Efficiency of crude oil markets: Evidences from informational entropy analysis

Energy Policy 41 (2012) 365–373

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Efficiency of crude oil markets: Evidences from informational entropy analysis Alejandro Ortiz-Cruz, Eduardo Rodriguez, Carlos Ibarra-Valdez, Jose Alvarez-Ramirez n ´noma Metropolitana-Iztapalapa, Apartado Postal 55-534, Me ´xico D.F., 09340, Me´xico Universidad Auto

a r t i c l e i n f o

abstract

Article history: Received 15 April 2011 Accepted 28 October 2011 Available online 17 November 2011

The role of crude oil as the main energy source for the global economic activity has motivated the discussion about the dynamics and causes of crude oil price changes. An accurate understanding of the issue should provide important guidelines for the design of optimal policies and government budget planning. Using daily data for WTI over the period January 1986–March 2011, we analyze the evolution of the informational complexity and efficiency for the crude oil market through multiscale entropy analysis. The results indicated that the crude oil market is informationally efficient over the scrutinized period except for two periods that correspond to the early 1990s and late 2000s US recessions. Overall, the results showed that deregulation has improved the operation of the market in the sense of making returns less predictable. On the other hand, there is some evidence that the probability of having a severe US economic recession increases as the informational efficiency decreases, which indicates that returns from crude oil markets are less uncertain during economic downturns. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Crude oil market Market efficiency Entropy

1. Introduction Energy and investment policy planning for many economic agents (e.g., firms and governments) requires an accurate knowledge of the complex mechanisms involved in price formation in crude oil markets and their relationship with economic dynamics and extreme (e.g., socio-political and meteorological) events. The possible relationships between crude oil prices and economic activity have been widely explored, suggesting that crude oil market dynamics have a direct effect on the economic cycle (Rasche and Tatom, 1977; Hamilton, 1983; Santini, 1985; Gisser and Goodwin, 1986; Rotemberg and Woodford, 1996; Carruth et al., 1998; Hamilton, 2003; Barsky and Kilian, 2004; Oladosu, 2009). Coincidences between high crude oil price increments and the outbreak of economic recessions have been documented (Hamilton, 1983; Mork, 1989), which suggest that crude oil price dynamics can be used as an indicative of the global economic activity. Other studies point towards that point towards permanent effects of crude oil prices on inflation and short-run but ˜ ado and asymmetric effects on production growth rates (Cun Perez de Gracia, 2003; Oladosu, 2009). Some studies have challenged the notion that movements of crude oil prices have significant effects on macroeconomic activity (Barsky and Kilian, 2004; Kilian, 2009). It has been also argued that the adverse

n

Corresponding author. Tel./fax: þ 52 55 58044600. E-mail address: [email protected] (J. Alvarez-Ramirez).

0301-4215/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enpol.2011.10.057

effects of positive shocks can be explained from the resulting tightening of monetary policy (Bernanke et al., 1997) and that crude oil prices are significantly influenced by fluctuations in the Kilian economic index through both long-run equilibrium conditions and short-run impacts (He et al., 2010). Summing up, research results have indicated that crude oil markets have an important impact on the performance of regional and global economies and that this influence has a complex and multifactorial nature. At the heart of the performance of crude oil markets is the concept of informational efficiency since oil price movements substantially affect, at different degrees and through different channels, the performance of most economic sectors and industries (Lescaroux and Mignon, 2008). The importance of the efficient market hypothesis (EMH) relies on the fact that in an efficient market all available and relevant information are fully and instantaneously reflected on the price of a market security so that no one can take advantage of this information (Fama, 1970, 1991). In this way, there are neither undervalued nor overvalued assets in an efficient market, and market price of financial assets constitutes a proper guide for capital budgeting and allocation. Arouri et al. (2010) have pointed out that market efficiency is desirable to asset pricing models and investor investment decision-making process; meanwhile it rests on strong assumptions such as, frictionless markets, information availability and transparency, investor rationality and arbitrage. Results regarding the fulfillment of the EMH should provide important insights in the dynamics of crude oil markets and their impact on worldwide and

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regional economies. In the following section, a literature review of the some salient contributions to the empirical analysis of crude oil market informational efficiency is provided. Subsequently, the main contribution of our work relative to existing results is described. The remainder of the paper is organized as follows. Section 2 provides a review of the main results reported in the scientific literature. Section 3 describes the empirical entropy method used to characterize the dynamics of crude oil prices and to explore the possibility of sudden changes in market efficiency. Section 4 presents the data. Section 5 discusses the results obtained. Section 6 concludes the paper and suggests some policy implications arising from the results.

2. Literature review Depending on the nature and source of the information, three forms of market efficiency can be established: strong form, semistrong form, and weak-form efficiency (Fama, 1970, 1991). The set of available information can be both public and private information for the strong-form efficiency. It is limited to all public information for the semi-strong form and to all past price movements for the weak form. Most empirical tests for the EMH focus on the weak form. In this way, market efficiency is related to the absence of arbitrage conditions since the crude market behavior cannot be predicted using the dynamics of past price returns. Green and Mork (1991) used the generalized method of moments to show that the weak-form efficiency does not hold for the whole sample period 1978–1985. However, some efficiency improvements over time were observed for sub-sample periods. Alvarez-Ramirez et al. (2002) used multifractal Hurst analysis to find that the market was consistent with the random-walk behavior only at scales of the order of days to weeks. By using detrended fluctuation analysis of logarithmic price differences, Serletis and Andreadis (2004) showed that the price behavior of North American energy markets can be explained from random fractional Brownian motion dynamics, so that prices has long-run dependences. Tabak and Cajueiro (2007) used R/S analysis to find evidences that crude oil markets are becoming more and more efficient over time, which suggests that market deregulation introduced positive effects in market operation. Recently, Alvarez-Ramirez et al. (2008) used methods based on detrended fluctuation analysis to show that crude oil markets are consistent with the EMH over long horizons, although timevarying autocorrelation can be exhibited for short time-scale. Maslyuk and Smyth (2008) used unit-root test to find that weekly crude oil price over the period 1991–2004 can be characterized as a random walk process with two structural breaks. Charles and Darne´ (2009) used variance ratio tests to show that Brent crude oil market is weak-form efficient while WTI crude oil market seems to be inefficient on the 1994–2008 subperiod. In contrast to Tabak and Cajueiro’s research report, this result suggests that the deregulation has not improved the efficiency on the WTI crude oil market in the sense of making returns less predictable. By means of multiscale fluctuation analysis, Wang and Liu (2010) suggested that short-term, medium-term and long-term behaviors were generally turning into efficient behavior over time. Lean et al. (2010) found no evidence of mean-variance and stochastic dominance oil spot and future prices, inferring that there is no arbitrage opportunity between these two markets and spot and futures oil markets are efficient and rational. AlvarezRamirez et al. (2010) studied lagged fractal autocorrelations of spot WTI prices to find that autocorrelations can be masked by delay effects. Arouri et al. (2010) used state space models to find evidence of short-term predictability in crude oil prices over time,

so the hypothesis of convergence towards weak-form informational efficiency should be rejected. Some approaches have focused on the predictability of crude oil prices. Shambora and Rossiter (2007) used artificial neural network models with moving average crossover inputs to analyze crude oil future markets and showed that prices can be forecasted in the long-run, casting doubts on the efficiency of crude oil markets. Results from nonlinear approaches have suggested that the crude oil market is not efficient as the price dynamics can be predicted to some extent by using nonlinear models (Wang and Yang, 2010), genetic algorithms (Fan et al., 2008) and wavelet decomposition (Jammazi and Aloui, in press). Overall, the empirical findings reported in the literature are still controversial since the results have been oriented to respond whether or not the crude oil price contains patterns that can be exploited to outperform the market. By recognizing the complexity of the problem, new results should be oriented to show not only that the market is efficient or not, but rather to provide a quantitative index of the crude oil market informational efficiency as well as to evaluate temporal changes that can be related to socioeconomic events. Next, concepts from informational entropy will be discussed to motivate their use for quantifying informational market efficiency in crude oil markets. Although entropy is a powerful concept to characterize the diversity of patterns contained in a time series, its use to analyze financial time series has been constrained to a limited number of research studies. It is apparent that Gulko (1999) firstly proposed the use of entropy concepts to study financial time series by showing that the maximum-entropy formalism, also called informational efficiency, makes the efficient market hypothesis operational and testable. This formalism is used to establish that entropic markets admit no arbitrage and support both the Ross arbitrage pricing theory and the Black–Scholes stock option pricing model. Darbellay and Wuertz (2000) demonstrated the usefulness of entropy concepts to characterize financial time series by showing that the salient advantage of the entropy approach resides in its ability to account for nonlinear dependences in the autocorrelation structure of the underlying system dynamics (Kaffashi et al., 2008; Hassan et al., 2011). Pincus and Kalman (2004) suggested that approximate entropy algorithm is suitable for analyzing financial time series as it can be applied to very short sequences and can be used as a marker of market stability. Recently, entropy concepts have been used to quantify market efficiency in foreign exchange and stock markets. Since entropy is an index of the quantity of information (measured in terms of pattern richness) contained in a time series, high entropy can be related to low predictability of the market dynamics and, hence, to high market efficiency. In contrast to previous approaches focusing on an all-or-nothing response to the efficiency question, entropy can provide a relative degree of the efficiency of a given market. Oh et al. (2007) used the global foreign exchange market indices in order to study the efficiency of various foreign exchange markets around the market crisis. It was found that the markets with a larger liquidity (e.g., European and North American) have higher market efficiency than those with a smaller liquidity. Zunino et al. (2009) showed that degree of stock market inefficiency is negatively correlated with the permutation entropy. Risso (2008, 2009) used Shannon entropy concepts on symbolic dynamics of stock indices to show that the probability of having a crash increases as entropy decreases. The aim of this work is to study the informational efficiency of crude oil markets by means of entropy analysis methods. The departing idea is that prices in efficient markets cannot be predicted because of the lack of intrinsic correlations and regular patterns. That is, the returns of a price trajectory for a market finding the weak-form of the EMH should correspond to uncorrelated stochastic

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noise. In turn, the returns should display maximum entropy content. Relative to existing results in the literature dealing with informational efficiency of crude oil markets (see, e.g., Serletis and Andreadis, 2004; Tabak and Cajueiro, 2007; Maslyuk and Smyth, 2008; AlvarezRamirez et al., 2010; Arouri et al., 2010), our contribution based on entropy analysis can be summarized as follows:

 An index of informational market efficiency is introduced in





terms of an entropy distance to randomness. Previous work has focused on studying informational market efficiency as an all-or-nothing question. In our approach, the informational efficiency index provides a mean for quantifying a distance from the weak form of informational efficiency. In line with the seminal concepts of adaptive markets hypothesis (Lo, 2004), the empirical analysis showed that the informational market efficiency for crude oil markets exhibits important temporal changes that depend on the time horizon. In turn, this implies that crude oil market participants (e.g., investors, governments, producers, etc.) adapt to endogenous and exogenous changing conditions, e.g., adjusting public budget or maximizing profit. A comovement between the crude oil market efficiency and US economic recessions is shown. This suggests that the efficiency of crude oil markets can be used as a proxy for the evolution of economic recessions.

An algorithm for entropy computation for finite data can be described as follows. A time series of length N sampled at time intervals Ts fX i g ¼ fx1 ,x2 ,:::,xN g

where xi ¼ xðt 0 þiT s Þ

ð1Þ

is considered. It is noted that the length N can be related to a time scale t ¼NTs. Two m-dimensional sequence vectors u(m)(i)¼ {xi,xi þ 1,y,xi þ m  1} and v(m)(j)¼{xj,xj þ 1,y,xj þ m  1}, iaj, 1ri, jrN  mþ1, are selected. These vectors u(m)(i) and v(m)(j) are called similar if their distance du,v ði,jÞ ¼ maxfuði þ kÞvðj þ kÞ9 : 0 rk r m1g

ð2Þ

is smaller than a specified tolerance e. For each of the N m þ1 vectors u(m)(i), the number of similar vectors v(m)(j) is given by measuring their respective distances. If nðmÞ is the number of i vectors v(m)(j) similar to u(m)(i), the relative frequency to find a vector v(m)(j), which is similar to u(m)(i) within a tolerance level e is given by C i ðm, e, tÞ ¼

nðmÞ i Nm

ð3Þ

where N m is the number of vectors vðmÞ ðjÞ auðmÞ ðiÞ that are potentially similar to u(m)(i). Next, one looks at the relative frequency of the logarithm of Ci(m,e,t), i.e.,

Fðm, e, tÞ ¼ It should be clear that the superiority of the entropy analysis over other analysis techniques (e.g., R/S scaling and detrended fluctuation analysis) and nonlinear modeling approaches (e.g., mean-reversion with jumps and ANN) is not claimed. Given that the crude oil market exhibits complex dynamics, different timeseries and model approaches provide different insights into the behavior of the market. In the present work, we are using entropy analysis because this technique (i) performs better than, e.g., Hurst analysis, for small length time series and (ii) allows the introduction of an index of market efficiency as a measure from randomness and to detect sharp changes in efficiency without specifying an evolution model. To the best of our knowledge, in the spirit of previous proposals (Lo, 2004; Zunino et al., 2009), this is the first attempt to quantify the degree of market efficiency rather than just addressing an all-or-nothing question.

367

Nm Xþ 1 1 ln C i ðm, e, tÞ Nm þ 1 i ¼ 1

ð4Þ

For finite N, the approximate entropy is estimated by the statistics AEðm, e, tÞ ¼

1 ½Fðm, e, tÞFðm þ 1, e, tÞ Ts

ð5Þ

In this way, lower values of AE(m,e,t) reflect more regular time series, while higher values are associated with less predictable (more complex) time series within the time scale t. The effects of the tolerance e, data length N and vector dimension m in the performance of the AE computation (Pincus, 1991). Stable statistics were found for N4500. On the other hand, the parameters m¼2 and e ¼0.15s, where s is standard deviation of the time series, are commonly used in applications. 3.2. Multiscale approximate entropy

3. Empirical analysis method Entropy is a basic concept used to quantify disorder and uncertainty of dynamic systems. Macroscopically, entropy can be related to the number and diversity of patterns and variations that a system can display for a very large set of trajectories. The Shannon and Kolmogorov–Sinai entropy concepts provide a way to characterize the gain of information by giving an index of disorder and uncertainty content.

In general, the complexity of real system is not constrained to a single scale. In this way, one can expect that entropy is scaledependent, meaning that a signal is more uncertain for certain time-scales and more irregular for others. The idea is that a multiscale entropy approach should provide an index that reflects the mean rate of creation of information at a given time-scale. As a result, the overall degree of predictability and uncertainty of a signal is assessed by considering the entropy values estimated for a pre-defined range of time-scales. A multiscale AE method involves two steps:

3.1. Approximate entropy A direct application of entropy concepts requires the availability of infinite data series with infinitely accurate precision and resolution. This is not possible in practice since measurements from real systems are sampled with limited resolution e and finite sampling rate 1/Ts. To alleviate this situation, approximate entropy statistics have been introduced (Pincus, 1991) to quantify regularity of time series of finite length. The approximate entropy (AE) computations are conceptually simple and are based on the likelihood that time series templates that are similar remain similar on next incremental comparisons. Hence, time series with large AE should have high uncertain fluctuations.

(i) A method allowing looking at representations of the system dynamics at different time scales. For a given the time-series X¼ {x1,x2,y,xN}, and a low-pass filter procedure LP(f) with cut-off frequency f, obtain the filtered time series as Y f ¼ LPðf ÞX,

where Y f ¼ fyf ,1 ,yf ,2 ,. . .,yf ,N g

ð6Þ

In this way, the new time-series Yf retains the complexity of the signal X for frequencies smaller than f, or time-scales higher than t ¼ 1/f. Different low-pass filtering operations LP(f) are available in commercial packages. In this work, as used in technical analysis by market practitioners to obtain

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long-term trending of financial signals, low-pass filtering is performed with moving-average filters. In this way, the filtered signal is obtained as yf ,i ¼

n 1X x n j ¼ 1 i þ j1

ð7Þ

The time scale is given by t ¼ Dtn and f¼1/t. Here, Dt is the sampling period. (ii) The quantification of the degree of irregularity of each time series Yf is accomplished using approximate entropy procedure described above. 4. Data and basic statistics As in the vast majority of recent empirical studies dealing with the weak form efficiency of crude oil markets, the data for empirical analysis consist of the daily closing spot prices over the market deregulation period starting in January, 1986. Although Brent and West Texas Intermediate (WTI) crude oils have different US dollar/barrel prices, some studies (e.g., Fattouh, 2010) have shown only slight differences in price dynamics. In this way, for convenience we focused only on the WTI price, which can be freely extracted from the Energy Information Administration (EIA) in the US Department of Energy (http:// tonto.eia.doe.gov). The data span from January 1st, 1986 to March 15, 2011 (6520 observations). Fig. 1a and b provides a graphical representation of the price and log differences time-series, respectively. The occurrence of some extreme events is pointed

160

Recession

Price, pt (USA $/barrel)

140 120 100 80 60

Iraq War Gulf War

Asian Financial Crisis

9/11

40 20 0 1986 1989 1992 1995 1998 2001 2004 2007 2010 0.15

log (Pt/Pt-1)

0.10

out for reference in result discussion. In the decade from 1998 to 2008 the oil price increased more than 700% in average. The spot price logarithmic differences are used for empirical analysis in this work. The mean and standard deviations of these differences are 0.024% and 2.32% daily, respectively. Skewness and kurtosis significant at 5% levels are –1.040 and 25.64, respectively, indicating that the daily log returns are highly non-normal. The Lagrange multiplier test indicated that ARCH(10)¼206.55, significant at 5% level, showing that the WTI prices display strong conditional heteroscedasticity.

5. Results and discussion The application of the multiscale procedure yields an entropy index that depends on time-scale, AE(t). This is illustrated by considering the logarithmic differences for the period from 1986 to 1989. Fig. 2 shows the signal obtained after low-pass filtering for n¼ 1 (daily, original signal), n ¼5 (weekly), n ¼20 (monthly) and n¼ 60 (quarterly) time scales. It is noted that the signals were normalized by their standard deviation. The resulting multiscale entropy behavior is displayed in Fig. 3. It is observed that the entropy exhibits a decreasing trend since the information content is reduced as the averaging length n increases. However, a local maximum is located at about 72 business days, suggesting that the market information is aggregated about quarterly time scales for the period 1986–1989. The multiscale entropy methodology is used in this work to study some stylized facts of the complexity and efficiency of crude oil markets in the recent 25 years. The method is based on monitoring the entropy variations with respect to scale and time in a sliding window of size Ns. In this way, as a prerequisite to perform entropy computations, a suitable sliding window size Ns should be selected. The entropy is an increasing function of the window size Ns since at least the existing patterns are not destroyed as Ns increases. To evaluate the effects of the window size, the entropy was computed over all windows of size Ns contained in the whole time series after low-pass filtered for a time-scale t ¼n business days. After averaging over all possible windows contained in a time series Yf, Fig. 4 presents the behavior of the entropy as a function of the window size Ns for daily, weekly, monthly and quarterly time-scales. For these selected time-scales, the entropy exhibits a well-defined decreasing order with respect to the time-scale t only for Ns 4500. It is noted that, for instance, for Ns o500, the average weekly entropy is higher than the average daily entropy. We selected Ns ¼600 business days (about 2.5 years) for entropy computations, which guarantees that the average entropy is decreasing with respect to t. A higher value of Ns could be chosen, although it can reduce the locality of the entropy fluctuations with respect to time.

0.05 5.1. Informational market complexity

0.00 Fig. 5 presents the entropy behavior with respect to time-scale

-0.05 -0.10 -0.15 1986 1989 1992 1995 1998 2001 2004 2007 2010 Time, t (years)

Fig. 1. (a) WTI price for the period January 1st, 1986 to March 15th, 2011. Some relevant events are marked, which presumably had an important impact in the dynamics of the market. (b) Logarithmic price differences for the same period than in panel (a).

t, for five selected three-year subperiods. For scales up to one quarter, the entropy is a decreasing function of the time-scale, which indicates a reduced information content as the time horizon is increased. In some cases, as in the 1986–1989 and 1993–1996 subperiods, prominent local maximums are observed for scales higher than one quarter. For instance, the subperiod 1993–1996 exhibits a local maximum at 200 business days, which suggests that the long-run price formation was largely affected by information flows aggregated at yearly time-scales. In contrast, many subperiods exhibit decreasing entropy over the whole timescale range. Gulko (1999) has pointed out that the greater the

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3

3 n=5

Filtered Signal, Yf /σ

n=1 2

2

1

1

0

0

-1

-1

-2

-2

-3 1986

1987

1988

1989

3

-3 1986

n = 20

Filtered Signal, Yf /σ

1987

1988

1989

1987 1988 Calendar Year

1989

3 n = 60

2

2

1

1

0

0

-1

-1

-2

-2

-3 1986

369

1987 1988 Calendar Year

1989

-3 1986

Fig. 2. Low-pass filtered sequences for logarithmic price difference for (a) daily (n¼ 1), (b) weekly (n¼ 5), (c) monthly (n¼ 20) and (c) quarterly (n ¼60) scales.

number of price patterns (maximum entropy), the harder the prediction of the price ongoing dynamics. Consequently, markets with low entropy levels are relatively easy to predict, in contrast to markets with high entropy for which forecasting should require more complex methods and algorithms.1 In this sense, markets with higher entropy levels are more complex than markets with low entropy levels. According with this notion, the results in Fig. 5 indicate the following features for the complexity, in terms of entropy content, of the crude oil market:

 The complexity of crude oil markets is scale-dependent. Fig. 5 showed that entropy of the logarithmic price differences non-trivially depends on the time-scale t. The general pattern is that entropy is higher for small time-scales while showing decrements for high time-scales. This behavior suggests that the crude oil price predictability is relatively higher in the long-run as the entropy displays relatively small values for high time-scales. Note that the entropy also evolves over time.

1 It should be recalled that entropy, as described in Section 3, is computed from combinatorial algorithms. In this way, entropy is an index of the factorial combination (NP computational complexity) of possible patterns.



For instance, the 2003–2006 subperiod exhibits higher entropy levels over the whole time-scale range than the other four subperiods. One can note that the crude oil market exhibited increased price pattern diversity during the subperiod 2003–2006, which was caused by many factors that included the robust growth of the US economy following the 2001 crisis, the sustained crude oil demand increase by Asian Pacific countries, the use of crude oil as an asset for financial speculation, etc. In contrast, the 2007–2010 subperiod corresponding to the current economic recession witnessed a severe entropy decrement over the whole time-scale range, which can correspond an important reduction of the market efficiency as the price dynamics become more predictable. The scale-dependent entropy pattern suggests the following crude oil market structure. In the short-run (from days to weeks) operation, the market exhibits the higher complexity (measured in terms of pattern diversity) as highest entropy values indicate poorly predictable price movements. The short-run market diversity of price dynamics arise from the combined effects of speculation (non-commercial positions), inventory level shocks (Merino and Ortiz, 2005), meteorological events (e.g., Katrina hurricane), etc. In contrast, in the longrun, for time-scales higher than one quarter, the market is

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1.4

1.6

1.4 1.2

Entropy, AE (τ)

Entropy, AE (τ)

1.2

1.0

0.8

1.0

0.8 1986-1989 1993-1996

0.6

1999-2002 0.6

2004-2007 0.4

0.4

0.2 1

10 100 Time-scale, τ (business days)

1

Fig. 3. Multiscale entropy pattern AE(t) for the subperiod 1986–1989 according to the low-pass filtered sequences in Fig. 2.

1.4 1.3 1.2 1.1 1.0 0.9

Two years

0.8 0.7 0.6

Daily Weekly Monthly Quaterly

0.5 0.4 200

400 600 Window Size, NW

800

10 Time-scale, τ (business days)

100

Fig. 5. Entropy behavior with respect to the time-scale t, for five selected threeyear subperiods.

oil market by promoting efficient resource allocation as high entropy values reduce the ability of speculators to exploit arbitrage conditions. On the other hand, long-run predictability should allow governments and companies planning production and budget in terms of a less volatile price projection as long-run oil price shocks can impact negatively the evolution of domestic economies (see, for instance, Farzanegan and Markwardt (2009) and Kasparian (2009)). In line with previous results derived from delayed autocorrelations (Alvarez-Ramirez et al., 2010), the results in Fig. 5 suggest that the short-run market dynamics are highly uncertain, induced by short-term (mean-reversion) dynamics related to 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). Regarding the entropy for high timescales, it is apparent that the decreased values could be induced by future markets that pull prices toward contract maturing prices within quarterly time horizons.

1.5

Entropy

2007-2010

1000

Fig. 4. Behavior of the entropy as a function of the window size Ns for daily, weekly, monthly and quarterly time-scales.

relatively more predictable as the long-term price dynamics depends of more structural factors, such as macroeconomic and political environments, sovereign production planning (i.e., number and capacity of Saudi and Venezuelan crude oil fields), long-term dollar volatility, etc. In this way, short-term high efficiency should contribute to the stability of the crude

The dependence of the crude oil market complexity with time and scale suggests that market participants adapt the information processing to specific time horizons to accomplish profit and energy planning tasks. This is in-line with the evolutionary approach (Lo, 2004), which states that the market complexity is a characteristic that varies continuously over time and across markets. 5.2. Informational market efficiency Some authors have related the content of informational entropy to market efficiency (Oh et al., 2007; Risso, 2008, 2009). The rationale behind the entropy-weak market efficiency relationship comes from the fact that price movements cannot be predicted by processing price movements by, e.g., technical analysis. Empirical tests for showing whether a market is efficient or not have focused on rejecting the hypothesis that logarithmic prices differences

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behave as a random walk process (i.e., logarithmic price differences reflect uncorrelated stochastic dynamics). In turn, this hypothesis implies that prices are unpredictable since no regular patterns can be found in market evolution. Campbell et al. (1997) argued that this approach is an idealization that is unattainable since market is operated by heterogeneous agents having different views and capacities for processing information (Giglio et al., 2008). Instead, the concept of relative market efficiency can be a more useful concept than the approach taken by the traditional literature. In other words, an index quantifying efficiency of markets should be relative, providing a measure of the level of predictability of a market dynamics. An alternative for testing the EMH can be given from an entropy standpoint if one considers that the difficulty of forecasting a market is directly related with the number and diversity of price movement (entropy content). To construct the benchmark for estimating a degree for market efficiency, the multiscale entropy pattern for samples of uncorrelated Gaussian noise sequences of size Ns ¼600 observations was considered. Fig. 6a shows the boundaries of benchmark entropy patterns for 10,000 samples. The upper and lower boundaries represent, respectively, the maximum and minimum entropy values for the whole set of sequences. In this way, the multiscale entropy pattern for any uncorrelated Gaussian noise sequence of size Ns ¼600 observations should be contained in the envelope shown in Fig. 6a. Note the fractal-like geometry of the envelope boundaries, reflecting the fact that even uncorrelated noise is a fractional dynamic system. The idea underlying a definition for informational market efficiency is that a real sequence of size Ns should contain maximal pattern diversity if its multiscale entropy pattern lies within the envelope in Fig. 6a. That is, the crude oil market will be fully (i.e., 100%) informationally efficient if the sequence of logarithmic differences is contained within the envelope.

Entropy, AE (τ)

1.4 1.2 1.0 0.8 0.6 0.4 1

10

371

In contrast, if the entropy pattern is below the lower boundary, the crude oil market will be only partially efficient. In this case, an index of informational market efficiency IIME(t) for a given time-scale t will be given IIME ðtÞ ¼

AEðtÞ  100 Bmin ðtÞ

ð8Þ

where AE(t) is the sequence entropy and Bmin ðtÞ is the lower boundary of the benchmark envelope in Fig. 6a. In this way, IIME(t)¼100% if AEðtÞ ZBmin ðtÞ. In contrast, IIME(t)o100% if the entropy pattern is not contained within the entropy benchmark. By introducing the definition for IIME(t), we avoid the all-or-nothing approach for market efficiency. In fact, the definition for IIME(t) allows more flexibility by providing a measure of the market efficiency. Fig. 6b shows the multiscale entropy pattern AE(t) for 100 subsamples of crude oil price logarithmic differences. It is noted that the entropy pattern is not necessarily contained into the benchmark region, so the market efficiency depends on time scale t. That is, the crude oil market can be fully efficient for certain time scales, and partially efficient for others. Also, the market efficiency depends also on time t. Fig. 7 shows the index of informational market efficiency IIME(t) with respect to time t and scale t. Reduced market efficiency (i.e., less than 100%) can be observed for the late 1980s when the crude oil market was adapting to deregulation conditions. Two other important periods of decreased market efficiency are observed for two periods in the early 1990s and the late 2000s. However, in general, the crude oil market has been informationally efficient along the recent two decades, which is in agreement with results obtained with Hurst analysis, which indicated that the crude oil market is consistent with the efficient market hypothesis (Alvarez-Ramirez et al., 2008). Similar to the conclusion drawn by Tabak and Cajueiro (2007), this result suggests that the market deregulation by 1986 improved the efficiency on the crude oil market in the sense of making returns less predictable. The above results confirm that the crude oil market is an adaptive evolutive system where efficiency is not constant but exhibits important time variations. As has been pointed out by Lim and Brooks (2011), the characteristics of the market microstructure, limits to arbitrage, psychological biases, noise trading and the existence of market imperfections are those potential factors that can be risen to periods of departure from market efficiency. Besides, the results confirm previous findings from unit root test showing that oil price dynamics can be characterized as a random walk process and that departs from efficiency are

100

1.4

Entropy, AE (τ)

1.2 1.0 0.8 0.6 0.4 1

10 Time-scale, τ (b-days)

100

Fig. 6. (a) Boundaries of benchmark entropy patterns for 10,000 samples of Gaussian uncorrelated noise sequences of length Ns ¼600 observations. (b) Entropy pattern compared to the benchmark for randomness.

Fig. 7. Index of informational market efficiency IIME(t) with respect to time t and scale t.

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significant and meaningful in terms of huge events that have impacted on world economy (Maslyuk and Smyth, 2008). 5.3. Crude oil market efficiency during US economic recessions Risso (2008) found evidences that the probability of having a financial crash in stock markets increases as the informational efficiency decreases. Fig. 7 showed that the crude oil market has been informationally efficient except for two periods corresponding to two large US economic recessions. In fact, important decrements in the informational market efficiency IIME(t) can be observed for the 1990:Q1–1993:Q2 and 2008:Q1–present subperiods, which correspond to the two most severe US economic crisis in the recent 25 years. The early 1990s recession lasted from July 1990 to March 1991 and had a  1.4% GDP decline (peak to trough). The recent recession (named the Great Recession) started in December 2007 and NBER officially declared it ended by June 2009. In this case, the GDP deterioration was more pronounced with 4.1% decline. It seems that these two recessions have had an important effect on crude oil market efficiency, which was reflected by severe entropy decrements over practically the whole time-scale range. During recession periods, the market dynamics are dominated by more regular price patterns as the diversity of participants and expectations is reduced by the effects of economic shortcomings. A brief economic downward was recognized in the early 2000s, which was explained by the collapse of the speculative dot-com bubble, a fall in business outlays and investments, and the September 11th attacks. The GDP decline was short, 0.3%, and effective monetary policies allowed a brief and shallow turmoil. Fig. 7 shows that this recession did not induce a significant change in the entropy dynamics of the crude oil market. In turn, this suggests that, since the recession did not introduce a major effect on energy markets, the origin of the early 2000s recession was not structural but induced by exogenous shocks that were absorbed accordingly. The results in Fig. 7 are in agreement with recent findings (Risso, 2008) that show that for different stock markets, the probability of having a crash increases as the informational efficiency decreases. In fact, severe US economic recessions have induced important efficiency decrements at different time-scales. In turn, these efficiency decrements could be related to decrements in the price entropy as the market undertakes preferential price directionality due to, e.g., reduction in demand and investment expectations. For instance, in the recent 2008 economic downturn, the crude oil suffered a sharp decrement from about 160 $/barrel to about 60 $/barrel induced mainly by a sink of the economic growth expectations. The crude oil price dynamics were governed more by negative adjustments in consumption and profits rather than by the action of many and different (e.g., noisy) market investors. As a consequence, the randomness content, and hence the diversity of patterns, in the crude oil price dynamics reduced in a large amount. If the crude oil market informational efficiency, measured in terms of entropy, is seen as a proxy for US recession durations, Fig. 6 indicates that a full recovery of the early 1990s downturn was achieved by 1993:Q2 (i.e., three years after the recession started) when the crude oil market efficiency was fully recovered and the economic growth was re-established.

6. Conclusions This work used multiscale entropy analysis to study the complexity and efficiency of crude oil markets in the recent 25 years. The idea behind the application of multiscale entropy methods is that the higher the entropy, the wider the diversity

of price fluctuations patterns. In turn, this implies that a market with high entropy values is more complex than those with smaller entropy values. In this way, the entropy method implemented on a rolling window scheme indicated: (a) The results are in-line with previous findings (Alvarez-Ramirez et al., 2008; Elder and Serletis, 2008; Arouri et al., 2010) in that we show evidence of time-varying market efficiency for daily crude oil returns over the deregulation period starting in 1986. (b) The multiscale entropy framework allowed the introduction of a scale-dependent index to quantify the degree of market efficiency. Except for two periods corresponding to the early 1990s and late 2000s US economic recessions, the efficiency index is 100% since the multiscale entropy pattern displays a behavior similar to that of uncorrelated noise. Given the scale-dependent complexity and efficiency of crude oil market, it would be in the interest of different market participants, from investors that search for profit opportunities within active investment strategies, to governments that should adjust public budget according to particular country conditions. For instance, in the aftermath of a recession triggering, the crude oil price becomes more predictable since firms will postpone long-term investment as they look for a certain directionality of the recession recovery (Bernanke, 1983). After a recession period, when economic activity is oriented to increase energy consumption, investors are prompted to look for long-term investment strategies. In turn, crude oil market complexity increases and price movements become more uncertain, which is reflected as increments of long-run entropy values. For global investors who allocate portions of their portfolios to crude oil products, the results imply that short- and mid-term benefits can be obtained from an active investment strategy during recessions and exogenous shocks when the market exhibits a high degree of predictability (low degree of price return patterns), while taking into account trading and transaction costs.

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