaotic process, microscopic changes in the value of the parameterŽs. of the system can result in very deviant time paths. This idea is later presented at greater length ŽBaumol and Benhabib, 1989.. 9 It is also noteworthy that short-term forecasting techniques, such as locally weighted regressions, are known to perform better for chaotic data than for random data Že.g. Hsieh, 1991.. 8 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 410 3. Testing for chaos The known tests for chaos attempt to determine from observed time, series data whether it is deterministic. There are three tests that we employ here: the correlation dimension of Grassberger and Procaccia Ž1983. and Takens Ž1984., and the BDS statistic of Brock et al. Ž1987., and a measure of entropy termed Kolmogorov᎐Sinai invariant, also known as Kolmogorov entropy. Among this group, Kolmogorov entropy probably represents the most direct test for chaos, measuring whether nearby trajectories separate as required by chaotic structure. However, this and other tests of SDIC Že.g. Lyapunov exponent. are known to provide relatively fragile results Že.g. Brock and Sayers, 1988.. Thus, the need for the alternate tests for chaos. We briefly outline the construction of the tests, but we do not address their properties at length, as they have been well established Žfor instance, Brock et al., 1987 and Brock et al., 1993.. 3.1. Correlation dimension Consider the stationary time series x t , t s 1...T.10 One imbeds x t in an mdimensional space by forming M-histories starting at each date t: x t2 s  x t , x tq14 , . . . , x tM s  x t , x tq1 , x tq2 , . . . x tqMy1 4 . One employs the stack of these scalars to carry out the analysis. If the true systern is n-dimensional, provided M G 2 n q l, the M-histories can help recreate the dynamics of the underlying system, if they exist ŽTakens, 1984.. One can measure the spatial correlations among the M-histories by calculating the correlation integral. For a given embedding dimension M and a distance ␧, the correlation integral is given by C M Ž ␧ . s lim  the number of Ž i , j . for which < x iM y x jM < F ␧ 4 rT 2 Ž1. Tª⬁ where <. < is the euclidean norm.11 For small values of ␧, one has C M Ž ␧ . ; ␧ D where D is the dimension of the system Žsee Grassberger and Procaccia, 1983.. The correlation dimension in embedding dimension M is given by D M s lim ␧ª0 lim  lnC M Ž ␧ . r ln␧ 4 Ž2. Tª0 and the correlation dimension is itself given by D s lim ln D M Ž3. Mª0 10 It is known that non-stationary processes can generate low dimensions even when not chaotic Že.g. Brock and Sayers, 1988.. To rule out non-stationarity as a ‘cause’ for low dimension, one may difference the original series if it contains a unit root. 11 In practice T is limited by the length of the data which in turn places limitations on the range of the values of ␧ and M to be considered. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 411 If the value of D M stabilizes at some value Ž D . as M increases, then D is the correlation dimension. If D M continues to rise as M is raised, then the system is to be regarded as stochastic, since for practical purposes, there is no difference between high-dimensional system and randomness. Furthermore, one’s computations can only be of finite resolution and data sets are of limited length, limiting the embedding level. On the other hand, if a stable low value of D M is obtained Žsubstantially lower than 10., there is evidence that the system is deterministic.12 A problem associated with the implementation of Eqs. Ž3. and Ž4. is that, with the limited length of the data, it will almost always be possible to select sufficiently small ␧ so that any two points will not lie within ␧ of each other Že.g. Ramsey and Yuan, 1987.. A popular approach to overcome this difficulty is to instead estimate the statistic SC M s  lnC M Ž ␧ t . y lnC M Ž ␧ ty1 .4  ln Ž ␧ t . y ln Ž ␧ ty1 .4 Ž4. for various values of M Že.g. Brock and Sayers, 1988.. The SC M statistic is a local estimate of the slope of the C M vs. ␧ function. Following Frank and Stengos Ž1989., we take the average of the three highest values of SC M for each embedding dimension. There are at least two ways to consider the SC M estimates. First, the original data may be subjected to shuffling, thus destroying any chaotic structure if it exists. If chaotic, the original series should provide markedly smaller SC M estimates than their shuffled counterparts Že.g. Scheinkman and LeBaron, 1989.13 Second, along with the requirement Žfor chaos. that SC M stabilizes at some low level as we increase M, we also require that linear transformations of the data leave the dimensionality unchanged Že.g. Brock, 1986.. For instance we would have evidence against chaos if AR errors provide SC M levels that are dissimilar to that from the original series. 3.2. BDS statistic BDS Ž1987. employ the correlation integral to obtain a statistical test that has been shown to have strong power in detecting various types of non-linearity as well as deterministic chaos. BDS show that if x t is IID with a non-degenerate distribution, M C M Ž ␧ . ª C 1 Ž ␧ . , as T ª ⬁ 12 Ž5. Grassberger and Procaccia Ž1983. determine the correlation dimension of the Logistic map at 1.00 " 0.02, the Henon map at 1.22 " 0.01, and the Mackey Glass equation at 7.5 " 0.15. For further discussion, see Brock et al. Ž1993.. 13 As discussed earlier, chaos is associated with lower dimensions than found in randomness. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 412 for fixed M and ␧. Employing this property, BDS show that the statistic W M Ž␧. s 'T w C MŽ M ␧ . y C 1 Ž ␧ . xr ␴ M Ž ␧ . Ž6. where ␴ M , the standard deviation of w.x, has a limiting standard normal distribution under the null hypothesis of IID. W M is termed the BDS statistic. Non-linearity will be established if W M is significant for a stationary series void of linear dependence. The absence of chaos will be implied if it is demonstrated that the non-linear structure arises from a known non-deterministic system. For instance, if one obtains significant BDS statistics for a stationary data series, but fails to obtain significant BDS statistics for the standardized residuals from an Auto Regressive Conditional Heteroskedasticity ŽARCH. model, it can be said that the ARCH process explains the non-linearity in the data, precluding low dimension chaos. Brock et al. Ž1993. employ this premise in their technique to test for chaos. The authors employ a two-step process. First, they extract the standardized residuals from an ‘appropriate’ ARCH-type model Žgenerally GARCH, Exponential GARCH, or GARCH in Mean.. Then the authors test for non-linear dependence on the standardized residials. An absence of non-linear dependence implies an absence of chaos, since the ARCH-type model has ‘captured’ the non-linearities in the data Žwith no remaining non-linearities to be explained by other processes ᎏ such as chaos.. Others who have employed the same methodology of precluding chaos when an ARCH-type process explains the non-linear behavior of the series include Hsieh Ž1989.; Hsieh Ž1991.. Brock et al. Ž1993. examine the finite sample distribution of the BDS statistic and find the asymptotic distribution will well approximate the distribution of the statistic when: the sample has 500 or more observations; the embedding dimension is selected to be 5 or lower; and ␧ is selected to be between 0.5 and 2 S.D. of the data. The authors also find that the asymptotic distribution does not approximate the BDS statistic very well when applied to the standardized residuals of ARCH-type models Žalso see Brock et al., 1987.. This is noteworthy as financial and commodity price movements are often found to have ARCH processes. The authors suggest bootstrapping the null distribution to obtain the critical values for the statistic when applying it to standardized residuals from these models. 3.3. Kolmogoro¨ entropy Kolmogorov entropy quantifies the concept of sensitive dependence on initial conditions. Imagine two trajectories representing time paths are extremely close so as to be indistinguishable to a casual observer. As time passes, however, the trajectories diverge so that they become distinguishable. Kolmogorov entropy Ž K . measures the speed with which this takes place.14 14 Correlation entropy estimates are based on the natural logarithms of the C M. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 413 Grassberger and Procaccia Ž1983. devised a measure for K, which is more implementable than earlier measures of entropy. The measure is given by K 2 s lim lim lim ln ␧ª⬁ mª⬁ Nª⬁ ž C M Ž␧. C Mq1 Ž ␧ . / . Ž7. If a time series is non-complex and completely predictable, K 2 ª 0 . If the time series is completely random, K 2 ª⬁ . That is, the lower the value of K 2 , the more predictable the system. For chaotic systems, one would expect 0 - K 2 - ⬁, at least in principle. 4. Evidence from energy futures markets We employ daily prices of the nearby Žexpiring. futures contracts written at New York Mercantile Exchange ŽNYMEX. on crude oil, heating oil, and unleaded gasoline.15 The commodities were selected for the study given their relatively long futures price histories Žsee Table 1 for the intervals studied.. To obtain a spliced, continuous price series for each commodity, we follow common practice in tracking a particular contract until the last day of the pre-expiration month, at which point the series switch to the next nearby contract Že.g. Adrangi and Chatrath, 1999.. We focus our tests on daily returns, which are obtained by taking the relative log of prices as in, R t s lnŽ PtrPty1 . = 100, where Pt represents the closing price Žat 15.10 h US central time. on day t.16 Table 1 presents the R t diagnostics for the four series. The series are found to be stationary employing the Augmented Dickey Fuller ŽADF. statistics. The series are found to exhibit linear and non-linear dependencies as indicated by the QŽ12. and Q 2 Ž12. statistics, and Autoregressive Conditional Heteroskedasticity ŽARCH. effects are strongly suggested by Engle’s ARCH chi-square statistic ŽEngle, 1982.. Thus, as expected, there are clear indications that non-linear dynamics are generating the price series. Whether these dynamics are chaotic in origin is the question that we turn to next. To eliminate the possibility that the linear structure or seasonalities may be responsible for the rejection of chaos by the tests employed, we first estimate autoregressive models for each of the three commodities with controls for possible seasonal effects, as in p Rt s is1 15 n Ý ␤ i R tyi q Ý ␥j Mjt q ␧ t , Ž8. js1 The data are obtained from the Futures Industry Institute, Washington, DC. We do not employ smoothing models to detrend the data, as we feel that the imposed trend reversions may erroneously be interpreted as structure Žsee Nelson and Plosser, 1982.. 16 414 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 Table 1 Return diagnostics a Obs. Crude 10r1r83 03r31r95 2887 Heating 1r02r85 03r31r95 2575 Unleaded 1r02r85 03r31r95 2575 Mean S.D. ADF ADFŽT. QŽ12. Q2 Ž12. ARCHŽ6. y0.015 2.12 y26.13b y26.13b 68.82b 177.32b 89.18b y0.015 2.12 y25.18b y25.18b 88.19b 185.16b 66.29b y0.004 2.35 y23.38b y23.38b 53.26b 203.87b 97.14b Interval a The table presents the return diagnostics for three commodity futures contracts Ždaily data.. Returns are given by R t s lnŽ PtrPty1 . ⭈ 100, where Pt represents closing price on day t. ADF, ADFŽT. represent the augmented Dickey᎐Fuller tests ŽDickey and Fuller, 1981. for unit roots, with and without trends, respectively. The QŽ12. and Q 2 Ž12. statistics represent the Ljung᎐Box Ž Q . statistics for autocorrelation of the R t and R 2t series respectively. The ARCHŽ6. satistic is the Engle Ž1982. test for ARCH Žof order 6. and is ␹ 2 distributed with 6 degrees of freedom b Represents a significance level of 0.01. where M jt represent the 12 month-of-the-year dummies. The lag length for each series is selected based on the Akaike Ž1974. criterion. The residual term Ž ␧ t . represents the price movements that are purged of linear relationships and seasonal influences. Table 2 reports the results from the OLS regressions. There is evidence of seasonal effects in each of the three returns. Returns are found to be significantly positive in August, and negative in the winter months of November through January. There is also significant linear structure in the returns, up to 10 lags for crude oil, heating oil, and unleaded gasoline contracts. The Q-statistics indicate that the residuals are free of linear structure. 4.1. Correlation dimension estimates Table 3 reports the correlation dimension Ž SC M . estimates for arious components of the returns’ series alongside that for the Logistic series developed earlier. We report dimension results for embeddings up to 20 in order to check for saturation.17 An absence of saturation provides evidence against chaotic structure. For instance, the SC M estimates for the Logistic map stay close to 1.00, even as we increase the embedding dimensions. Moreover, the estimates for the Logistic series 17 Yang and Brorsen Ž1993., who also calculate correlation dimensions for various commodity futures, compute SC M only up to M s 8. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 415 Table 2 Linear structure and seasonality a Crude Heating oil Unleaded gas Rty 1 Rty 2 Rty 3 Rty 4 Rty 5 Rty 6 Rty 7 Rty 8 R Rty 10 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 0.014 Ž0.78. y0.019 Žy1.03. y0.089b Ž480. y0.039c Ž2.08. y0.079b Ž4.26. y0.001 Žy0.07. 0.023 Ž1.25. y0.084b Ž4.50. ᎐ ᎐ y0.164 Žy1.24. y0.328c Žy2.35. 0.090 Ž0.69. 0.226d Ž1.62. 0.073 Ž0.53. y0.157 Žy1.14. 0.031 Ž0.22. 0.240d Ž1.77. 0.154 Ž1.09. y0.065 Žy0.49. y0.239d Ž1.74. y0.082 Žy0.61. 0.037d Ž1.89. y0.005 Žy0.03. y1.060b Žy5.40. y0.006 Žy0.33. y0.075b Žy3.79. y0.045b Žy2.57. 0.016 Ž0.79. y0.091 Žy4.61. 0.009 Ž0.45. 0.048c Ž2.44. 0.444b Žy3.13. y0.399b Žy2.72. y0.008 Žy0.06. 0.108 Ž0.74. 0.005 Ž0.04. y0.091 Žy0.63. 0.180 Ž1.24. 0.373b Ž2.61. 0.245d Ž1.67. y0.023 Žy0.17. y0.079 Žy0.53. y0.084 Žy0.58. 0.033d Ž1.68. 0.009 Ž0.49. y0.091b Žy4.62. y0.004 Žy0.02. y0.018 Žy0.91. y0.048c Žy2.44. y0.004 Žy0.19. 0.071b Žy3.57. ᎐ ᎐ 0.045 Ž0.29. y0.241 Žy1.49. 0.381c Ž2.52. 0.377c Ž2.30. y0.006 Žy0.04. y0.285d Žy1.78. y0.164 Žy1.01. 0.259d Ž1.64. y0.074 Žy0.45. 0.102 Ž0.65. y0.409c Žy2.45. y0.171 Žy1.05. R2 QŽ12. LMŽ1. 0.025 14.23 0.73 0.037 12.29 0.81 0.024 12.23 0.06 a The coefficients and residual diagnostics are from the OLS regressions of returns on prior returns and 12 month-of-year dummies. The lag-length was selected based on the criterion of Akaike Ž1974.. The QŽ12. statistic represents the Ljung᎐Box Ž Q . statistics for autocorrelation in the residuals. The Lagrange multiplier test ŽLM. is for the null of one degree of autocorrelation less than the ␹ 2 statistic with one degree of freedom. Statistics in Ž . are t-values. b Represents a significance level of 0.01. c Represents a significance level of 0.05. d Represents a significance level of 0.10. do not change meaningfully after AR transformation. Thus, as should be expected, the SC M estimates are not inconsistent with chaos for the Logistic series. For the three commodity series, on the other hand, the SC M estimates provide evidence against chaotic structure. For instance, if one examines the estimates for the crude oil returns alone, one could Žerroneously. make a case for low dimension chaos: the SC M statistics seem to ‘settle’ at much lower than 10. However, the estimates are substantially higher for the seasonally adjusted residual series. Thus, the correlation dimension estimates suggest that there is no chaos in crude oil prices. Patterns for the other two contracts are even more pronounced in the sense 416 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 Table 3 Correlation dimension estimates a Ms 5 10 15 20 Logistic Logistic AR 1.02 0.96 1.00 1.06 1.03 1.09 1.06 1.07 Crude returns Crude ARŽ8. Crude ARŽ8.,S Crude shuffled 2.82 2.88 2.88 3.84 4.12 3.95 4.75 6.11 4.38 5.06 4.82 13.52 7.02 7.91 9.53 17.28 Heating oil returns Heating oil ARŽ10. Heating oil ARŽ10.,S Heating oil shuffled 4.16 4.13 4.13 3.97 7.78 8.13 7.89 7.79 7.20 6.80 7.62 7.12 12.37 14.10 14.42 13.71 Unleaded gas returns Unleaded gas ARŽ8. Unleaded gas ARŽ8.,S Unleaded gas shuffled 4.06 4.09 4.02 3.77 7.18 7.25 7.45 7.54 6.34 10.04 10.55 6.74 13.95 9.29 14.82 15.56 a The table reports SC M statistics for the logistic series Ž w s 3.750, n s 2000., daily crude oil, heating oil, and unleaded gasoline returns, and their various components over four embedding dimensions: 5, 10, 15, 20. ARŽ p . represents autoregressive Žorder p . residuals, ARŽ p .S represents residuals from autoregressive models that correct for month-of-year effects in the data. that the correlation dimension estimates do not settle for any of the estimated models. It is notable, however, that while the SC M estimates for the ARŽp. series are similar to the estimates for seasonally corrected ARŽp.,S series for heating oil, the estimates are quite dissimilar for crude oil and unleaded gasoline. Note that the estimates are generally much higher for the ARŽp.,S series in the crude and unleaded gasoline series, especially for the higher dimensions Žnotably, M s 20.. Thus, the correlation dimension estimates are found to be fairly sensitive to controls for seasonal effects. This has important implications for futurc tests for chaos employing correlation dimension. 4.2. BDS test results Following the technique in Brock et al. Ž1993. and Hsieh Ž1991., we also test for chaos by examining whether the non-linear dependence noted in the data can be explained away by alternate non-chaotic processes. As in Brock et al. Ž1993. and Hsieh Ž1991., we first fit the data to alternate ARCH-type systems known to satisfactorily explain the non-linear dependence in several financial and commodity price series. Then, the standardized residuals are tested for non-linearities employ- B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 417 Fig. 1. Kolmogorov Entropy Estimates. ing the BDS test statistic. As in Brock et al. Ž1993., if ARCH-type models ‘explain’ the non-linearities in the data, we conclude that the non-linear process is nonchaotic. Table 4 reports the BDS statistics for wARŽp.,Sx series, and standardized residuals Ž ␧r6h. from three sets of ARCH-type models with their respective variance equations,GARCH Ž1,1.: h t s ␣ 0 q ␣ 1 ␧ 2ty1 q ␤ 1 h ty1 q ␤ 2 TTMt , Ž9. Exponential GARCHŽ1,1.: log Ž h t . s ␣ 0 q ␣ 1 ␧ ty1 h ty1 q ␣2 ␧ ty1 h ty1 q ␤ 1 log Ž h ty1 . q ␤ 2 TTMt , Ž 10. Asymmetric component GARCH Ž1,1.: h t s qt q ␣ Ž ␧ 2ty1 y qty1 . q ␤ 1Ž h ty1 y qty1 . q ␤ 2 Ž ␧ 2ty1 y qty1 . d ty1 q ␤ 3TTMt qt s ␻ q ␳ Ž qty1 y ␻ . q ␾ Ž ␧ 2ty1 y h ty1 . , Ž 11. 418 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 where d ty1 s 1 if ␧ t - 0; 0 otherwise, the return equation which provides ␧ t is the same as in Eq. Ž8., and TTM represents time-to-maturity Žin days. of the futures contract.18 The time to maturity variable is intended to control for any maturity effects in the series ŽSamuelson, 1965..19 The BDS statistics are evaluated against critical values obtained by bootstrapping the null distribution for each of the GARCH models Žsee Appendix A.. The estimates from the above variance equations are discussed later in the paper. The BDS statistics strongly reject the null of no non-linearity in the ARŽp.,S errors for crude and heating oil futures, however, the evidence on unleaded futures is weaker. This evidence, that the energy futures have non-linear dependencies, is consistent with the findings in Table 1, and in Lichtenberg and Ujihara Ž1988. and Chwee Ž1998., among others. The BDS statistics for the standardized residuals from the ARCH-type models, however, indicate that the source of the non-linearity in the three commodities is not chaos. For instance, for the three contracts, the BDS statistics are dramatically lower Žrelative to those for the ARŽp.,S errors. for all the standardized residuals, and consistently insignificant for the GARCH Ž1,1. model. On the whole, the BDS test results further support the notion that the non-linear dependence in energy futures is explained by dynamics other than chaos. The evidence is compelling that the non-linear dependencies in commodity futures arise from ARCH-type effects, rather than from complex, chaotic structures. Finally, it is also noteworthy that the asymmetric GARCH models also performed well for each contract, and the exponential GARCH model seemed to satisfactorily fit the crude oil and unleaded gasoline series. 4.3. Entropy estimates Fig. 1 plots the Kolmogorov entropy estimates Žembedding dimension 15᎐32. for the Logistic map Ž w s 3.75, x 0 s 0.10., wARŽp.,Sx for crude oil, heating, and unleaded gasoline series.20 The entropy estimates for a shuffled crude oil return series are also presented 18 The return equation from the ARCH-type systems provided coefficients similar to those in Table 2. We also estimated another familiar model, GARCH in Mean ŽGARCHM.. The BDS statistics from the GARCHM and GARCH Ž1,1. models were found to be very similar. In the interest of brevity, we do not provide the results from the GARCHM model. The GARCH model is due to Bollerslev Ž1986., the exponential model ŽEGARCH. is from Nelson Ž1991., and the asymmetric component ARCH model is a variation of the Threshold GARCH model of Rabemananjara and Zakoian Ž1993.. 19 It is noteworthy that, in each of the above models, the TTM variable is found to be significant and in support of the Samuelson hypothesis: volatility Žconditional variance. rises as one approaches contract maturity. 20 Correlation entropy may not be singularly reliable for differentiating chaotic from stochastic processes. The correlation entropy will converge to infinity Žfor stochastic processes. only if ␧ tends to zero. In practice, K 2 is estimated for finite ␧. For this reason, most researchers seek a preponderance of evidence for chaos rather than relying on the entropy measure alone. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 419 Table 4 BDS statistics a Panel A: Crude ␧r␴ ARŽ8., S, E. 0.50 1.00 1.50 2.00 M 2 3 4 5 14.76b 16.03b 16.78b 16.21b 20.13b 20.79b 21.00b 19.75b GARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 y0.93 y1.27c y1.67 y2.09b y0.74 y1.07 y1.44 y1.83c y0.34 y0.23 y0.82 y1.27 y1.39 y0.63 y0.17 y0.86 EGARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 y1.17 y1.24 y1.36 y1.41 y1.02 y1.10 y1.06 y0.94 y0.09 y0.28 y0.39 y0.24 0.74 0.44 0.20 0.23 AGARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 0.64 0.38 y0.04 y0.53 0.57 0.22 y0.16 y0.60 1.82 1.17 0.52 0.04 2.98 2.05 1.10 0.36 Panel B: Heating oil ␧r␴ ARŽ10., S, E. 0.50 1.00 1.50 2.00 26.26b 24.89b 23.95b 22.23b 34.90b 28.89b 26.07b 23.62b 9.12b 11.22b 13.17b 13.51b 11.81b 14.11b 16.08b 16.42b 13.10b 16.50b 18.15b 18.25b 17.08b 18.67b 19.63b 19.28b GARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 0.09 0.11 0.08 0.03 0.29 0.14 0.11 0.03 0.83 0.53 0.46 0.04 1.15 0.78 0.70 0.62 EGARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 1.65 2.00 2.75b 3.06b 2.18 2.42 3.43b 3.95b 2.52b 2.92b 3.90b 4.66b 2.69 3.11 4.13b 4.92b AGARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 0.31 0.50 0.52 0.56 0.75 0.71 0.65 1.87b 1.55 1.17 1.11 1.19 2.02c 1.48c 1.41c 1.46c 420 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 Table 4 Ž Continued. Panel C: Unleaded gas ␧r␴ ARŽ8., S, E. 0.50 1.00 1.50 2.00 M 2 3 4 5 0.75 1.23 0.90 0.47 0.91 2.42c 2.05c 1.15 0.77 3.24b 3.17 1.87d 0.56 3.62b 4.12b 2.61 GARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 y0.40 y0.44 y0.44 y0.41 y0.17 0.24 0.01 0.01 y0.66 0.93 0.42 0.24 y0.57 1.03 0.32 0.04 EGARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 y1.12 y1.16 y0.86 y0.58 y1.01 y0.71 y0.41 y0.16 y0.08 y0.22 y0.06 y0.03 y0.54 y0.32 y0.26 y0.18 AGARCHŽ1,1. S.E. 0.50 1.00 1.50 2.00 y1.84 y1.79 y1.97 y1.82 y1.30 y0.90 y1.18 y1.12 y0.62 y0.32 y0.88 y0.97 y0.17 y0.09 y0.89 y1.06 a The figures are BDS statistics for ARŽ p .,S residuals, and standardized residuals ␧r6h from three ARCH-type models. The BDS statistics are evaluated against critical values obtained from Monte Carlo simulation ŽAppendix A.. b Represents a significance level of 0.01. c Represents a significance level of 0.05. d Represents a significance level of 0.10. for comparison. The estimates for the Logistic map and the shuffled series provide the benchmarks for a known chaotic, and a generally random series. The entropy estimates for the wARŽp.,Sx crude oil, heating oil and unleaded gasoline series show little signs of ‘settling down’ as do those for the Logistic map. They behave much more like the entropy estimates for the shuffled series: a general rise in the K2 statistic as one increases the embedding dimension. The plots in Fig. 1 reaffirm the correlation dimension and BDS test results: there is no evidence of low dimension chaos in energy futures prices. 4.4. ARCH and maturity effects in futures markets It is apparent from the BDS statistics presented in Table 4, that the GARCH Ž1,1. model effectively explained the non-linearities in the three contracts. Furthermore, we can reexamine the Samuelson hypothesis on the relationship between B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 421 Table 5 ARCH dynamics in crude oil, heating oil, and unleaded gasoline futures a Constant ␧ty 1 hty 1 TTM LL Chi-square Crude w ht x Heating oil w ht x Unleaded gas w ht x 0.029b Ž1.73. 0.11b Ž13.75. 0.88b Ž107.04. y0.007 Žy0.52. y5339.77 1752.96b 0.24b Ž7.05. 0.086b Ž12.90. 0.89b Ž97.21. y0.014b Žy5.96. y5060.54 924.22b 0.51b Ž10.19. 0.12b Ž14.56. 0.86b Ž88.92. y0.03 Žy11.82. y5310.60 276.32b a The maximum likelihood estimates are from GARCH Ž1,1. models fitted to the three futures returns. The variance parameters estimated are from Eqs. Ž9. and Ž11.. Statistics in Ž. are t-values. TTM represents time to maturity in days. The Chi-square test statistic is given by 2wLLŽGARCH.-LLŽOLS.x, where LL represents the Log-likelihood function. b Represents a significance level of 0.01. contract maturity and variance employing the appropriately modelled variance structure. The Samuelson hypothesis implies that the volatility in futures pricechanges increases as a contract’s delivery date approaches. If the Samuelson hypothesis were to be valid, proper valuation of futures and futures options would require that the term-structure of the volatility be estimated Žalso see Bessembinder et al., 1996.. Table 5 reports the maximum likelihood estimation results for the three energy contracts. In the interest of brevity, we do not present the results from the mean equations. The results indicate strong ARCH effects in all contracts. The Samuelson hypothesis is clearly supported for both the heating oil and unleaded gasoline contracts: the time to maturity ŽTTM. variable is negative and significant in the two variance equations. The same negative relationship is shown for the crude oil contract, but is not statistically significant. Thus, there is strong evidence that as we get closer to maturity Žas TTM falls., the conditional variance Ž h t . increases. However, it is notable that while TTM is found to be significant in the variance equation, this variable did not play a large role as a ‘control variable’ in the tests for chaos: the BDS statistics remained almost unchanged when we employed standardized residuals from models without TTM. In other words, the correlation᎐integration based tests for chaos are not as sensitive to controls for TTM. It should be noted that the evidence on strong ARCH-type effects Žand a lack of chaos. is robust to controls for structural changes in the data. For instance, given that 1986 and 1991 witnessed tremendous volatility in oil prices, we also considered extensions to the above ARCH type models that included controls for the abnormal price shocks over the 2 years. We formed a dummy variable which takes on the value of 1 for those years Ž1986 and 1991. and zero otherwise. The ARCH type models for the three contracts were reestimated, and their standardized residuals extracted. The BDS statistics for the standardized residuals were virtually identical to those reported in Table 2. These results are available from the authors. 422 B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 5. Conclusion Financial research in the past two decades has focused on the chaotic behavior of various price series for a number of reasons. Several research papers document evidence of non-linearity and chaos for various financial and economic variables. Chaotic dynamics are necessarily non-linear, and may be quite suitable in explaining sudden movements and wide fluctuations in stock prices, exchange rates, and myriad of other financial and economic series may not be properly captured by linear models. Furthermore, there is growing belief that technical analysis may succeed in forecasting short-term price behavior of chaotic time series. For instance, there is some evidence that simple and common technical trading rules Že.g. the heads-over-shoulders trading rule. will provide a short-term advantage to the trader when the price series is chaotic. Employing daily data for over a decade, we conduct a battery of tests for the presence of low-dimensional chaotic structure in the crude oil, heating oil, and unleaded gasoline futures prices. Daily returns data from the nearby contracts are diagnosed employing correlation dimension tests, BDS tests, and tests for entropy. While we find strong evidence of non-linear dependence in the data, the evidence is not consistent with chaos. Our test results indicate that ARCH-type processes may explain the non-linearities in the data. We also make a case that employing seasonally adjusted price series is important to obtaining robust results via the existing tests for chaotic structure. The evidence that the three oil commodities are driven by ARCH-type processes Žand not chaos. is robust to controls for the oil shocks of 1986 and 1991. For the three contracts several ARCH-type models adequately explain non-linear dependencies, and the maturity-effect in futures prices. The GARCH Ž1,1. results for the three futures contracts provide evidence in favor of the Samuelson hypothesis: volatility in futures returns increases as contracts approach maturity. However, the tests for chaos were found not to be as sensitive to controls for futures contract-maturity as they were to controls for seasonality. Appendix A. Simulated critical values for the BDS test statistic The figures represent the simulated values of the BDS statistic from Monte Carlo simulations of 2000 observations each. The simulations generated the 250 replications of the GARCH model Ž ␣ 1 s 0.10, ␤ 1 s 0.80., the exponential GARCH model Ž ␣ 1 s 0.05, ␣ 2 s 0.05, ␤ 1 s 0.80., and the asymmetric component model Ž ␣ s 0.05, ␤ s 0.10, ␳ s 0.80, ␾ s 0.05.. BDS statistics for four embedding dimensions and ␧ s 0.5, 1, 1.5 and 2 S.D. of the data were then computed for the 2500 simulated series. The critical values represent the 97.5th and 2.5th percentile of the distribution of the simulated statistics. B. Adrangi et al. r Energy Economics 23 (2001) 405᎐425 423 ␧r␴ M 0.5 GARCH Ž1,1. Ž97.5% critical values. 2 1.62 3 1.76 4 2.35 5 2.42 1.0 1.5 2.0 1.53 1.63 2.21 2.28 1.42 1.45 2.16 2.25 1.25 1.44 1.97 2.10 Exponential GARCH Ž97.5% critical values. 2 2.75 2.54 3 3.30 3.07 4 3.48 3.31 5 3.66 3.47 2.10 2.42 2.66 2.97 1.83 2.38 2.56 2.61 y1.78 y2.49 y2.81 y3.08 y1.74 y2.26 y2.55 y2.64 1.02 1.17 1.22 1.31 0.80 0.93 1.00 1.07 Asymmetric Component GARCH Ž2.5% critical values. 2 y2.86 y2.29 3 y3.51 y2.89 4 y3.64 y3.01 5 y3.67 y3.12 Asymmetric Component GARCH Ž97.5% critical values. 2 1.40 1.13 3 1.47 1.27 4 1.62 1.28 5 1.82 1.40 References Aczel, A.D., Josephy, N.H., 1991. The chaotic behavior of foreign exchange rates. Am. Econ. 35, 16᎐24. Adrangi, B., Chatrath, A., 1999. Margin requirements and futures activity: evidence from the soybean and corn markets. J. Futures Markets 19, 433᎐455. Akaike, H., 1974. A new look at statistical model identification. IEEE Trans. Auto. Control 19, 716᎐723. Baumol, W.J., Benhabib, J., 1989. 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