Crude oil market efficiency and modeling: Insights from the multiscaling autocorrelation pattern

Energy Economics 32 (2010) 993–1000

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Energy Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n e c o

Crude oil market efficiency and modeling: Insights from the multiscaling autocorrelation pattern Jose Alvarez-Ramirez a,b,⁎, Jesus Alvarez a, Ricardo Solis b a b

Departamento de Ingenieria de Procesos e Hidraulica, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F., 09340, Mexico Departamento de Economia, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F., 09340, Mexico

a r t i c l e

i n f o

Article history: Received 10 February 2010 Received in revised form 27 April 2010 Accepted 28 April 2010 Available online 6 May 2010 JEL classification: C14 Keywords: Crude oil Autocorrelations Multiscaling pattern Mean reversion

a b s t r a c t Empirical research on market inefficiencies focuses on the detection of autocorrelations in price time series. In the case of crude oil markets, statistical support is claimed for weak efficiency over a wide range of timescales. However, the results are still controversial since theoretical arguments point to deviations from efficiency as prices tend to revert towards an equilibrium path. This paper studies the efficiency of crude oil markets by using lagged detrended fluctuation analysis (DFA) to detect delay effects in price autocorrelations quantified in terms of a multiscaling Hurst exponent (i.e., autocorrelations are dependent of the time scale). Results based on spot price data for the period 1986–2009 indicate important deviations from efficiency associated to lagged autocorrelations, so imposing the random walk for crude oil prices has pronounced costs for forecasting. Evidences in favor of price reversion to a continuously evolving mean underscores the importance of adequately incorporating delay effects and multiscaling behavior in the modeling of crude oil price dynamics. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Fair valuation of projects and securities for the crude oil industry, including exploration, extraction, distribution and chemical transformation, requires accurate stochastic modeling to describe the observed complex dynamics of prices and returns. For instance, the famous BlackScholes option pricing formula assumes that the commodity price follows a geometric Brownian motion (Smith and McCardle, 1998). In this model, prices are expected to grow at some constant drift rate with the variance in future spot prices increasing in proportion to time. If prices increase (resp., decrease) more than anticipated in one time period, all future forecasts are increased (resp., decreased) proportionally. The attractiveness of this modeling approach is that it leads to closed-form solutions that can be easily used in practice. The underlying idea behind the Brownian motion assumption is that, after removing a constant drift, the dynamics of the (logarithmic) price differences can be described as an uncorrelated process standing in for any and all sources of uncertainty in the price history of the commodity. In turn, this concept is linked to the idea that valuation fairness is possible because returns cannot be predicted for any time horizon. In financial theory, this is known as the efficient-market hypothesis (EMH), which asserts that financial markets are informationally ⁎ Corresponding author. Departamento de Ingenieria de Procesos e Hidraulica, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F., 09340, Mexico. Tel./fax: + 52 55 58044650. E-mail address: [email protected] (J. Alvarez-Ramirez). 0140-9883/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.eneco.2010.04.013

efficient in the sense that prices on traded assets already reflect all known information1, and instantly change to reflect new information (Fama, 1970). Therefore, according to theory, it is impossible to consistently outperform the market by using any information that the market already knows, except through luck. In its weak form, the EMH states that future prices cannot be predicted by analyzing price from the past. In this way, excess returns cannot be earned in the long run by using investment strategies based on historical share prices or other historical data. In turn, this implies that prices exhibit no serial dependencies, meaning that there are no patterns to asset prices, and so future price movements are determined entirely by information not contained in the price series. Such assumptions should imply that prices must follow a random walk. In recent years, investors and researchers have disputed the EMH both empirically and theoretically. The normal occurrence of human errors in reasoning and information processing (e.g., overconfidence, overreaction and information bias) has been used as a suitable framework by behavioral economist to explain information imperfections in financial and commodity markets (Kahneman and Tversky, 1979, 2000). On the other hand, empirical analysis has provided mixed results, although evidence hinting to market efficiency is poorly supported (Chen et al., 2003; Charles and Darne, 2009). Speculative bubbles can be considered as anomalies where the market often appears to be driven by buyers operating on irrational exuberance, 1 Information or news in the EMH is defined as anything that may affect prices that is unknowable in the present and thus appears randomly in the future.

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who take little notice of underlying value. These bubbles are typically followed by an overreaction of frantic selling, allowing shrewd investors to buy stocks at bargain prices (Lo and MacKinlay, 2001). The efficiency of crude oil markets is a subtle issue given that the market configuration involves governments, large-scale producers, consumers and investors. In this way, one could expect that the underlying market dynamics exhibit important deviations from the EMH. This paper focuses on the market efficiency issue by exploring the presence of autocorrelations in historic crude oil price dynamics. In contrast to existing results in the open literature, the present work considers delay effects acting in the formation of crude oil prices. The results found with this approach indicate that deviations from efficiency and the type of model to describe return dynamics are dependent of the forecasting horizon. 1.1. Literature review Empirical studies of testing for the efficiency of crude oil markets have been oriented to exhibit the presence of long-term dependencies in historical spot price sequences. Alvarez-Ramirez et al. (2002) implemented multifractal Hurst analysis on crude oil prices, showing that the market was consistent with the random-walk behavior only at scales of the order of days to weeks. Serletis and Andreadis (2004) showed that price dynamics of North American energy markets can be explained from random fractal structures. Tabak and Cajueiro (2007) using the R/S method found evidence that crude oil markets are becoming more and more efficient over time. However, a drawback of the R/S method is its sensitivity to trends and short-term dependency (Lo, 1991), which can lead to biased estimation of long-term correlations. To overcome this problem, Alvarez-Ramirez et al. (2008) used detrended fluctuation analysis (DFA), a method suited for non-stationary series plagued with trends (Peng et al., 1994), to show that the crude oil markets are consistent with the EMH over long horizons, although time-varying autocorrelation can be exhibited for short time scales. Shambora and Rossiter (2007) used artificial neural network models to show that prices can be forecasted, casting doubts on the efficiency of crude oil markets. Wang and Liu (2010) used multiscale DFA to show that short-term, medium-term and longterm behaviors were generally turning into efficient behavior over time. Also, multifractality analysis suggested that small fluctuations of WTI crude oil market are persistent; however, large fluctuations display high instability, both in the short- and long-terms. Summing, up, empirical evidences have indicated that crude oil markets are not uniformly efficient (i.e., over the whole time scale range) and that prices can be forecasted to some extent using empirical models (e.g., ANNs). In this form, traditional valuation approaches based on the geometric Brownian motion assumption can yield biased results. Hence, unbiased valuation of projects and securities for the crude oil industry should consider the market complexity2 and propose alternative forecasting models. Recently, a number of authors motivated by theoretical arguments have considered the use of mean-reverting (Ornstein-Uhlenbeck) (Pindyck, 1999; Smith and McCardle, 1999; Tvedt, 2002) and Poisson-based jump dynamics with structural change (Lee et al., 2010) processes as being more appropriate for crude oil market modeling. A detailed understanding of historical price dynamics should be combined with theoretical arguments to develop accurate stochastic models for forecasting of crude oil prices and valuation of financial and investment instruments (Pindyck, 1999). Although recent results have shed some light on, e.g., the existence of time-varying autocorrelations and the role of fractal integrating processes, the problem deserves more detailed analysis with improved methods. In 2 Complexity of crude oil markets emerges as a reflect of the interaction of numerous participants (producers, governments, consumers, etc.) and the effect of exogenous (economical, climatic, political, etc.) events (Hamilton, 2008).

principle, results obtained in this line should improve modeling considerations and enhance theoretical arguments. This work uses a lagged DFA (Alvarez-Ramirez et al., 2009) to characterize the multiscale pattern of crude oil prices. The motivation for using the lagged version of the DFA is that, similar to traditional autocorrelation analysis, maximal long-term correlations can be found for non-zero lags, reflecting delayed effects in the underlying stochastic process. Similar to the approach by Wang and Liu (2010), a scale-dependent Hurst exponent (multiscale pattern) is computed to explore changes of the autocorrelation strength induced by shocks persisting in different time scales. In contrast to previously reported results, the analysis show that the price dynamics are hardly consistent with the EMH, exhibiting important departures from a geometric Brownian motion assumption. Interestingly, the multiscaling pattern is not continuous, but shows two jumps at one-quarter and one-year time scales, which can be related to the intrinsic cycles of financial markets. The discussion is extended by incorporating arguments from the crude market structure (Hamilton, 2008) for explaining the autocorrelation pattern. The paper is organized as follows. The data description is made in Section 2. The methodology, based on lagged DFA, is provided in Section 3. The empirical analysis is made in Section 4. The results discussion is given in Section 5. Conclusions are shown in Section 6. 2. Data We study daily closing price data of the West Texas Intermediate (WTI) with the date from January 1st. 1986 to December 31, 2009. The data are obtained from Energy Information Administration (EIO) in the U.S. Department of Energy (http://tonto.eia.doe.gov). The

Fig. 1. (a) WTI price dynamics for the period ranging from 1996 to 2009. Some salient events that affected the market evolution are indicated. The current financial and economical recession induced the largest price variation in the recent three decades. Exponential price growth, indicated by a dotted line, can be observed for the period from 2000 to 2008, which is probably related to the high energy demand by China's fast economical growth. (b) Logarithmic price differences showing bursting periods associated to high price volatility.

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graphical representation of the price dynamics is illustrated in Fig. 1a. For subsequent reference, some salient events that affected the market evolution are indicated. The current financial and economical recession induced the largest price variation in the recent three decades. Exponential price growth, indicated by a dotted line, can be observed for the period from 2000 to 2008, which is probably related to the high energy demand by China's fast economical growth. Let Pt denote the crude oil price on day t. The daily price return, rt, is calculated as its logarithmic difference, rt = log(Pt) − log(Pt − 1)∙. The returns are exhibited in Fig. 1b, showing complex behavior with random components. 3. Methodology

predictability of a time series. In fact, the Hurst exponent can be seen as an index of the relative tendency of a time series to either regress to a longer term mean value or cluster in a direction. In this way, the time series {xt} is uncorrelated if H(θ) = 0.5, meaning that the next move cannot be predicted. In contrast, the time series is positively autocorrelated (i.e., persistent)4 if H(θ) N 0.5, and negatively autocorrelated (i.e., anti-persistent)5 if H(θ) b 0.5. Positive and negative deviations of H(θ) from 0.5 imply deviations from a purerandom process. In this way, minimum randomness is found for θmax, where   θmax = arg maxθ HðθÞ−0:5

j

The DFA is a method designed to estimate long-range correlations for nonstationary sequences (Peng et al., 1994)3. The DFA can be modified to account for lagged autocorrelation effects (AlvarezRamirez et al., 2009). For completeness in the presentation, a brief description of the method is given as follows. Suppose that xt is a series of length N, and that this series is of compact support. The support is defined as the set of the indices t with nonzero values xt. The value of xt = 0 is interpreted as having no value at this time t. • Step 1. Determine the profile

It is possible that the fluctuation function F(τ, θ) is not described as a power-law of the form Eq. (3). In fact, the dynamics of complex systems is characterized by oscillatory components interacting on many time scales. Systems of this type are referred as multiscaling. In such case, one can maintain the power-law framework by computing a scale-dependent (i.e., local) Hurst exponent H(τ, θ) as the logarithmic derivative of F(τ, θ) at the time scale τ: Hðτ; θÞ =

t =1

For convenience, define the profile Zi = Yi + θ, i = 1, …, N − θ. Note that Zi is lagged θ times from Yi. • Step 2. Divide the profiles Yi and Zi into Nτ = int[(N − θ)/τ] nonoverlapping segments of equal length τ. • Step 3. Calculate the local trends Ỹ(v − 1)τ + i and Z̃(v − 1)τ + i for each of the Nτ segments by a least-square fit of the series. Then determine the lagged variance 1 τ ∑ τ i=1

jY

˜

ðv−1Þτ + i −Y ðv−1Þτ + i

jjZ

˜

ðv−1Þτ + i −Z ðv−1Þτ + i

j

ð1Þ The trends Ỹ(v − 1)τ + i and Z̃(v − 1)τ + i can be computed from a linear, quadratic or higher order polynomial fit of the profile for each segment. Note that by taking Nτ = int[(N − θ)/τ] nonoverlapping segments in Step 2, the values of Z(v − 1)τ + i are well-defined for all i 2 [1, τ] and all θ ≥ 0. • Step 4. Average over all segments to obtain the fluctuation function ( Fðτ; θÞ =

1 Nτ 2 ∑ V ðτ; θÞ Nτ v = 1 v

)1 = 2 ð2Þ

• Step 5. Determine the scaling behavior of the fluctuation functions by analyzing the log-log plots F(τ; θ) versus the scale τ. If the series xt are long-range power-law correlated, F(τ; θ) increases, for large values of τ, as a power-law Fðτ; θÞ∼τ

HðθÞ

ð4Þ

3.2. Multiscaling

i

Yi = ∑ ½xt −hxi; i = 1; …; N

2

j

and Hmax = H(θmax) can be seen as an index of maximal autocorrelations.

3.1. Lagged DFA

Vv ðτ; θÞ =

995

ð3Þ

where H(θ) [0, 1] will be referred as θ -lagged scaling or Hurst exponent. The lagged scaling exponent is defined from predictability considerations since the Hurst exponent is an index of the 3 The reader is referred to the Physionet Internet site http://physionet.net for a detailed description of the method and a list of publications showing the use of DFA to characterize correlations.

dlogðFðτ; θÞÞ dlogðτÞ

ð5Þ

The local Hurst exponent H(τ, θ) reflects the scaling properties of the signal xt in a neighborhood of the time scale τ. Given the sampled and noisy nature of F(τ, θ), the derivative in Eq. (5) cannot be computed directly. Instead, αβ-filters (Kingsley and Quegan, 1997) are used, which provide accurate approximation and fast computation of Hloc(τ, θ). Similarly, minimum randomness is obtained for the scaledependent lag θmax ðτÞ = arg fmaxθ Hðτ; θÞ−0:5g corresponding to the maximum Hurst exponent Hmax(τ) = H(τ, θmax). 4. Empirical results Fig. 2a shows that the zero-lag fluctuation function F(τ, 0) cannot be described as a sole power-law of the form given by Eq. (3)6. This means that the dynamics of the logarithmic price differences are multiscaling. The asymptotic Hurst exponent values indicate that the price differences are weakly persistent for small time scales, of the order of days, which is in line with previous results (Alvarez-Ramirez et al., 2008). On the contrary, the dynamics are antipersistent for interannual time scales, suggesting that the long-term dynamics of crude oil prices are mean-reverting7. More detailed results are found when computing the multiscaling pattern H(τ) with αβ-filtering as shown in Fig. 2b. Negative autocorrelations are observed for onemonth scales, indicating that the differences for one-month averaged prices are also antipersistent and so can be modeled as a meanreversion process. In contrast, uncorrelated behavior is observed for one-year scales, suggesting that yearly averaged prices cannot be predicted. Maximal persistence is found for time scales of the order of 4 If the last move was up, the next one most likely will be up, and vice versa. This behaviour is sometimes called mean-reversion which means future values will have a tendency to return to a longer term mean value. The strength of this mean reversion increases as H approaches 0. 5 If the last move was up, the next one most likely will be down. That is the time series is trending. The larger the H value is, the stronger the trend. 6 The software used for computations was author developed by the authors. A Fortran version is available under request. 7 Mean-reversion means that future values will have a tendency to return to a longterm mean value.

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Fig. 2. (a) Zero-lag DFA results showing that the fluctuation function cannot be described as a unique power law for the whole time-scale range. (b) Local Hurst exponent computed with αβ-filters. One-month and one-year scales are indicated to stress the multiscaling nature of crude oil prices.

Fig. 3. Maximum Hurst exponent for (a) 25 and (b) 150, showing that maximum correlations can be found at non-zero lags.

6–8 months, with weak positive autocorrelations. For these scales, price differences can be modeled as a fractional Brownian motion with H ≈ 0.6. The crude oil price differences tend to significant antipersistence (H → 0.25) for interannual scales, meaning that a positive difference is more likely to be followed by a negative difference and vice versa. These results show that the crude oil market dynamics are quite complex, with many mechanisms interacting on many time scales. 4.1. Delayed autocorrelations Lag effects are contained in the price dynamics if the Hurst exponent is a non-trivial function of the lag θ. Fig. 3a and b shows the Hurst exponent as a function of the lag θ for the time scales τ = 25 and τ = 150 trading days. In the former case, the Hurst exponent variations are within the antipersistence domain with an apparent slow convergence to uncorrelated behavior (H = 0.5) for large lags. In this case, the maximal distance from H = 0.5 is found for zero lag, indicating that the price differences are not affected by delay effects for small time scales. In contrast, the Hurst exponent fluctuations are contained within the persistence domain for τ = 150. In this case, maximal autocorrelations, with Hmax ≈ 0.77, are found for θmax = 5 trading days. Fig. 4a presents the multiscaling pattern Hmax(τ) for the logarithmic price differences. The corresponding maximal lag θmax is shown in Fig. 4b. It is worth noting that maximal autocorrelations (i.e., zero lags) are exhibited for scales smaller than one quarter. In this way, the price dynamics are not affected by delay effects for time scales smaller than one quarter. An interesting result is the presence of two discontinuities at one-quarter and one-year time scales, which are induced by the effects of non-zero lags affecting the autocorrelation structure. In fact, θmax goes from zero to positive values at τ = 1

Fig. 4. (a) Multiscaling pattern and (b) corresponding delays exhibiting discontinuities at one-quarter and one-year scales. In the short- and long-term the behavior is related to negative autocorrelations, which is consistent with mean-reversion dynamics.

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quarter. The second discontinuity is related to a sudden change of θmax from moderate (about 20 trading days) to high lag values (about 100 trading days). The discontinuity at one quarter corresponds to a transition from antipersistence to persistence, and the discontinuity at one year reflects a transition from persistence to antipersistence. This suggests that price differences can be described as a fractional Brownian motion, with H ≈ 0.7, for time scales between one and four quarters. Data sampling frequency can have an important effect in the scaling pattern as sampling acts as a low-pass filter that removes highfrequency effects. The stability of the multiscaling pattern shown in Fig. 4a was tested by considering weekly and monthly data. The results presented in Fig. 5 show that the one-quarter discontinuity is removed for both sampling frequencies, suggesting that this discontinuity is the effect of intraweek dynamics. In contrast, the one-year discontinuity is retained, hinting to long-term persistence of seasonal effects. 4.2. Temporal changes of the scaling pattern To test for temporal changes of the scaling pattern, the multiscaling computations were carried for four six-year subperiods. Fig. 6 exhibits the zero-lag fluctuation function for two subperiods, showing differences in the scaling pattern. For instance, F(τ) is convex for 1986–1991 and concave for 1992–1997, which is reflected in the local Hurst exponent values. Fig. 7 shows the scaling pattern for four subperiods, showing that the position of the discontinuities changes with time. For instance, the first discontinuity is moved left to 0.2 years for the subperiod 1996–1991. In contrast, persistence is not displayed for the subperiod 1992–1997. In this case, the Hurst exponent approaches uncorrelated behavior (H = 0.5) for interquarter scales. The 1998–2003 period shows only a small range (from 0.14 to 0.22 years) of scales with persistent behavior. Afterwards, the Hurst exponent shows a fast decrement to negatively autocorrelated dynamics. 5. Discussion 5.1. Market structure Fig. 4 showed that the scaling properties of crude oil returns can be divided into three time-scale regimes divided by two discontinuities at one-quarter and one-year scales; namely, (a) short-times scales ranging from days to weeks where autocorrelations are negative (mean-reversion), (b) medium-term or intraquarter scales charac-

Fig. 5. Multiscaling pattern for weekly and monthly sampled data. The one-quarter discontinuity is removed by the effects of higher-frequency sampling. This suggests that the short-term mean-reversion dynamics are induced by the response of the system to short-term shocks.

Fig. 6. Fluctuation function for the 1986–1991 and 1992–1997 subperiods. Important differences in the shape of F(τ) are noted, which have implications in the multiscaling pattern.

terized by positive autocorrelations, and (c) long-term scales, including interannual dynamics, for which dynamics are affected by strong negative autocorrelations. Short-term mean-reversion dynamics are related to damping of, e.g., daily, and weekly shocks, with stochastic convergence to an equilibrium price trend imposed by medium- and long-term supply-demand mechanisms (Wang and Liu, 2010). On the other hand, it is apparent that interquarter positive autocorrelations are induced by futures markets that pull prices toward a more correlated (contract maturing) price. Finally, longterm dynamics are linked to shocks that are dissipated within yearly scales. These shocks arise from economical (e.g., financial crisis), technological (e.g., drilling methods for deep sea conditions) and social (e.g., Iraq War) events. The stronger mean-reversion degree (represented by smaller Hurst exponent) for long-term scales as compared with that for short-term scales can be explained from the elasticity of energy markets (Hamilton, 2008). Elasticity of demand

Fig. 7. Multiscaling pattern for four subperiods. Differences in the discontinuities position are easily observed.

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and supply in the short term is very low8 while the elasticity in the long term is much larger. Once a shock on supply (or demand) occurred, the change of demand (or supply) would be very limited due to technological, production, transportation and consumption limitations. The results are consistent with the view that long-term oil prices are determined globally (Mohammadi, 2009). The consequence is that the oil price would keep steadily upward or downward momentum (Wang and Liu, 2010). In the long-term the market is more elastic because producers and consumers dispose of sufficient time to adjust their expectations to existing supply and demand dynamics. In this way, the crude oil prices are adjusted in the upward or downward direction to reflect the economical, geopolitical and production conditions. Within a general framework, our results indicate that oil price dynamics may indeed emerge due to the nonlinear interplay between different participant types. 5.2. Is the crude oil market weakly efficient? Empirical results on the efficiency of crude oil markets are controversial. Some studies indicated that the market is becoming efficient over time (Tabak and Cajueiro, 2007), efficient in the longterm but with short-term autocorrelations (Alvarez-Ramirez et al., 2008) and efficient over limited periods for small and large fluctuations (Wang and Liu, 2010). Our results provide further doubts on the efficiency of crude oil market. By considering the effects of delayed dynamics, the results in Fig. 4 exhibit important departures from uncorrelated dynamics, suggesting the existence of autocorrelations that can be exploited for arbitrage. Except for a period within the 1990s decade, price differences cannot be explained from the uncorrelated process framework. In fact, the crude oil market was weakly efficient during the 1990s decade (Fig. 7) when the Hurst exponent approached 0.5 for medium-term (intrannual) scales. Interestingly, the 1990s decade witnessed a stable worldwide economical growth with the emergence of new economical regions having an increasing role in the output contribution. It is apparent that this growth was mounted on an efficient crude oil energy market with stable price level and supply volume. However, it seems that the fast growth of the Asian region starting in the late 1990s disturbed this pattern, leading to increasing energy demand accompanied by an apparent decay of crude oil long-term reserves (Hamilton, 2008). The crude oil market in the recent years, including those of the recent recession, is characterized by significant positive autocorrelations (Hmax ≈ 1) for interquarter dynamics, implying important deviation from efficiency. In this way, the existence of exploitable inefficiencies is apparent in the recent years (Charles and Darne, 2009), induced maybe by China's oil demand moving prices toward an upper equilibrium price (Leung, 2010). The results in Fig. 7 deserves a further analysis. It was shown that the crude oil market was efficient during the period 1992–1997. It is apparent that the market efficiency was related to a stable growth of the global economy with an increased energy demand that was equilibrated by producers and supply mechanisms. However, the emergence of several economies (e.g., China, India, etc.) as important crude oil demanders disrupted the market equilibrium introduced strong price swings induced by the uncertainty of crude oil supply. As new technologies (drilling in even deeper seawaters) were incorporated and existing crude oil fields were over-exploited (e.g., Mexican Cantarell field), the crude oil market achieved a fragile equilibrium although affected by strong uncertainties on future supply mechanisms (Hamilton, 2008). As a consequence, crude oil prices increased up to historic values. Interestingly, the strong correlations for one quarter to one year time scales during the period 2004–2009 are still found for the recent 2.5 years, suggesting that such surge of positive 8 Constraints in, e.g., the adjustment of crude oil production and transportation limit the supply elasticity.

correlations was caused by the global recession. In fact, the current global recession imposed a downsizing of energy consumption which, in turn, induced a drastic reduction of spot and futures crude oil prices (from about 160 to about 60 $/bbl). 5.3. Modeling implications Taken collectively, the empirical results indicate that crude oil markets exhibit complex dynamics, with stochastic (relative to fluctuation scaling) price properties depending on time-scale, sampling frequency and time. The scaling pattern in Fig. 4 showed that the fractal scaling properties of logarithmic price differences is a function of the time-scale. This empirical result provides some insights in modeling for forecasting purposes. Indeed, the main indication is that the structure of a given model for crude oil price dynamics depends on the time-horizon considered for forecasting. Some modeling guidelines, drawn from the multiscaling pattern, can be summarized as follows: a) Short-term forecasting. For time-scales smaller than one-quarter, the price differences exhibited negatively autocorreleted dynamics with Hmax ≈ 0.4. This suggests that mean-reversion models (Pindyck, 1999) can be used for modeling price dynamics intended for short time horizons, from days to weeks. The Hurst exponent decreases to achieve a minimum value (Hmax ≈ 0.35), indicating that the mean-reversion rate increases with the time-scale and that the dynamics are dominated by dissipative effects linked to the stability of an equilibrium price. For times-scales higher than about one month the Hurst exponent increases, suggesting a weakness of the mean-reversion strength induced by longer timescale effects (probably caused by equilibrium price drift). If πt = log (Pt). a mean-reversion model structure for interday dynamics can be described as follows: dπt = κðϕπ −πt Þdt + σπ dzπ

ð6Þ

where ϕπ represents an equilibrium price trend (Pindyck, 1999). The mean-reversion coefficient κ represents the rate of speed at which the price πt reverts to its equilibrium πeq (i.e., the rate at which short-run deviations are dissipated). Here, σπ is the shortrun volatility of logarithmic differences and the shocks zt are independently and identically distributed as normal. It is apparent from Fig. 6 that the mean-reversion behavior for short time-scales has been present in the crude oil market in the recent three decades. The equilibrium path ϕπ can be deterministic or stochastic. For instance, Pindyck (1999) proposed ϕπ = ao + a1t, where ao and a1 satisfy geometric Brownian motions. b) Medium-term forecasting. Forecasting for time horizons of the order of weeks and months should consider the one-quarter discontinuity observed in the multiscaling pattern (Fig. 4). In the medium-term, the dynamics of the price differences are dominated by positively autocorrelated mechanisms, which achieve their larger effects for time-scales of the order of 6-8 months (Hmax ≈ 0.7). Significant delay effects are also observed, with delays of the order of 10-15 trading days (i.e., two to three calendar weeks). By following Schwartz and Smith (2000), the mid-term positively autocorrelated dynamics can be attributed to stochastic drift of the equilibrium price. While ϕt in Eq. (6) is a fixed equilibrium drift line that can suffice for short time-scales (Pindyck, 1999), it cannot retain the effects of persistent, dependent, shocks in the evolution of crude oil markets. It can be assumed that the construction of the crude oil price is given by the contribution of a stochastic medium-term trend and shortterm deviations representing the effects of autonomous random shocks. Let πt = χt + ξt, where χt is referred to as the short-term deviation in prices and ξt the equilibrium price level. The short-run

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deviations χt are governed by negatively autocorrelated dynamics (Fig. 4) and so revert toward zero, i.e., dχt = −κχt dt + σχ dzχ

ð7Þ

Schwartz and Smith (2000) modeled the equilibrium price level ξt as a Brownian motion plus a price drift given by dξt = ϕξ dt + σξ dzξ

ð8Þ

where dzξ and dzχ are correlated increments with dzξdzχ = ρξχdt. The Brownian motion behavior hold only if H = 0.5. However, results in Fig. 4 shows that the medium-term dynamics are not uncorrelated, but present delayed (θmax ≈ 10–15 trading days or 2–5 calendar weeks) positive autocorrelations (Hmax ≈ 0.7). On the other hand, Fig. 5 shows that medium-term autocorrelation structure is retained for weekly averaged prices. The above suggests a multiscale model structure where the equilibrium level is updated with weekly averaged prices to avoid including delay effects. Also, dzχ should satisfy a fractional process with Hmax ≈ 0.7. In this way, the parameters of Eq. (7) shall be estimated from daily data and the parameters for Eq. (8) from weekly averaged data. More sophisticated models incorporating constant drift reduction and convenience yield9 dynamics (Pindyck, 1999; Schwartz and Smith, 2000; Bernard et al., 2008 and references therein) should consider the effects commented above; namely, 2–3 week delays and positive autocorrelations for medium-term (interquarter) scales. c) Long-term forecasting. Several crude oil participants, including governments, energy producers and large consumers, regularly perform interannual price forecasting for general purposes of strategic planning and for specific investment decisions related to resource exploration, reserve development and production. To this end, weekly and monthly averaged data are commonly used. This is in line with the results in Fig. 5 showing that models using weekly and monthly averaged data for interannual forecasting should consider medium-term (for scales smaller than one year) positive autocorrelations and long-term negative autocorrelations (mean-reversion). It is worth noting that the mean-reversion strength for interannual scales (Hmax ≈ 0.2) is stronger than for interday and interweek scales (Hmax ≈ 0.4). The model structure given by Eqs. (7) and (8) for medium-term forecasting is not compatible with this multiscale autocorrelation pattern. An alternative is to use mean-reversion models with medium-term positively autocorrelated fluctuations where the dynamics of χt are given by Eq. (7) and the price level is given by ξt = ψt + ζξ, where ψt is a trend and dζξ = σζ dzζ

ð9Þ

where, according to Fig. 4, dzζ is a correlated increment with H ≈ 0.6. Here, we used linear models to discuss the potential implications of the multiscaling pattern for price modeling. However, it has been indicated that crude oil price dynamics are multifractal (AlvarezRamirez et al., 2002), suggesting the presence of nonlinear price adjustment mechanisms. In this way, more accurate crude oil price modeling can also consider nonlinear mean-reversion models, similar to the ones used in real exchange rates (Taylor et al., 2001). Nontraditional models based on neural networks (Yu et al., 2008) and soft 9 The convenience yield (Pindyck, 2001) can be defined as the flow of goods and services that accrue to the owner of a spot commodity (a physical inventory) but not to the owner of a futures contract (a contract for future delivery).

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computing (Ghaffari and Zare, 2009) should retain the main multiscale features of crude oil market dynamics. 6. Conclusions In this paper, we used detrended fluctuation analysis (DFA) with lagged autocorrelations to analyze the scaling properties of daily crude oil prices. We focused on the estimation of the multiscaling pattern determined by the variations of the Hurst exponent with respect to time-scales. The results and their discussion suggested the following conclusions: (a) The multiscaling pattern is not continuous, showing two discontinuities at one-quarter and one-year scales. These discontinuities indicate different sources of price fluctuations, from speculative effects to fundamental supply and demand shocks. (b) The crude oil market present important deviations from efficiency. In contrast to previous results that suggested efficient behavior for large time-scales, our results indicate positive or negative autocorrelations that might be masked by delay effects. (c) The application range of a forecasting model is limited by the time horizon. In turn, the multiscaling pattern within the considered time horizon provide important insights in the model structure and sampling (i.e., daily, weekly or monthly) frequency. (d) Negative autocorrelations over a wide time-scale range indicate that mean-reversion with continuously evolving mean is a suitable modeling framework for forecasting purposes. Within this view, our results indicate that changes in the real price of crude oil have historically tended to be permanent, difficult to predict and governed by very different regimes at different periods of time (Hamilton, 2008). Reaching beyond the multiscaling pattern issue and the implications for the validity of the EMH, our results illustrate the importance of relying on time series analysis for establishing model structures, formulating investment time horizons (quarter, years or longer) and defining sampling frequencies. In principle, results in this line should be combined theoretical considerations for the construction of accurate models incorporating both empirical evidence and economical fundamentals. For instance, future approaches should incorporate the recent perception by market participants of the importance of reserves uncertainty and exhaustion of production fields. References Alvarez-Ramirez, J., Cisneros, M., Ibarra-Valdez, C., Soriano, A., 2002. Multifractal Hurst analysis of crude oil prices. Physica A 313, 651–670. Alvarez-Ramirez, J., Alvarez, J., Rodriguez, E., 2008. Short-term predictability of crude oil markets: a detrended fluctuation analysis approach. Energy Economics 30, 2645–2656. Alvarez-Ramirez, J., Rodriguez, E., Echeverria, J.C., 2009. Using detrended fluctuation analysis for lagged correlation analysis of nonstationary signals. Physical Review E 79, 057202. Bernard, J., Khalaf, L., Kichian, M., McMahon, S., 2008. Oil Prices: Heavy Tails, Mean Reversion and the Convenience Yield, Quebec Ministry of Finance. Chen, K.C., Gup, B.E., Pan, M.-S.h., 2003. International stock market efficiency and integration: a study of eighteen nations. Journal of Business Finance & Accounting 24, 803–813. Charles, A., Darne, O., 2009. The efficiency of the crude oil markets: evidence from variance ratio tests. Energy Policy 37, 4267–4271. Fama, E., 1970. Efficient capital markets: a review of theory and empirical work. Journal of Finance 25, 383–417. Ghaffari, A., Zare, S., 2009. A novel algorithm for prediction of crude oil price variation based on soft computing. Energy Economics 31, 531–536. Hamilton, J.D., 2008. Understanding Crude Oil Prices: NBER Working Paper, No. w14492. Kahneman, D., Tversky, A., 1979. Prospect theory: an analysis of decisions under risk. Econometrica 47, 313–327. Kahneman, D., Tversky, A. (Eds.), 2000. Choices, values and frames. Cambridge University Press, New York. Kingsley, S., Quegan, S., 1997. Understanding Radar Systems. McGraw-Hill, New York. Lee, Y.H., Hu, H.N., Chiou, J.S., 2010. Jump dynamics with structural breaks for crude oil prices. Energy Economics 32, 343–350. Leung, G.C.K., 2010. China's oil use, 1990–2008. Energy Policy 38, 932–944. Lo, A.W., 1991. Long-term memory in stock market prices. Econometrica 59, 1279–1313. Lo, A., MacKinlay, C., 2001. A Non-random Walk Down Wall Street. Princeton Press, Princeton, NJ.

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