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Are the crude oil markets becoming weakly efficient over time? A test for time-varying long-range dependence in prices and volatility
Energy Economics 29 (2007) 28 – 36 www.elsevier.com/locate/eneco
Are the crude oil markets becoming weakly efficient over time? A test for time-varying long-range dependence in prices and volatility☆ Benjamin M. Tabak a,⁎, Daniel O. Cajueiro b a
b
Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar. DF 70074-900 Universidade Católica de Brasίlia – Mestrado em Economia de Empresas. SGAN 916, Módulo B – Asa Norte. DF 70790-160 Brazil Received 14 June 2006; accepted 22 June 2006 Available online 14 August 2006
Abstract This paper analyzes the efficiency of crude oil markets (Brent and West Texas Intermediate) by means of estimating the fractal structure of these time series. We test for time-varying degrees of long-range dependence using the Rescaled Range Hurst analysis and find evidence that this market has become more efficient over time. These results are robust for controlling for short-term autocorrelation by means of a shuffling procedure. © 2006 Elsevier B.V. All rights reserved. Keywords: Long-range dependence; Oil prices; Volatility
1. Introduction Crude oil is a commodity of fundamental importance. It is considered one of the most important commodities of the world, whose consumption is about 75 million of barrels per day. Therefore, the understanding of the dynamics of its price time series seems to be crucial, since it may allow to assess the potential impacts of its shocks in several economies and on other financial assets. Crude oil prices have presented, as well as any other commodity, large variations in times of shortage or oversupply. The prices of crude oil have been strongly influenced by wars, embargoes and revolutions. In particular, the following facts affected strongly crude oil prices: (1) The Yom ☆ The opinions expressed in this paper are those of the authors and do not necessarily reflect those of the Central Bank of Brazil. ⁎ Corresponding author. Tel.: +55 61 4143092; fax: +55 61 4143045. E-mail address: [email protected] (B.M. Tabak).
0140-9883/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.eneco.2006.06.007
B.M. Tabak, D.O. Cajueiro / Energy Economics 29 (2007) 28–36
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Kippur War which started with an attack on Israel by Syria and Egypt in 1973 and curtailed the production of crude oil by 5 million barrels per day; (2) The Iranian revolution which yielded in the lost of about 3 million barrels per day between 1978 and 1980; (3) The uncertainty associated to the Iraqi invasion of Kuwait in 1990 and the subsequent Gulf War; (4) The beginning of USA military action in Iraq on March of 2003. The regulation of prices performed mainly by USA and OPEC countries also played an essential role in this scenery. Several facts are chronologically important in this context: (1) Formation of the OPEC (Organization of Petroleum Export Countries) in 1960 to represent the interests of the petroleum supply side; (2) Due to the limited production of crude oil in the United States in 1971, the power of control of crude oil was shifted from USA to OPEC; (3) The USA imposing of prices controls on domestically produced oil in 1973–1974 to lessen the impact of the embargo. Actually, in spite of much effort performed by OPEC to set production small enough to stabilize crude oil prices, many of these attempts did not work as OPEC expected. Moreover, one may realize that the price series of crude oil is formed by cycles – some of them caused by shocks related to the success of OPEC's policy for price stabilization and other of them caused by shocks related to OPEC's failure. The main goal of this paper is to investigate whether or not this market is weak efficient. Moreover, following the novel approach introduced in Cajueiro and Tabak (2004a), we want to test if the crude oil market has presented a movement toward efficiency over time. So, in order to deal with the above problem, we calculate the Hurst's exponent H over time using moving time windows of fixed length.1 If H N 0.5, then the market is not weakly efficient2 since it possesses long memory. Additionally, the detection of the presence of long-range dependence in financial data is essential to facilitate optimal portfolio allocation decisions, since its presence causes several drawbacks in modern finance: (1) the optimal consumption and portfolio decisions may become extremely sensitive to the investment horizon (Mandelbrot, 1997); (2) the methods used to price financial derivatives based on martingale models (the most common models, e.g. the Black– Scholes model (Black and Scholes, 1973) are not useful anymore; (3) since the usual tests based on the Capital Asset Pricing Model and Arbitrage Pricing Theory (Black et al., 1972) do not take into account long-range dependence, they cannot be applied to series that present such behavior. We seek for the presence of long-range dependence in both crude oil prices and volatility. Additionally, for comparison purposes, our data includes two different crude oil markets: Brent (London) and West Texas Intermediate (New York). We expect to find the presence of long memory in these crude oil time series, but with intensity that decreases over time – which means that this market is becoming more efficient. The presence of long memory in these price time series can be justified if the shocks that took place in these time series decay at a hyperbolic rate rather than exponential as one would assume for short-term processes. On the other hand, the motivation that make us to believe that the crude oil market is becoming more efficient over time stems from the fact that this market has undergone a major structural change due to policy changes that attempted to increase the efficiency of the North American energy industry [see Serletis and Andreadis (2004) and Serletis and Rangel-Ruiz (2004) for a nice discussion of these changes and also Winston (1993)]. Therefore, the methodology using a time-varying Hurst exponent (Cajueiro and Tabak, 2004a) may show whether 1 From the statistical physics, we know that a given market is said to present the long range dependence phenomena with persistent behavior if the Hurst's exponent H N 0.5, with anti-persistent behavior is if H b 0.5 and random walk behavior if H = 0.5. In usual situations, financial time series presents Hurst's exponent equal or higher than 0.5. 2 The weak form of market efficiency states that under the information contained on the set formed by past returns, future returns are unpredictable (Fama, 1970).
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the crude oil market has become more efficient in the 1990s (deregulated period) if compared to the 1980s (regulated period). The paper is organized along the following lines. Section 2 provides a brief literature review. While Section 3 discusses the methodology employed in the paper, Section 4 describes the data. Section 5 presents empirical results. Finally Section 6 concludes the paper. 2. Brief literature review This paper is not the first one that investigates whether or not the crude oil market is efficient. To understand the purpose of the seminal papers in this subject, one should take the period from 1973 to 1974 to until at least the end of 1985 in to account. In this period, one could see the coexistence of two crude oil markets: (1) one market that represented the biggest part of the crude oil traded on the world and was sold under long term contracts at the so-called OPEC official prices, which were adjusted only infrequently and (2) a smaller, although increasing market, which quantities were traded at market determined spot prices. The coexistence of these two markets raised the issue of whether the official prices behaved like efficient contract prices. Therefore, some works tried to cope with this issue. For details, see Gately et al. (1977), Verleger (1982) and Green and Mork (1991). A more recent literature is concerned with the existence of chaotic and nonlinear structures in oil markets series. If these time series are chaotic or present deterministic non-linear structures, then technical analysis can be useful for prediction purposes and the weak form of efficiency does not play role in these markets. In particular, while Panas and Ninni (2000) test whether the oil markets are chaotic and find evidence (using correlation dimension entropies and Lyapunov exponents) that oil products such as NAPHTA, MOGAS PREM., SULFUR FO 3.5% and FO 1.0%, GASOIL and MOGAS REG. UNL. are chaotic time series, Adrangi et al. (2001) find evidence of non-linear dependencies for crude oil, heating oil and unleaded gasoline futures prices, but do not find evidence of chaos in these time series. Although the literature which seeks for the presence of long-range dependence phenomena in equity returns is very wide and this phenomena is seem as a stylized fact,3 to the best of our knowledge, only Alvarez-Ramirez et al. (2002) and Serletis and Andreadis (2004) have provided empirical evidence of long-range dependence in crude oil prices with persistent structure. In fact, while Alvarez-Ramirez et al. (2002) have studied the Brent crude oil prices, West Texas Intermediate crude oil prices and Dubai crude oil prices, Serletis and Andreadis (2004) have studied the West Texas Intermediate crude oil prices and Henry Hub natural gas prices. On the other hand, in a variety of papers, however, there is considerable evidence suggesting that not only changes in log of prices may possess long-range dependence but that also volatility is persistent. Most studies such as Cajueiro and Tabak (2005a,b) have found that Hurst exponents for volatility returns are above 0.8 in most cases, suggesting strong volatility long memory. However, to the best of our knowledge such a study has not been done on crude oil market. This paper pretends also to fill in this gap. Therefore, this paper will extend the studies of Alvarez-Ramirez et al. (2002) and Serletis and Andreadis (2004) by evaluating the dynamic nature of these time series comparing the Brent and 3 Cajueiro and Tabak (2004a,b) and Matteo et al. (2003) have shown that financial time series usually present the long range dependence phenomena but its degree is associated to the degree of the development of the given financial market. Moreover, Cajueiro and Tabak (2004a,b), employing a moving window with fixed length of 1008 observations, also have shown that the intensity of the long range dependence phenomena has been decreasing over time meaning that these markets are converging toward efficiency.
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WTI crude oil markets and also performing such studies to focus on volatility. Again, we are not only interested in the detection of the long-range dependence phenomena, but also how it evolves over time. We intend to capture underlying structural changes in the dynamics of these series, to compare the deregulated and regulated period for crude oil markets and finally to verify if these markets of crude oils are converging toward efficiency. Moreover, knowing that the presence of the short-range dependence may affect our results (see Section 3), we apply a procedure that cleans our results of the influence of this phenomenon. 3. Methodology The methodology that we employ here is based on the estimation of the Hurst exponent H by the R/S analysis (Hurst, 1951) when applied to log return time series. More explicitly, let X(t) be theprice of a commodity on a time t and r(t) be the logarithmic return denoted by rðt Þ ¼ ln xðtþ1Þ xðtÞ .The R/S statistic is the range of partial sums of deviations of times series from its mean, rescaled by its standard deviation. So, consider a sample of continuously compounded asset returns {r1, r2, …, rτ} and let ¯r τ denote the sample mean Ps 1 r where τ is the time span considered. Then the R/S statistic is given by t¼1 t s " # t t X X 1 max ðrk − ¯ ðrk − ¯r s Þ ð1Þ ðR=SÞs u r s Þ− min 1VtVs Ss 1VtVs k¼1 k¼1 where sτ is the usual standard deviation estimator " #1 2 1X 2 Ss u ðrt − r¯s Þ s t
ð2Þ
Hurst found that the rescaled range, R/S, for many records in time is very well described by the following empirical relation: ðR=SÞs ¼ ðs=2ÞH
ð3Þ
The major problem of the R/S method is its sensitive to short-term memory. Therefore, in order to correct for short-range dependence we have applied the R/S analysis to blocks of shuffled data, i.e., one picks a random permutation of the data series within blocks of predetermined size (in general, small size blocks) and applies the R/S analysis to this shuffled data.4 In this work, we use blocks of size 20. Finally, considering the approach proposed in Cajueiro and Tabak (2004a), we estimate the Hurst exponent (using the R/S analysis and the correction for short memory) for subsample windows of 1008 observations (approximately 4 years of data) to test for time-varying degrees of long-range dependence. 4. Data The crude oil prices data were taken from the Bloomberg database. The data is given in US dollar per barrel for the Brent (North Sea–Europe) and West Texas Intermediate (WTI) Cushing (US). The data constitutes of daily closing prices over the period from May 16, 1983 to July 28, 4 This is justified due to the Lo's (1991) critique that the R/S statistics is sensitive to the presence of short range dependence and the effect of random permutations in these small blocks is necessary to destroy any particular structure of autocorrelation within these blocks.
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Fig. 1. Daily Brent crude oil prices.
2004 for WTI (a total of 5276 observations) and from May 16, 1983 to July 29, 2004 for Brent (a total of 5277 observations). Figs. 1 and 2 provide a graphical representation of these series. 5. Empirical results We have found a very interesting pattern regarding the long-range dependence behavior of the crude oil market by studying two major indices, the Brent and the West Texas Intermediate (WTI).
Fig. 2. Daily WTI crude oil prices.
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Fig. 3. Time-varying Hurst exponents for Brent crude oil returns.
Empirical results suggest that the 1980s were much more inefficient than the 1990s, i.e., Hurst exponents calculated for changes in crude oil prices are on average 0.624 and 0.572 for Brent and WTI for the 1980s while these numbers drop to 0.532 and 0.509 on average for the deregulated period (1990s). In Fig. 3 we show the graph of time-varying Hurst exponents evaluated for moving windows of fixed length of 1008 observations (approximately 4 years) at a time. We also plot a 95% confidence interval. The date in the x-axis stands for the beginning of the sample used in the
Fig. 4. Time-varying Hurst exponents for WTI crude oil returns.
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Fig. 5. Time-varying Hurst exponents for Brent crude oil volatility.
estimation of the Hurst exponents. Therefore, for a date May-83 the Hurst exponents were evaluated for the sample beginning in May-83 and ending 4 years later (May-87) and so forth. The change in the dynamics of these Hurst exponents is striking. Hurst exponents tend to fluctuate in the 0.60–0.65 range in the 1980s period, decreasing substantially in the latter period. In Fig. 4 Hurst exponents are in the range 0.55–0.6 in the former period and decrease substantially in the latter period and after 1994 these Hurst exponents seems to be gravitating around 0.5 (random walk without long-range dependence). These results seem to be quite different from the ones presented in Serletis and Andreadis (2004) because our time series were corrected for short-term autocorrelation, which is presented in these returns. Nonetheless, qualitative results are in line with previous findings that there are periods in which crude oil prices seem to possess long-range dependence. An important issue that has not been addressed before is whether volatility for crude oil prices has long memory. We present a similar graph in Fig. 5 for volatility of Brent returns. We use absolute returns as a proxy for volatility following Cajueiro and Tabak (2005,b).5 Although there is some evidence of a small downward trend there seems to be cycles in which persistence changes. However, the main finding is that volatility is highly persistence and volatility models such as GARCH or EGARCH cannot be used to estimate and forecast crude oil prices volatility. Furthermore, this has an important implication for option pricing, as the Black and Scholes model (1973) cannot be used to price crude oil options as it does note take into account such features. A similar result is found for the volatility of WTI (Fig. 6). However, interestingly Hurst exponents are higher for the volatility of WTI crude oil prices if compared to Brent. Furthermore, although there seems to be a downward trend Hurst exponents are very far from 0.5. These latter results on Hurst exponents for volatility of crude oil prices suggest that volatility is a highly persistent process, which has important implications. Shocks to volatility
5
However, qualitative results are similar if we use squared returns.
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Fig. 6. Time-varying Hurst exponents for WTI crude oil volatility.
tend to die rather slowly if compared to what GARCH and E-GARCH models would forecast. 6. Conclusions This paper examines the dynamics properties of crude oil prices using a novel approach for testing for long-range dependence. Our testing procedure consists in “shuffling” returns for testing for long-range dependence in a “rolling sample”. This methodology allows ascertaining whether long memory effects found in previous studies are in place for more recent data and whether the deregulation of the crude oil market that took place in the 1980s has provoked changes in these time series dynamics. Our results indicate that although a fractal structure has been detected for crude oil prices in previous research, when conducting a time-varying study the crude oil market has become more efficient over time. Furthermore, the WTI crude oil prices seems to be more weak form efficient than Brent prices. A novel finding is that volatility for these crude oil prices is highly persistent and GARCH or E-GARCH models for these times series are misspecified. Furthermore, although Hurst exponents for these time series have decreased substantially over time they are still well above 0.5, in the 0.6–0.7 range, which is substantially high. Our results are consistent with the findings of Serletis and Andreadis (2004), which show that crude oil prices possess long-range dependence. However, the degree of long-range has decreased in the recent period for both mean and volatility returns. References Adrangi, B., Arjun, C., Dhanda, K.K., Raffiee, K., 2001. Chaos in oil prices? Evidence from future markets. Energy Economics 23, 405–425.
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Alvarez-Ramirez, J., Cisneros, M., Ibarra-Valdez, C., Soriano, A., 2002. Multifractal Hurst analysis of crude oil prices. Physica A 313, 651–670. Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659. Black, F., Jensen, M., Scholes, M., 1972. The capital asset pricing model: some empirical tests. In: Jensen, M. (Ed.), Studies in the Theory of Capital Markets Praeger. New York. Cajueiro, D.O., Tabak, B.M., 2004a. The Hurst's exponent over time: testing the assertion that emerging markets are becoming more efficient. Physica A 336, 521–537. Cajueiro, D.O., Tabak, B.M., 2004b. Ranking efficiency for emerging markets. Chaos, Solitons and Fractals 22, 349–352. Cajueiro, D.O., Tabak, B.M., 2005a. Testing for time-varying long range dependence volatility for emerging markets. Physica A 346, 577–588. Cajueiro, D.O., Tabak, B.M., 2005b. Ranking efficiency for emerging markets II. Chaos, Solitons and Fractals 23, 671–675. Fama, E.F., 1970. Efficient capital markets: a review of theory and empirical work. Journal of Finance 25, 383–417. Gately, D., Kyle, J.F., Fisher, D., 1977. Strategies for OPEC's pricing decisions. European Economic Review 10, 208–230. Green, S.L., Mork, K.A., 1991. Toward efficiency in the crude-oil market. Journal of Applied Econometrics 6, 45–66. Hurst, E., 1951. Long term storage capacity of reservoirs. Transactions on American Society of Civil Engineering 116, 770–799. Lo, A.W., 1991. Long-term memory in stock market prices. Econometrica 59, 1279–1313. Mandelbrot, B., 1997. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer, New York. Matteo, T.D., Aste, T., Dacorogna, M.M., 2003. Scaling behaviors in differently developed markets. Physica A 324, 183. Panas, E., Ninni, V., 2000. Are oil markets chaotic? A non-linear dynamic analysis. Energy Economics 22, 549–568. Serletis, A., Andreadis, I., 2004. Random fractal structures in North American energy markets. Energy Economics 26, 389–399. Serletis, A., Rangel-Ruiz, R., 2004. Testing for common features in North American energy markets. Energy Economics 26, 401–414. Verleger, P.K., 1982. Oil Markets in Turmoil. Ballinger, Cambridge, MA. Winston, C., 1993. Economic deregulation: days of reckoning for microeconomists. Journal of Economic Literature 31, 1263–1289.
Are the crude oil markets becoming weakly efficient over time? A test for time-varying long-range dependence in prices and volatility☆ Benjamin M. Tabak a,⁎, Daniel O. Cajueiro b a
b
Banco Central do Brasil, SBS Quadra 3, Bloco B, 9 andar. DF 70074-900 Universidade Católica de Brasίlia – Mestrado em Economia de Empresas. SGAN 916, Módulo B – Asa Norte. DF 70790-160 Brazil Received 14 June 2006; accepted 22 June 2006 Available online 14 August 2006
Abstract This paper analyzes the efficiency of crude oil markets (Brent and West Texas Intermediate) by means of estimating the fractal structure of these time series. We test for time-varying degrees of long-range dependence using the Rescaled Range Hurst analysis and find evidence that this market has become more efficient over time. These results are robust for controlling for short-term autocorrelation by means of a shuffling procedure. © 2006 Elsevier B.V. All rights reserved. Keywords: Long-range dependence; Oil prices; Volatility
1. Introduction Crude oil is a commodity of fundamental importance. It is considered one of the most important commodities of the world, whose consumption is about 75 million of barrels per day. Therefore, the understanding of the dynamics of its price time series seems to be crucial, since it may allow to assess the potential impacts of its shocks in several economies and on other financial assets. Crude oil prices have presented, as well as any other commodity, large variations in times of shortage or oversupply. The prices of crude oil have been strongly influenced by wars, embargoes and revolutions. In particular, the following facts affected strongly crude oil prices: (1) The Yom ☆ The opinions expressed in this paper are those of the authors and do not necessarily reflect those of the Central Bank of Brazil. ⁎ Corresponding author. Tel.: +55 61 4143092; fax: +55 61 4143045. E-mail address: [email protected] (B.M. Tabak).
0140-9883/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.eneco.2006.06.007
B.M. Tabak, D.O. Cajueiro / Energy Economics 29 (2007) 28–36
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Kippur War which started with an attack on Israel by Syria and Egypt in 1973 and curtailed the production of crude oil by 5 million barrels per day; (2) The Iranian revolution which yielded in the lost of about 3 million barrels per day between 1978 and 1980; (3) The uncertainty associated to the Iraqi invasion of Kuwait in 1990 and the subsequent Gulf War; (4) The beginning of USA military action in Iraq on March of 2003. The regulation of prices performed mainly by USA and OPEC countries also played an essential role in this scenery. Several facts are chronologically important in this context: (1) Formation of the OPEC (Organization of Petroleum Export Countries) in 1960 to represent the interests of the petroleum supply side; (2) Due to the limited production of crude oil in the United States in 1971, the power of control of crude oil was shifted from USA to OPEC; (3) The USA imposing of prices controls on domestically produced oil in 1973–1974 to lessen the impact of the embargo. Actually, in spite of much effort performed by OPEC to set production small enough to stabilize crude oil prices, many of these attempts did not work as OPEC expected. Moreover, one may realize that the price series of crude oil is formed by cycles – some of them caused by shocks related to the success of OPEC's policy for price stabilization and other of them caused by shocks related to OPEC's failure. The main goal of this paper is to investigate whether or not this market is weak efficient. Moreover, following the novel approach introduced in Cajueiro and Tabak (2004a), we want to test if the crude oil market has presented a movement toward efficiency over time. So, in order to deal with the above problem, we calculate the Hurst's exponent H over time using moving time windows of fixed length.1 If H N 0.5, then the market is not weakly efficient2 since it possesses long memory. Additionally, the detection of the presence of long-range dependence in financial data is essential to facilitate optimal portfolio allocation decisions, since its presence causes several drawbacks in modern finance: (1) the optimal consumption and portfolio decisions may become extremely sensitive to the investment horizon (Mandelbrot, 1997); (2) the methods used to price financial derivatives based on martingale models (the most common models, e.g. the Black– Scholes model (Black and Scholes, 1973) are not useful anymore; (3) since the usual tests based on the Capital Asset Pricing Model and Arbitrage Pricing Theory (Black et al., 1972) do not take into account long-range dependence, they cannot be applied to series that present such behavior. We seek for the presence of long-range dependence in both crude oil prices and volatility. Additionally, for comparison purposes, our data includes two different crude oil markets: Brent (London) and West Texas Intermediate (New York). We expect to find the presence of long memory in these crude oil time series, but with intensity that decreases over time – which means that this market is becoming more efficient. The presence of long memory in these price time series can be justified if the shocks that took place in these time series decay at a hyperbolic rate rather than exponential as one would assume for short-term processes. On the other hand, the motivation that make us to believe that the crude oil market is becoming more efficient over time stems from the fact that this market has undergone a major structural change due to policy changes that attempted to increase the efficiency of the North American energy industry [see Serletis and Andreadis (2004) and Serletis and Rangel-Ruiz (2004) for a nice discussion of these changes and also Winston (1993)]. Therefore, the methodology using a time-varying Hurst exponent (Cajueiro and Tabak, 2004a) may show whether 1 From the statistical physics, we know that a given market is said to present the long range dependence phenomena with persistent behavior if the Hurst's exponent H N 0.5, with anti-persistent behavior is if H b 0.5 and random walk behavior if H = 0.5. In usual situations, financial time series presents Hurst's exponent equal or higher than 0.5. 2 The weak form of market efficiency states that under the information contained on the set formed by past returns, future returns are unpredictable (Fama, 1970).
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the crude oil market has become more efficient in the 1990s (deregulated period) if compared to the 1980s (regulated period). The paper is organized along the following lines. Section 2 provides a brief literature review. While Section 3 discusses the methodology employed in the paper, Section 4 describes the data. Section 5 presents empirical results. Finally Section 6 concludes the paper. 2. Brief literature review This paper is not the first one that investigates whether or not the crude oil market is efficient. To understand the purpose of the seminal papers in this subject, one should take the period from 1973 to 1974 to until at least the end of 1985 in to account. In this period, one could see the coexistence of two crude oil markets: (1) one market that represented the biggest part of the crude oil traded on the world and was sold under long term contracts at the so-called OPEC official prices, which were adjusted only infrequently and (2) a smaller, although increasing market, which quantities were traded at market determined spot prices. The coexistence of these two markets raised the issue of whether the official prices behaved like efficient contract prices. Therefore, some works tried to cope with this issue. For details, see Gately et al. (1977), Verleger (1982) and Green and Mork (1991). A more recent literature is concerned with the existence of chaotic and nonlinear structures in oil markets series. If these time series are chaotic or present deterministic non-linear structures, then technical analysis can be useful for prediction purposes and the weak form of efficiency does not play role in these markets. In particular, while Panas and Ninni (2000) test whether the oil markets are chaotic and find evidence (using correlation dimension entropies and Lyapunov exponents) that oil products such as NAPHTA, MOGAS PREM., SULFUR FO 3.5% and FO 1.0%, GASOIL and MOGAS REG. UNL. are chaotic time series, Adrangi et al. (2001) find evidence of non-linear dependencies for crude oil, heating oil and unleaded gasoline futures prices, but do not find evidence of chaos in these time series. Although the literature which seeks for the presence of long-range dependence phenomena in equity returns is very wide and this phenomena is seem as a stylized fact,3 to the best of our knowledge, only Alvarez-Ramirez et al. (2002) and Serletis and Andreadis (2004) have provided empirical evidence of long-range dependence in crude oil prices with persistent structure. In fact, while Alvarez-Ramirez et al. (2002) have studied the Brent crude oil prices, West Texas Intermediate crude oil prices and Dubai crude oil prices, Serletis and Andreadis (2004) have studied the West Texas Intermediate crude oil prices and Henry Hub natural gas prices. On the other hand, in a variety of papers, however, there is considerable evidence suggesting that not only changes in log of prices may possess long-range dependence but that also volatility is persistent. Most studies such as Cajueiro and Tabak (2005a,b) have found that Hurst exponents for volatility returns are above 0.8 in most cases, suggesting strong volatility long memory. However, to the best of our knowledge such a study has not been done on crude oil market. This paper pretends also to fill in this gap. Therefore, this paper will extend the studies of Alvarez-Ramirez et al. (2002) and Serletis and Andreadis (2004) by evaluating the dynamic nature of these time series comparing the Brent and 3 Cajueiro and Tabak (2004a,b) and Matteo et al. (2003) have shown that financial time series usually present the long range dependence phenomena but its degree is associated to the degree of the development of the given financial market. Moreover, Cajueiro and Tabak (2004a,b), employing a moving window with fixed length of 1008 observations, also have shown that the intensity of the long range dependence phenomena has been decreasing over time meaning that these markets are converging toward efficiency.
B.M. Tabak, D.O. Cajueiro / Energy Economics 29 (2007) 28–36
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WTI crude oil markets and also performing such studies to focus on volatility. Again, we are not only interested in the detection of the long-range dependence phenomena, but also how it evolves over time. We intend to capture underlying structural changes in the dynamics of these series, to compare the deregulated and regulated period for crude oil markets and finally to verify if these markets of crude oils are converging toward efficiency. Moreover, knowing that the presence of the short-range dependence may affect our results (see Section 3), we apply a procedure that cleans our results of the influence of this phenomenon. 3. Methodology The methodology that we employ here is based on the estimation of the Hurst exponent H by the R/S analysis (Hurst, 1951) when applied to log return time series. More explicitly, let X(t) be theprice of a commodity on a time t and r(t) be the logarithmic return denoted by rðt Þ ¼ ln xðtþ1Þ xðtÞ .The R/S statistic is the range of partial sums of deviations of times series from its mean, rescaled by its standard deviation. So, consider a sample of continuously compounded asset returns {r1, r2, …, rτ} and let ¯r τ denote the sample mean Ps 1 r where τ is the time span considered. Then the R/S statistic is given by t¼1 t s " # t t X X 1 max ðrk − ¯ ðrk − ¯r s Þ ð1Þ ðR=SÞs u r s Þ− min 1VtVs Ss 1VtVs k¼1 k¼1 where sτ is the usual standard deviation estimator " #1 2 1X 2 Ss u ðrt − r¯s Þ s t
ð2Þ
Hurst found that the rescaled range, R/S, for many records in time is very well described by the following empirical relation: ðR=SÞs ¼ ðs=2ÞH
ð3Þ
The major problem of the R/S method is its sensitive to short-term memory. Therefore, in order to correct for short-range dependence we have applied the R/S analysis to blocks of shuffled data, i.e., one picks a random permutation of the data series within blocks of predetermined size (in general, small size blocks) and applies the R/S analysis to this shuffled data.4 In this work, we use blocks of size 20. Finally, considering the approach proposed in Cajueiro and Tabak (2004a), we estimate the Hurst exponent (using the R/S analysis and the correction for short memory) for subsample windows of 1008 observations (approximately 4 years of data) to test for time-varying degrees of long-range dependence. 4. Data The crude oil prices data were taken from the Bloomberg database. The data is given in US dollar per barrel for the Brent (North Sea–Europe) and West Texas Intermediate (WTI) Cushing (US). The data constitutes of daily closing prices over the period from May 16, 1983 to July 28, 4 This is justified due to the Lo's (1991) critique that the R/S statistics is sensitive to the presence of short range dependence and the effect of random permutations in these small blocks is necessary to destroy any particular structure of autocorrelation within these blocks.
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Fig. 1. Daily Brent crude oil prices.
2004 for WTI (a total of 5276 observations) and from May 16, 1983 to July 29, 2004 for Brent (a total of 5277 observations). Figs. 1 and 2 provide a graphical representation of these series. 5. Empirical results We have found a very interesting pattern regarding the long-range dependence behavior of the crude oil market by studying two major indices, the Brent and the West Texas Intermediate (WTI).
Fig. 2. Daily WTI crude oil prices.
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Fig. 3. Time-varying Hurst exponents for Brent crude oil returns.
Empirical results suggest that the 1980s were much more inefficient than the 1990s, i.e., Hurst exponents calculated for changes in crude oil prices are on average 0.624 and 0.572 for Brent and WTI for the 1980s while these numbers drop to 0.532 and 0.509 on average for the deregulated period (1990s). In Fig. 3 we show the graph of time-varying Hurst exponents evaluated for moving windows of fixed length of 1008 observations (approximately 4 years) at a time. We also plot a 95% confidence interval. The date in the x-axis stands for the beginning of the sample used in the
Fig. 4. Time-varying Hurst exponents for WTI crude oil returns.
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Fig. 5. Time-varying Hurst exponents for Brent crude oil volatility.
estimation of the Hurst exponents. Therefore, for a date May-83 the Hurst exponents were evaluated for the sample beginning in May-83 and ending 4 years later (May-87) and so forth. The change in the dynamics of these Hurst exponents is striking. Hurst exponents tend to fluctuate in the 0.60–0.65 range in the 1980s period, decreasing substantially in the latter period. In Fig. 4 Hurst exponents are in the range 0.55–0.6 in the former period and decrease substantially in the latter period and after 1994 these Hurst exponents seems to be gravitating around 0.5 (random walk without long-range dependence). These results seem to be quite different from the ones presented in Serletis and Andreadis (2004) because our time series were corrected for short-term autocorrelation, which is presented in these returns. Nonetheless, qualitative results are in line with previous findings that there are periods in which crude oil prices seem to possess long-range dependence. An important issue that has not been addressed before is whether volatility for crude oil prices has long memory. We present a similar graph in Fig. 5 for volatility of Brent returns. We use absolute returns as a proxy for volatility following Cajueiro and Tabak (2005,b).5 Although there is some evidence of a small downward trend there seems to be cycles in which persistence changes. However, the main finding is that volatility is highly persistence and volatility models such as GARCH or EGARCH cannot be used to estimate and forecast crude oil prices volatility. Furthermore, this has an important implication for option pricing, as the Black and Scholes model (1973) cannot be used to price crude oil options as it does note take into account such features. A similar result is found for the volatility of WTI (Fig. 6). However, interestingly Hurst exponents are higher for the volatility of WTI crude oil prices if compared to Brent. Furthermore, although there seems to be a downward trend Hurst exponents are very far from 0.5. These latter results on Hurst exponents for volatility of crude oil prices suggest that volatility is a highly persistent process, which has important implications. Shocks to volatility
5
However, qualitative results are similar if we use squared returns.
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Fig. 6. Time-varying Hurst exponents for WTI crude oil volatility.
tend to die rather slowly if compared to what GARCH and E-GARCH models would forecast. 6. Conclusions This paper examines the dynamics properties of crude oil prices using a novel approach for testing for long-range dependence. Our testing procedure consists in “shuffling” returns for testing for long-range dependence in a “rolling sample”. This methodology allows ascertaining whether long memory effects found in previous studies are in place for more recent data and whether the deregulation of the crude oil market that took place in the 1980s has provoked changes in these time series dynamics. Our results indicate that although a fractal structure has been detected for crude oil prices in previous research, when conducting a time-varying study the crude oil market has become more efficient over time. Furthermore, the WTI crude oil prices seems to be more weak form efficient than Brent prices. A novel finding is that volatility for these crude oil prices is highly persistent and GARCH or E-GARCH models for these times series are misspecified. Furthermore, although Hurst exponents for these time series have decreased substantially over time they are still well above 0.5, in the 0.6–0.7 range, which is substantially high. Our results are consistent with the findings of Serletis and Andreadis (2004), which show that crude oil prices possess long-range dependence. However, the degree of long-range has decreased in the recent period for both mean and volatility returns. References Adrangi, B., Arjun, C., Dhanda, K.K., Raffiee, K., 2001. Chaos in oil prices? Evidence from future markets. Energy Economics 23, 405–425.
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