Correlation structure and principal components in global crude oil market
Yue-Hua Dai, Wen-Jie Xie, Zhi-Qiang Jiang, George J. Jiang, Wei-Xing Zhou
CCorrelation structure and principal components in global crude oil market
Yue-Hua Dai a , Wen-Jie Xie a,b , Zhi-Qiang Jiang a , George J. Jiang c , Wei-Xing Zhou a,b, ∗ a School of Business, East China University of Science and Technology, Shanghai 200237, China b Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China c Department of Finance and Management Science, Washington State University, Pullman, U.S.A.
Abstract
This article investigates the correlation structure of the global crude oil market using the daily returns of 71 oil pricetime series across the world from 1992 to 2012. We identify from the correlation matrix six clusters of time seriesexhibiting evident geographical traits, which supports Weiner (1991)’s regionalization hypothesis of the global oilmarket. We find that intra-cluster pairs of time series are highly correlated while inter-cluster pairs have relatively lowcorrelations. Principal component analysis shows that most eigenvalues of the correlation matrix locate outside theprediction of the random matrix theory and these deviating eigenvalues and their corresponding eigenvectors containrich economic information. Specifically, the largest eigenvalue reflects a collective e ff ect of the global market, otherfour largest eigenvalues possess a partitioning function to distinguish the six clusters, and the smallest eigenvalueshighlight the pairs of time series with the largest correlation coe ffi cients. We construct an index of the global oilmarket based on the eigenfortfolio of the largest eigenvalue, which evolves similarly as the average price time seriesand has better performance than the benchmark 1 / N portfolio under the buy-and-hold strategy. JEL classification:
G1, C15
Keywords:
Crude oil, Principal component analysis, Correlation structure, Regionalization, Geographicalinformation, Eigenvalue
1. Introduction
Crude oil is the life blood of our modern industrial society and a unique strategic resource that is of crucialimportance to any economy. The prices of crude oil are driven by the supply / demand imbalance and the uncertaintyof this imbalance which causes increased speculations (Alvarez-Ramirez et al., 2002; He et al., 2009; Kaufmann andUllman, 2009; Sornette et al., 2009). There is huge literature devoting to the study of the dynamics of crude oil pricesand their mutual relationships. A considerable portion of the literature focuses on the co-movement and convergenceof oil prices at di ff erent regions. Adelman (1984) asserts that the global market of crude oils is unified as “one greatpool”. Conversely, Weiner (1991) argues that the crude oil markets are regionalized which challenges the “one greatpool” hypothesis of Adelman (1984). These two competing hypotheses have stimulated many debates and extensivestudies (Rodriguez and Williams, 1993; Weiner, 1993; Rodriguez and Williams, 1994).The majority of empirical studies support the “one great pool” hypothesis. In this line, di ff erent econometric meth-ods have been adopted and di ff erent “definitions” of market unification have been implicitly assumed. Sauer (1994)incorporates cointegration relationships into multivariate time series models to examine the extent of regionalization inthe global market for crude oil imports and finds that the empirical results support Adelman (1984)’s “one great pool”assertion. G¨ulen (1997, 1999) performs cointegration tests for co-movement of monthly and weekly prices on threegroups of crude oil of similar quality and finds that the world crude oil market is unified over the 1980-95 period,rejecting the regionalization hypothesis. Bentzen (2007) finds bidirectional causality among these major crude oil ∗ Corresponding author. Address: 130 Meilong Road, P.O. Box 114, School of Business, East China University of Science and Technology,Shanghai 200237, China, Phone: +
86 21 64253634.
Email addresses: [email protected] (George J. Jiang), [email protected] (Wei-Xing Zhou)
Preprint submitted to Elsevier May 21, 2014 a r X i v : . [ q -f i n . S T ] M a y rices (OPEC, Brent and WTI) and argues that the regionalization hypothesis of the global oil market is thus rejected.Fattouh (2010) investigates the dynamics of crude oil price di ff erentials using the two-regime threshold autoregressive(TAR) method of Caner and Hansen (2001) and finds strong evidence of threshold e ff ects in the adjustment process tothe long-run equilibrium. Since the crude oil prices are linked, Fattouh (2010) argues that the oil market is “one greatpool” at the very general level. Reboredo (2011) examines the dependence structure between crude oil benchmarkprices using copulas and finds evidence of significant symmetric upper and lower tail dependence between crude oilprices. He states that crude oil prices are linked with the same intensity during bull and bear markets, thus favoringthe “one great pool” hypothesis over the regionalization hypothesis.Kaufmann and Ullman (2009) argue there is no room for innovations in world oil prices to enter the market if theworld oil market is unified and there would be no causal relationships between prices for di ff erent crude oils. Theypoint out that changes may first appear in the price of one or more benchmark crude oils and subsequently spreadthrough the global market. They find evidence of Granger causality from benchmark markers to other crude oil mar-kets. Akhmedjonov and Lau (2012) study the monthly energy prices of four energy products for 83 Russian regionsusing the Exponential Smooth Auto-Regressive Augmented Dickey-Fuller unit root test and find no evidence of afully integrated national energy market in Russia. Liu et al. (2013) examine the regionalization issue by investigatingthe integration between China’s and four major crude oil markets with a threshold error correction model and findonly unidirectional volatility spillover running from benchmark markets to China’s oil market. Their results do notfavor the “one great pool” hypothesis.Our work contributes to this literature by uncovering the correlation structure of the global crude oil market withprincipal component analysis (Jolli ff e, 2002) and random matrix theory (Mehta, 2006). The principal componentanalysis has been widely applied in finance (see, for example, Kritzman et al., 2011; Billio et al., 2012, and refer-ences therein). However, principal component analysis is less adopted in the studies of energy markets. Chantziaraand Skiadopoulos (2008) perform principal component analysis on the prices of crude oil, heating oil and gasolineon the New York Mercantile Exchange and crude oil futures on the International Petroleum Exchange and show thatretained principal components have limited power in predicting the prices.On the other hand, random matrix theory has been applied extensively in studying multiple financial time series(Laloux et al., 1999; Plerou et al., 1999; Kwapien and Drozdz, 2012). Random matrix theory is in essence equivalentto principal component analysis because both of them deal with the correlation matrix and its eigenvalues. Under theframework of random matrix theory, if the eigenvalues of the real time series di ff er from the prediction of randommatrix theory, there must exist hidden economic information in those deviating eigenvalues. For stock markets, thereare several deviating eigenvalues in which the largest eigenvalue reflects a collective e ff ect of the whole market andother largest eigenvalues can be used to identify industrial sectors (Plerou et al., 2002) or clusters of stocks with strongcross-correlations (Shen and Zheng, 2009). Di ff erent information can be extracted for housing markets Meng et al.(2014) and global stock markets Song et al. (2011). Such kind of analysis enables us to uncover market driving forces(Shapira et al., 2009) as well as mutual and common influence (Garas and Argyrakis, 2007).We apply principal component analysis and random matrix theory to investigate the correlation structure of theglobal crude oil market by using 71 spot price time series from di ff erent countries. We are able to identify six clustersof oil price time series that have clear geographic traits. This finding supports the regionalization hypothesis of Weiner(1991). We also extract rich economic information from the deviating eigenvalues, suggesting that the global crudeoil market has a very distinct correlation structure. This rest of this paper is organized as follows. Section 2 describesthe data sets and presents the summary statistics. Section 3 investigates the cross-correlation structure of the globalcrude oil market. Section 4 studies the correlation matrix and explores the economic information contents embeddedin the principal components and the smallest eigenvalues. We conclude in Sec. 5.
2. Data description
We retrieved from the Bloomberg database 71 daily spot price time series of crude oil in various markets all overthe world. The spot price time series cover a period from October 1992 to December 2012. These crude oil price timeseries di ff er in several aspects. The oil markets may locate in di ff erent countries or regions including the main crudeoil export countries such as Iran and Saudi Arab or di ff erent places in a country. Particularly, some oil price series2
992 1996 2000 2004 2008 20120100200 t (cid:2) P (cid:3) , I 〈 P 〉 I t (cid:2) r (cid:3) Figure 1: (Color online) Evolution of average daily prices, the constructed index based on the eigenportfolio of the largest eigenvalue, and averagelogarithmic returns of crude oils. are recorded according to their di ff erent crude oil qualities (including density and sulfur content). For the originaldata, we intersect them and then complement price of the missing data same as that of the previous day. If data ofthe previous day is still missing, we complement them with that of the day before previous day, the rest is done in thesame manner. The labels of di ff erent crude oils, their corresponding ticker names, and the length of time series arepresented in Table 1. The logarithmic return of the i th crude oil price series over a time scale ∆ t is calculated as follows: r i ( t ) = ln P i ( t ) − ln P i ( t − ∆ t ) , (1)where P i ( t ) denotes the price of i th crude oil at time t . Note that the label i for each time series is given in the firstcolumn of Table 1. In this work, we present the results for daily returns with ∆ t = ∆ t = > (cid:104) P ( t ) (cid:105) = (cid:80) i = P i ( t ) and the corresponding averagelogarithmic return (cid:104) r ( t ) (cid:105) = (cid:80) i = r i ( t ). The most significant pattern in the price evolution is the boom and bust of ahuge bubble around 2008, mainly caused by speculations due to market uncertainty (Sornette et al., 2009). The timeseries of the average returns evidently exhibits the volatility clustering phenomenon. We also find that the distributionof the average returns is leptokurtic with the excess kurtosis being 10.02 and left skewed with the skewness being-0.091. 3 . Cross correlation structure We calculate the pairwise cross-correlation coe ffi cients between any two crude oil return time series. For simplic-ity, the original returns for each crude oil time series are standardized as follows: g i ( t ) = r i ( t ) − (cid:104) r i ( t ) (cid:105) σ i , (2)where (cid:104)·(cid:105) denotes the time average of a given time series and σ i = (cid:112) (cid:104) r i ( t ) (cid:105) − (cid:104) r i ( t ) (cid:105) is the standard deviation of r i ( t ).The cross-correlation coe ffi cients c i j are computed as follows: c i j = (cid:104) g i ( t ) g j ( t ) (cid:105) . (3)By definition, c i j ranges from -1 to 1, where c i j = c i j = − c i j = r i ( t ) and r j ( t ).Fig. 2(a) shows the resulting correlation matrix. The correlation structure of the global crude oil market exhibitsintriguing features. There are dense blocks of very high cross-correlations with c i j close to 1. This feature stems fromthe geographic closeness of the time series, such as the block around 60 for the Middle East. We also find that crudeoil prices in Asia and Pacific region correlate to those in the Middle East, but have very low correlations with theAmerican and European markets. The main reason is that crude oil in Asian and Pacific regions is imported from theMiddle East areas with prices being determined by their o ffi cials.During the sample period, the average correlation coe ffi cient (cid:104) c (cid:105) = .
57 is much higher than most stock markets(Plerou et al., 2002; Pan and Sinha, 2007; Shen and Zheng, 2009), which indicates that the crude oil markets are morevulnerable to risks than stock markets.
In order to extract the clusters of time series in an objective way, we adopt modern algorithms for communitydetection in complex networks. For the correlation matrix C , following the idea of Lancichinetti and Fortunato (2012)and Meng et al. (2014), we combine the box clustering algorithm of Sales-Pardo et al. (2007) and the consensusclustering method of Lancichinetti and Fortunato (2012) to search for clusters of crude oil time series.We first determine the optimal ordering of C by identifying the largest elements in C close to the backwarddiagonal i = − j , where the simulated annealing approach is adopted to minimize the cost function Q = (cid:88) i , j = C i j | i − j | . (4)We then use a greedy algorithm to partition clusters of time series (Sales-Pardo et al., 2007). We repeat this procedure200 times and obtain 200 partitions. We construct an a ffi nity matrix A (cid:48) whose element A (cid:48) i j is the number of partitionsin which i and j are assigned to the same cluster, divided by the number of partitions 200. Furthermore, we apply theclustering method to the a ffi nity matrix A (cid:48) , resulting in a final partition A (Lancichinetti and Fortunato, 2012). Wefinally rearrange the order of states in C to be the same as in A (Meng et al., 2014).The resultant rearranged correlation matrix is illustrated in Fig. 2(b), in which six clusters of time series areidentified. In the last column of Table 1, we provide the cluster information of each time series with the clusterslabeled 1 to 6 from left to right in Fig. 2(b). In addition, we color the world map based on the clusters, as shown in inFig. 2(c). We assign each cluster a unique color and a region or a country is colored if its crude oil market belongs toa specific cluster.There are two arrows in the colored map, one in blue and the other in yellow. The blue arrow stands for Cluster6 where the crude oils exported from Mideast area to Northwest Europe, while the yellow arrow represents Cluster 4where the crude oils exported from Mideast area to Asia-Pacific area. The rest time series form Cluster 2 in MiddleEast. Cluster 1 contains the time series in Asia and Australia, which are uncorrelated with all other time series asshown in Fig. 2(b). Cluster 3 mainly contains time series in North America and Cluster 5 in Europe and Nigeria.As shown in Table 1, there are only a few exceptions. Therefore, the correlation structure of the global oil marketexhibits remarkable geographical traits, which is reminiscent of the global stock markets (Song et al., 2011) and theUS housing market (Meng et al., 2014). 4 abel L a b e l
10 20 30 40 50 60 7010203040506070 0.10.20.30.40.50.60.70.80.91
Label L a b e l (a) (b)(c) Figure 2: (Color online) Cross-correlation structure among 71 crude oil price time series all over the world. (a) Correlation matrix of the dailylogarithmic returns. (2) Clusters identification of the oil price time series using the box clustering algorithm and the consensus clustering method.(c) Colored map for the six clusters. c ij P ( c i j ) Figure 3: Distribution of correlation coe ffi cients. ffi cients We plot the distribution of correlation coe ffi cients in Fig. 3. Four remarked peaks can be easily identified. Thisfeature stems from the fact that the global oil market is less integrated, as shown in Fig. 2. Such multi-modal patterns inthe distribution of correlation coe ffi cients have not been observed in financial markets. Usually, we observe unimodaldistributions for financial markets, including the global stock markets (Song et al., 2011) and the US housing market(Meng et al., 2014) where we observe similar geographical traits.The right peak corresponds to the largest correlation coe ffi cients between intra-cluster time series, which are alongthe back diagonal in Fig. 2(b). The second peak from the right highlights the correlations between inter-cluster timeseries for clusters 3, 4 and 5. In certain sense, Fig. 2(b) illustrates that these three clusters can be viewed as a largecluster. The left peak contains correlation coe ffi cients between cluster 1 and clusters 3, 4, 5 and 6. The second peakfrom the left represents the correlation coe ffi cients between inter-cluster time series other than those in the left peak.We can see that Fig. 3 unfolds a picture of the global oil market with a remarked feature of localization.
4. Eigenvalues and information content
For the correlation matrix C of each crude oil price series, we can calculate its eigenvalues, C = U Λ U T , (5)where U denotes the eigenvectors, Λ is the eigenvalues of the correlation matrix, whose density f c ( λ ) is defined asfollows Laloux et al. (1999), f c ( λ ) = N dn ( λ ) d λ , (6)where n ( λ ) is the number of eigenvalues of C that are less than λ . If M is a T by N random matrix with zero meanand unit variance, f c ( λ ) is self-averaging. In particular, in the limit N → ∞ , T → ∞ and Q = T / N ≥ f c ( λ ) of eigenvalues λ of the random correlation matrix M has a close form (Edelman,1988; Sengupta and Mitra, 1999; Laloux et al., 1999): f c ( λ ) = Q πσ √ ( λ max − λ )( λ − λ min ) λ (7)with λ ∈ [ λ min , λ max ], where λ maxmin is given by λ maxmin = σ (1 + / Q ± (cid:112) / Q ) , (8)6 λ P ( λ ) λ P ( λ ) Figure 4: Empirical distribution of eigenvalues of the correlation matrix C with the large eigenvalues in the inset. The solid line is the theoreticaldistribution expressed in Eq. (7) from the random matrix theory predictions. and σ is equal to the variance of the elements of M (Sengupta and Mitra, 1999; Laloux et al., 1999). In our case, Q = / = . σ is equal to 1 in our normalized data. If C is a random matrix, the largesteigenvalue λ RMTmax = . λ RMTmin = . N eigenvalues of the correlation matrix C , λ max = λ > λ > · · · > λ > λ = λ min . Wefind that the largest eigenvalue λ max = .
76 and the smallest eigenvalue λ min = . λ and λ falling in the theoretical curve. The five largest eigenvalues are greater than λ RMTmax and the other 64 eigenvalues are smaller than λ RMTmin . The deviation of the empirical distribution from the theoreticalprediction implies that the correlation matrix C is not a random matrix and there is economic information embeddedin the deviating eigenvalues. To uncover the information contents in the deviating eigenvalues, we construct eigenportfolios for each eigenvalue.For λ k , we have R k ( t ) = u Tk · r (cid:80) Ni = u i , k = (cid:80) Ni = u i , k N (cid:88) i = u i , k r i ( t ) , (9)where u Tk · r is the projection of the normalized return vector r on the k -th eigenvector u k .Figure 5(a) illustrates the relationship between the eigenportfolio R and the mean return (cid:104) r (cid:105) . There is an evidentlinear relationship with a very high R-square of 0.97. Hence, the largest eigenvalue reflects a common factor thatdrives the global crude oil market. Because the largest eigenvalue explains λ / N = .
63% of the variation of theprice fluctuations, this common factor implies a collective e ff ect of the whole market mode. In addition, Fig. 5(b)shows that all the components of the first eigenvector have the same sign and are not randomly distributed. However,no evident relationship between the mean return and the eigenportfolios is observed for other eigenvalues, whichmeans that other deviating eigenvalues do not bear any market-wide e ff ects. These observations are also reported forstock markets (Plerou et al., 2002).Because the largest eigenvalue reflects the global movement of the crude oil markets, we can construct an indexfor the global crude oil market based on the eigenportfolio of the largest eigenvalue: I ( t ) = (cid:104) P (0) (cid:105) exp T (cid:88) t = R ( t ) (10)where (cid:104) P (0) (cid:105) = . (cid:2) r (cid:3) R Label u −0.2 −0.1 0 0.1 0.2−0.4−0.200.20.4 (cid:2) r (cid:3) R −0.2 −0.1 0 0.1 0.2−4−20246 (cid:2) r (cid:3) R −0.2 −0.1 0 0.1 0.2−50510 (cid:2) r (cid:3) R −0.2 −0.1 0 0.1 0.2−1−0.500.51 (cid:2) r (cid:3) R (a) (b)(c) (d)(e) (f) Figure 5: Testing collective market e ff ect. (a) Relationship between the mean returns (cid:104) r (cid:105) and the eigenportfolio R . (b) Components of theeigenvector u of the largest eigenvalue λ . (c) Relationship between the mean returns (cid:104) r (cid:105) and the eigenportfolio R . (d) Relationship between themean returns (cid:104) r (cid:105) and the eigenportfolio R . (e) Relationship between the mean returns (cid:104) r (cid:105) and the eigenportfolio R . (f) Relationship between themean returns (cid:104) r (cid:105) and the eigenportfolio R . the index performs better than the average price under the buy and hold strategy because the index is always greaterthan the average price. ff ect For stock markets, other deviating eigenvalues that are greater than λ RMTmax contain partitioning information ofindustrial sectors (Plerou et al., 2002). Following this idea, we show the four eigenvectors in the left panel of Fig. 6. Wecan already see clusters of components with similar heights. These clusters of components correspond to the clustersidentified in Fig. 2. To further illustrate this point, we plot in the right panel of Fig. 6 the reordered eigenvectors basedon the six clusters.
Having unveiled the latent information carried by the deviating eigenvalues greater than λ RMTmax and their corre-sponding eigenvectors, we now turn to investigate the hidden information in the smallest eigenvalues. According toPlerou et al. (2002), the smallest eigenvalues and their corresponding eigenvectors stand for the high-correlation timeseries pairs of stocks. We find similar information in the global crude oil market.Figure 7 shows the eigenvectors of the four smallest eigenvectors. In each plot of u , u and u , we observe apair of components with opposite signs and significant large magnitudes. We locate the price label of each componentand calculate the corresponding correlation coe ffi cient between each pair of return time series. The two time series for u are Sirtica and Zueitina, whose correlation coe ffi cient c , = . ffi cients.The pair for u are Escravos and Pennington, whose correlation coe ffi cient c , = . c i j ’s and the pair for eigenvector u are Arab Medium to USA and Arab Heavy to USA with c , = .
20 40 60−0.4−0.200.20.4
Label u Cluster u Label u Cluster u Label u Cluster u Label u Cluster u (a) (b)(c) (d)(e) (f)(g) (h) Figure 6: (Color online) Partitioning e ff ect of the eigenvectors u (a, b), u (c, d), u (e, f), and u (g, h) associated with the four deviatingeigenvalues λ to λ . Left panel: Raw eigenvectors; Right panel: Reordered eigenvectors based on the six clusters.
20 40 60 80−1−0.500.51 c =0.9941 c =0.9938c =0.9937 c =0.9925c =0.9906 Label u Arab Medium to USA → Arab Heavy to USA → c =0.9953 Label u ← Escravos ← Penningtonc =0.9960
Label u ← Sirtica ← Zueitinac =0.9992
Label u Figure 7: (Color online) Eigenvectors u , u , u , and u of the four smallest eigenvalues λ to λ . The situation for u is less clear and complicated, because there are several “bars” that outstand. We can nev-ertheless identify several pairs for the longest bars: Amna and Zueitina with c , = . ffi cient, Amna and Sirtica with c , = . ffi cient, and BonnyLight and Brass River with c , = . ffi cient. We also find that the fifth largest is c , = . c , = . u is small.Of these time series in the pairs, seven series (23, 27, 28, 29, 30, 31 and 33) belong to Cluster 5, while two (53and 56) belong to Cluster 3. This finding is reasonable because time series in the same cluster usually have largecorrelation coe ffi cient as shown in Fig. 2(b).
5. Conclusions
We have investigated the correlation structure of the global crude oil market including 71 time series of daily oilprices all over the world by performing principal component analysis. Several empirical statistical properties of theglobal oil market have been unveiled.We found that the distribution of correlation coe ffi cients between pairs of daily returns is not unimodal, but multi-modal. We also identified six clusters of price time series from the correlation matrix, which have evident geographicaltraits. The multi-modality of the correlation coe ffi cient distribution is caused by the geographical traits of the pricetime series and the peaks correspond well to the distinct intra-cluster and inter-cluster correlations.We found that most eigenvalues of the correlation matrix locate outside the prediction of the random matrix theory,where the five largest eigenvalues are greater than the theoretical maximal eigenvalue λ RMTmax and 64 small eigenvaluesare less than the theoretical minimal eigenvalue λ RMTmin . We showed these deviating eigenvalues and their correspondingeigenvectors embed certain economic information. Specifically, the largest eigenvalue reflects a collective e ff ect ofthe global market, other four largest eigenvalues bear geographical information that is capable of identifying the sixclusters, and the smallest eigenvalues usually have very large components in the eigenvectors which corresponds topairs of time series with the largest correlation coe ffi cients in the correlation matrix.We constructed eigenportfolios based on the eigenvectors. It is found that the returns of the eigenportfolio for thelargest eigenvalue correlate strongly to the average returns of the oil prices, while other eigenportfolios do not exhibitsignificant correlations with the average returns. Inspired by this feature, we proposed to construct an index for theglobal crude oil market based on the eigenportfolio of the largest eigenvalue. This index evolves very similarly to theaverage price time series and outperforms the latter under the buy-and-hold strategy.10 cknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 11075054), Shanghai(Follow-up) Rising Star Program (Grant No. 11QH1400800), and Fundamental Research Funds for the Central Uni-versities.
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