آشفتگی در قیمت نفت؟ شواهدی از بازارهای آتی

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.

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 (see Brock, 1986 for a survey). Such random-looking but deterministic series have been termed ‘chaotic’ in the literature (e.g. Devaney, 1986 and 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 (e.g. Stutzer, 1980, Benhabib and Day, 1981 and Benhabib and Day, 1982), with researchers such as Brock and Sayers (1988) employing relatively new techniques to test the null of chaos in a number of macroeconomic series (such as the US unemployment rate).1 The evidence of chaos in economic time series such as GNP and unemployment has thus far been weak (e.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 (1988) apply a non-linear cobweb model to US crude oil prices; Frank and Stengos (1989) estimate the correlation dimension and Kolmogorov entropy for gold and silver spot prices; Blank (1991) estimate the Lyapunov exponent for soybean futures; DeCoster et al. (1992) apply correlation dimension to daily sugar, silver, copper and coffee futures prices; Yang and Brorsen (1993) employ correlation dimension and the Brock, Dechert, and Scheinkman (BDS) test on several futures markets.2 Why is the evidence of chaos stronger for commodity prices than for economic time series? Baumol and Benhabib (1989) suggest that disaggregated variables (such 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 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 and/or chaotic (see for example, Bohan, 1981, Brush, 1986 and 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 (see LaBaron, 1991, Brock et al., 1992, Taylor, 1994, Blume et al., 1994, Chang and Osler, 1995 and Kumar, 1992, among others). Clyde and Osler (1997) 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: (i) relatively long price histories are examined;6 (ii) unlike 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; (iii) a wider range of ARCH-type models are considered as explanations to the non-linearities; and (iv) 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 (1986) and Exponential GARCH model of Nelson (1991) 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

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.