Dependency, centrality and dynamic networks for international commodity futures prices

International Review of Economics and Finance 67 (2020) 118–132

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International Review of Economics and Finance journal homepage: www.elsevier.com/locate/iref

Dependency, centrality and dynamic networks for international commodity futures prices Fei Wu a, Wan-Li Zhao b, Qiang Ji c, d, Dayong Zhang e, * a

Department of Economics, University of Birmingham, Birmingham, B15 2TT, UK School of Economics & Management, Beihang University, Beijing, 100191, China Center for Energy and Environmental Policy research, Institutes of Science and Development, Chinese Academy of Sciences, Beijing, 100190, China d School of Public Policy and Management, University of Chinese Academy of Sciences, Beijing, 100049, China e Research Institute of Economics and Management, Southwestern University of Finance and Economics, Chengdu 611130, China b c

A R T I C L E I N F O

A B S T R A C T

Keywords: Commodity futures prices Crude oil Dependency network Financialisation Time-varying

This paper adopts a network approach to measure dependency among a set of international commodity futures prices. We first use partial correlations to construct a static dependency network for a vector of variables, and then illustrate within-system connections in a minimum spanning tree (MST) to evaluate the centrality of the variables. Rolling-window estimation is then applied to address time variations in both dependency and centrality networks. We show that crude oil price plays a pivotal role in connecting together components in the networks and there is clear evidence of time-varying within-system dependency. Our method demonstrates a new and easy-to-apply way to investigate dependency. The empirical results provide new evidence to the recent intensive discussions on financialisation in energy and commodity markets.

1. Introduction A recent trend in the energy and commodity market research is the prompting of the concept of commodity financialisation (for example, Cheng & Xiong, 2014; Zhang & Broadstock, 2018). This concept per se clearly indicates an inter-disciplinary nature, encompassing a number of important implications for relevant research. The global commodity market has experienced dramatic fluctuations especially in the post-financial crisis era, with rising contagion risks across various commodity prices. As one of the key considerations in policy-making and business decision making, commodity price movements have been closely watched by policy makers and analysts. Among all of the changes, financialisation, as a fresh attribute of the commodity market in recent years (Cheng & Xiong, 2014), has endowed new characteristics of commodity price co-movement, which draw keen interests from academics. Among the various newly emerged topics in the thriving literature of commodity finance, a core question concerns how the integration of commodity markets evolves over time and whether exogenous economic or financial shocks increase risk contagion across different commodities. Concurrent with the ever-evolving commodity markets and increasingly complex market conditions, the relevant literature has extended from a bivariate to a multivariate framework. While earlier studies mainly aim to examine the relationship between two commodities, recent studies seek to investigate the interrelationships within the whole commodity system encompassing all types of commodities. New directions of this vine of research include the exploration of a unified framework that can be adopted to

* Corresponding author. E-mail address: [email protected] (D. Zhang). https://doi.org/10.1016/j.iref.2020.01.004 Received 1 July 2019; Received in revised form 1 January 2020; Accepted 15 January 2020 Available online 23 January 2020 1059-0560/© 2020 Elsevier Inc. All rights reserved.

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analyze the system-wide interrelationship and to identify the most important commodities in the commodity system. Studying the various issues related to financialisation of commodity markets inevitably requires the understanding of market microstructure changes in the futures market (for example, Ji & Zhang, 2018). Financialisation of the commodity markets raises the concern of information spillovers in the commodity system. While commodity prices are traditionally viewed as being mainly driven by fundamental market forces such as supply and demand, the introduction and deepening of financialisation highlights the increasing importance of market information, which should be reflected in commodity prices, consistent with the Efficient Market Hypothesis (EMH) proposed by Fama (1970). It is central to the research of commodity market financialisation to investigate information linkages and spillovers across a number of asset classes (for example, Ji, Geng, & Tiwari, 2018b). Motivated by these new trends in both the commodity markets and the related literature, this study seeks to explore the dependency network of a set of commodity futures prices and the time-varying characteristics of the network structure, which play a key role in facilitating risk/information spillovers and price co-movements in a dynamic fashion. This evolving dependency structure of commodity prices is highly relevant to the topics of commodity financialisation. By answering this question, we explicitly fit our study into the growing body of the literature on commodity financialisation, and contribute to the development of this fast evolving interdisciplinary filed by providing new empirical evidence and novel insights into several conceptual issues. Methodologically, this study follows a recent work in the finance literature, Sandoval, Mullokandov, and Kenett (2015), to construct a dependency network based on partial correlations. Sandoval et al. (2015) provide a simple yet effective way of analyzing the within-system interdependence. A further advantage of this method is that the construction of the dependency network is essentially based on correlation analysis, which is a commonly applied method in portfolio construction. With this commonality, our findings provide useful and easy-to-understand implications to investors. The third main contribution of this study is that it adopts graph theory and renders an intuitive illustration of how information spillovers work in the system. Using this approach, we construct a centrality network, which provides a robustness check and serves as a complement to the dependency network. It further identifies the clustering characteristics of different commodity types and discloses the most important commodity product in the system. The rest of this paper is structured as follows: Section 2 reviews relevant literature; Section 3 briefly introduces the methodology applied; Section 4 presents data and provides key descriptive statistics of the sample; Section 5 reports main empirical results with relevant discussions, and the last section concludes. 2. Literature review Recently, there have emerged a number of new stylized facts in the global commodity markets, such as the strong price comovements among major commodities (Fernandez, 2015; Luo & Ji, 2018), and large price fluctuations (Arezki, Loungani, van der Ploeg, & Venables, 2014; Fernandez, Gonzalez, & Rodriguez, 2018). These issues raise important questions for macroeconomic policy-making and academic research. Increased co-movements in commodity prices indicate higher systemic risk (Zhang & Broadstock, 2018), which poses challenges to regulators in terms of causing economic instability and investors who seek for efficient hedging strategies and optimal portfolio composition. Consistent with the concept of commodity financialisation, De Nicola, De Pace, and Hernandez (2016) provide evidence that co-movements in commodity markets are positively associated with volatility of stock market returns, especially after 2007. Zhang and Broadstock (2018) document the substantially increasing interconnectedness among commodity prices since the 2008 global financial crisis. The authors try to explain this structural change by taking into account the possible impacts of commodity financialisation on market fundamentals, as it is generally agreed that this price synchronization cannot simply be explained by economic fundamentals (Pindyck & Rotemberg, 1990). On the other hand, considerable price fluctuations in global commodity markets have been observed especially after the 2008 global financial crisis. One noticeable example is crude oil, whose price volatility has been unprecedentedly high (Ji, Zhang, & Zhang, 2019a), evidenced by the slump of Brent spot price from a historical high of 139.38 dollars per barrel on 27 June 2008 to 34.45 on 24 December 2008.1 Volatility of commodity prices may be related to macroeconomic uncertainties (for example, Brooks, Prokopczuk, & Wu, 2015; Zhang, Lei, Ji, & Kutan, 2018a), but price change of this magnitude is obviously beyond what can be explained by fundamental shocks such as the standard demand and supply factors. Among all major commodities in the global markets, crude oil is a very special one and has often been considered to be the most important driving factor of the price movements in commodity markets. For example, Ahmadi, Behmiri, and Manera (2016) find that volatility in agricultural and metal commodity prices does react to oil price shocks and the reactions were significantly stronger after the 2007-08 world food price crisis and the 2008 global financial crisis, respectively. Ji and Fan (2012) demonstrate the core position of the crude oil market in the commodity market as a whole, as it exhibits significant volatility spillovers on non-energy commodity markets. Crude oil price’s critical role in affecting price movements in the commodity markets can be attributed to several plausible reasons, as revealed by some recent researches. First, as one of the main exhaustible natural resources, oil as a type of fuel is hardly substitutable. It acts as the foundation of the modern industrial economy, and plays an irreplaceable role in household consumption (Zhang, Broadstock, & Cao, 2014). Second, oil has been shown to be one of the most important factors affecting economic growth (Zhang, 2008), and stock market performance (Broadstock, Cao, & Zhang, 2012). These types of influence are found prevalent around the world. As empirical evidence shows that crude oil price shocks clearly have profound impacts on the global economic conditions, they

1

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consequentially affect the dynamics in commodity markets. Third, oil price is often used as a benchmark price for other commodities, for example, natural gas, with a large volume of literature discussing the concept of oil-indexation in natural gas prices (for example, Zhang, Shi, & Shi, 2018b,c; Zhang & Ji, 2018). This benchmarking role also implies that oil price and its shocks are central to studying the price interactions in the commodity markets. In terms of methodology, standard time series approaches such as multivariate GARCH models, cointegration analysis, vector autoregressive models (VAR), and Granger causality tests are often used in the existing literature (e.g., Ji & Fan, 2016a). These econometric methods are proved to be very useful instruments in detecting cross-market relationships, although each has its limitations. One notable shortcoming these models commonly have is the failure of capturing the network structure linking together a set of variables, in this case, commodity markets. This complex and possibly time-varying network structure can be an important determinant of how variables influence each other and to what extent they may move together. The significance of this network factor should not be understated, which, however, is not explicitly modelled or considered in the standard time series models. To address the critical role of the network structure in determining cross-variable linkages and information spillovers, Diebold and Yilmaz (2009, 2014) extend the standard VAR model by incorporating a network perspective. By repackaging the forecasting error variance decomposition of the estimated VAR model, they show the power of network typology in explaining systemic interactions in an informative yet easy-to-understand fashion. With these advantages, this approach has quickly gained its popularity and become one of the most widely used instruments in recent literature (for example, Ji & Zhang, 2019; Ji et al., 2018; Ma, Zhang, Ji, & Pan, 2019; Song, Ji, Du, & Geng, 2019; Wu, Zhang, & Zhang, 2019; Zhang & Fan, 2018). Nevertheless, its application is restricted by the dimensionality issue (for example, Zhang & Fan, 2018). With such a concern, this study proposes to use an alternative way, the network approach, to rectify the problem embedded in the VAR model. Specifically, this study follows the methodology in Sandoval et al. (2015) to construct a dependency network based on partial correlations, which is briefly explained in Section 3. 3. Methodology The dependency network (for example, Sandoval et al., 2015) is essentially a model-free technique that relies solely on the calculation of partial correlations (Livingston & Stanley, 1972). In other words, the method does not require us to specify any econometric model (such as VAR) in advance for the estimation of interdependence. Instead, the dependency within a system (or among a vector of variables) is simply calculated from the conditional joint distributions across all variables. Another advantage of this method over the existing VAR-based approaches and other connectedness measures is that it is not restricted by the number of variables, which is often a major concern in time series models in terms of dimensionality. For simplicity, in this study we use only conditional correlations (partial correlations) to construct the dependency network. As a simple but powerful tool, the dependency network can render rich and comprehensive information on dependence effect, intermediate effect, and asymmetric effect, which are of great importance to quantify the dynamic network structure and investigate co-movements in a system consisting of a relatively large number of variables. As the method is based on correlation analysis, it is also in the spirit of the standard finance and investment theories/models, thus able to provide practical implications for market participants. 3.1. Modelling network dependency Following the methodology in Sandoval et al. (2015), our first step is to calculate partial correlations. Equation (1) describes the measurement of the first order partial correlation between variable i and j conditional on variable k. Cði; jÞ  Cði; kÞCðj; kÞ PCði; jjkÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1  C 2 ði; kÞ 1  C 2 ðj; kÞ

(1)

where PCði; jjkÞ is the first order partial correlation between variable i and j given a third variable k, Cði; jÞ is the correlation (measured by the Kendall rank correlation coefficient, or Kendall’s tau) between variable i and j. The difference between Cði; jÞ andPCði; jjkÞ is the dependency effect of variable k on the correlation Cði; jÞ, which is defined as dði; jjkÞ  Cði; jÞ  PCði; jjkÞ. Whendði; jjkÞ equals zero, variable k has no impacts on the correlation between i and j, and thus variable i’s correlation with variable j does not depend on variable k. The higher thedði;jjkÞis, the more dependency effect variable k has on variable i. Notably, we replace the standard Pearson correlation used in Sandoval et al. (2015) by Kendall’s Tau rank correlation to better reflect the notion of non-linear dependency. We then expand the three-variable case to a system containing N variables. Like in a standard portfolio formation case, we calculate correlations between asset i and each of the rest assets (excluding i itself and variable k). The total dependency effect of variable k on variable i can be defined as the average of the dependency effects of variable k on the correlations between variable i and all the other N2 variables in the system, written as: Dði; kÞ ¼

N 1 X dði; jjkÞ; j 6¼ k; i N  2 j¼1

(2)

Generally, Dði; kÞ is not equal to Dðk;iÞ, implying that the dependency effect is asymmetric, and the influence of variable i on variable k is different from the influence of variable k on variable i. We can accordingly construct a directed and asymmetric dependency network using the net dependency of all pairs, which is calculated as the difference between the bidirectional dependency effects, or Eik  Dðk; iÞ  Dði; kÞ. A directed edge can be drawn from i to k if and only ifEik > 0. 120

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3.2. Modelling centrality An alternative approach of capturing the network structure is to use graph theory. Specifically, we construct a minimum spanning tree (MST) based on the correlations calculated in Section 3.1, and then draw a centrality network to capture the core-periphery structure of the system and identify the variables playing the most critical linking roles in the system. MST only contains the most important correlations and information by a certain filtering rule. By construction, it intuitively reflects the simplest, core relationship among the variables in a system (Eom & Park, 2017; Ji, Bouri, Roubaud, & Kristoufek, 2019b; Zeng, Xie, Yan, Hu, & Mao, 2016). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Following Ji and Fan (2016b), we convert the Kendall Tau rank correlations into the metrices of distance mdij ¼ 2ð1  Cði; jÞÞ, and then use the Prim’s algorithm to construct the MST of this system to connect all nodes in a graph with minimum possible total edge weight and no loops. In the centrality network in the form of an MST, the normalized tree length, which is defined as the average of all the edges length, is used to measure the degree of system integration. We also use the survival ratio of edges to measure the stability of the network structure over time, and use three centrality measures named degree centrality, closeness centrality and betweenness centrality to identify the most important nodes in the network. The detailed formulas are written as follows. ! Normalized tree length:L t ¼

Survival ratio:σ ðt; kÞ ¼

X 1 eij;t N  1 eij 2MSTt

(3)

1 jEðtÞ \ Eðt  1Þ⋯Eðt  k þ 1Þ \ Eðt  kÞj N1

Degree centrality:kðiÞt ¼

n X

(4)

(5)

aij;t

j¼1

Closeness centrality:CCðiÞt ¼

X Rij;t ;

j ¼ 1; 2…N; j 6¼ i

(6)

ði;jÞ

Betweenness centrality:BðiÞt ¼

X σ jlðiÞ;t 2 ; i 6¼ j 6¼ l NðN  1Þ ðj;lÞ2MST σ jl;t

(7)

t

where eij is the edge length in the MST between node i and j; σ ðt; kÞ is the k–step survival ratio of edges in the MST; E(t) is the set of edges in the MST at time t; \ is the intersection operator; and j⋯j gives the number of elements in the set. aij ¼ 1 if and only if vertex i and j have an edge in the MST, or otherwise zero; Rij is the shortest path from i to j in the MST; σ jlðiÞ is the number of the shortest paths from j to l passing through i, and σ jl is the number of the shortest paths from j to l. A more important node in the MST is likely to be connected to more nodes, and more often act as a bridge to link together other nodes, as reflected in the values of degree, closeness and betweenness (Wu et al., 2019). Table 1 Summary statistics of commodity futures prices (returns). Commodity type

Mean

Std. Dev.

Skewness

Kurtosis

Jarque-Bera

Cocoa Coffee Corn Cotton Lean Hogs Live Cattle Orange Juice Soybean Sugar Wheat Aluminum Copper Gold Nickel Silver Crude Oil Heating Oil Natural Gas Gasoline

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.018 0.019 0.019 0.018 0.021 0.011 0.021 0.014 0.021 0.021 0.014 0.018 0.012 0.023 0.021 0.023 0.020 0.031 0.024

0.341 0.108 0.613 1.130 0.161 0.257 0.050 0.154 0.051 0.197 0.104 0.080 0.339 0.029 0.893 0.112 0.321 0.549 0.123

6.060 5.096 13.589 20.581 22.720 12.772 6.589 5.775 6.126 5.331 4.972 6.991 8.717 5.748 10.286 7.895 8.201 8.139 10.226

1409.869*** 637.011*** 16295.466*** 45061.650*** 55784.398*** 13732.471*** 1848.592*** 1117.908*** 1402.647*** 801.941*** 564.186*** 2287.478*** 4753.879*** 1083.278*** 8071.383*** 3442.938*** 3938.819*** 3960.482*** 7497.132***

Notes: The table reports the descriptive statistics for the returns of the 19 commodity futures at daily frequency from 20 October 2005 to 31 December 2018. Jarque-Bera denotes the Jarque-Bera statistics for normality. *** indicates rejection of the null hypothesis at the 1% level. 121

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4. Data and descriptive statistics To investigate the dependence in the commodity market system, we use the data of commodity futures prices collected from the Commodity Research Bureau (CRB) database. The CRB index is a futures price index compiled by the Commodity Research Bureau of the United States based on the prices of 19 basic economic sensitive commodities in the world market. As the CRB index includes the price fluctuations of the core commodities, it reflects the dynamic information of the world’s major commodity prices as a whole, and is widely used to observe and analyze price fluctuations and macroeconomic conditions in the commodity market, and to some extent, to predict the future trends in the markets. There are 19 commodity futures in our sample, which are the key components of the CRB index. These futures include four energy products (crude oil, heating oil, natural gas, and gasoline), five metal products (aluminum, copper, gold, nickel, and silver), and ten agricultural products (cocoa, coffee, corn, cotton, lean hogs, live cattle, orange juice, soybean, sugar, and wheat). Data in our sample are t daily frequency, and the sample period ranges from 20 October 2005 to 31 December 2018 (subject to data availability), which results in a total of 3443 observations. The whole sample period can be roughly divided into three distinctive stages, namely, the commodity market boom period (2005–2008), the global financial crisis period and its aftermath (2008–2012), and the post-financial crisis period (2013–2018). In general, the sample’s time span covers almost all major changes in the global commodity markets. Table 1 reports some basic descriptive statistics of the sample data. It is worth clarifying that all price data are converted into returns (for the sake of stationarity). Average daily returns in the sample period are very low for all commodity types. It is interesting to notice that energy commodity futures are generally riskier than other categories (as evidenced by their higher standard deviations). Among all of the commodities, natural gas futures exhibit the highest volatility. While returns of the majority of commodity futures are negatively skewed, the natural gas series have the highest positive skewness. All series demonstrate Leptokurtosis (with excess kurtosis and fat tails) and they are all non-normally distributed, as corroborated by the Jarque-Bera test statistics. Fig. 1 plots a heat-map to show the pairwise correlations across all commodities. The darker the color is in the map, the lower the pairwise correlation is. Graphic evidence in the map shows that the energy sector has visibly higher correlations with other categories, which is followed by the metals. Although there are ten agricultural commodities in the system, most of them have relatively low correlations with other agricultural commodities, except that only wheat and corn have correlation above 0.5. Among the metal commodities, silver and gold series are highly correlated, plausibly due to that they are the only two precious metal products in the CRB index.

Fig. 1. Heat-map for correlations. 122

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Cor

Cot

Lea

Liv

Ora

Soy

Sug

Whe

Alu

Cop

Gol

Nic

Sil

Cru

Hea

Nat

Gas

From

0.000 0.010 0.005 0.007 0.002 0.004 0.001 0.006 0.007 0.005 0.007 0.008 0.007 0.006 0.008 0.008 0.007 0.003 0.006 0.107 0.094

0.015 0.000 0.012 0.013 0.005 0.008 0.005 0.014 0.018 0.012 0.011 0.012 0.013 0.011 0.014 0.013 0.011 0.005 0.009 0.201 0.047

0.008 0.015 0.000 0.020 0.006 0.011 0.002 0.030 0.017 0.047 0.014 0.014 0.012 0.012 0.014 0.016 0.016 0.011 0.016 0.282 0.002

0.009 0.013 0.015 0.000 0.003 0.008 0.004 0.015 0.013 0.013 0.011 0.012 0.009 0.011 0.011 0.012 0.011 0.002 0.011 0.184 0.065

0.001 0.002 0.002 0.001 0.000 0.006 0.002 0.002 0.001 0.002 0.002 0.002 0.001 0.002 0.001 0.002 0.002 0.001 0.002 0.034 0.074

0.003 0.004 0.005 0.004 0.009 0.000 0.003 0.005 0.004 0.004 0.004 0.004 0.001 0.003 0.003 0.004 0.003 0.002 0.003 0.070 0.073

0.001 0.002 0.000 0.001 0.001 0.002 0.000 0.002 0.001 0.001 0.002 0.001 0.001 0.001 0.002 0.002 0.001 0.001 0.002 0.023 0.070

0.013 0.021 0.039 0.024 0.007 0.013 0.010 0.000 0.018 0.031 0.024 0.029 0.021 0.023 0.026 0.031 0.032 0.013 0.029 0.406 0.070

0.009 0.015 0.012 0.012 0.002 0.006 0.003 0.010 0.000 0.011 0.010 0.012 0.007 0.009 0.009 0.011 0.010 0.004 0.008 0.158 0.075

0.008 0.013 0.034 0.014 0.005 0.008 0.002 0.019 0.012 0.000 0.010 0.010 0.009 0.008 0.010 0.011 0.010 0.006 0.010 0.199 0.054

0.015 0.017 0.017 0.017 0.009 0.011 0.009 0.023 0.016 0.014 0.000 0.046 0.025 0.043 0.031 0.026 0.023 0.008 0.021 0.370 0.036

0.019 0.022 0.020 0.023 0.012 0.013 0.005 0.033 0.024 0.018 0.059 0.000 0.036 0.058 0.046 0.036 0.031 0.008 0.030 0.494 0.138

0.013 0.015 0.012 0.011 0.001 0.003 0.004 0.016 0.010 0.010 0.020 0.023 0.000 0.016 0.047 0.017 0.016 0.002 0.014 0.251 0.063

0.011 0.013 0.012 0.015 0.008 0.006 0.006 0.019 0.013 0.010 0.035 0.037 0.017 0.000 0.022 0.020 0.018 0.004 0.017 0.281 0.034

0.019 0.023 0.019 0.019 0.005 0.007 0.008 0.027 0.016 0.016 0.034 0.041 0.080 0.029 0.000 0.027 0.025 0.006 0.023 0.423 0.092

0.025 0.026 0.027 0.025 0.013 0.015 0.011 0.042 0.025 0.021 0.036 0.042 0.029 0.033 0.034 0.000 0.089 0.023 0.081 0.597 0.222

0.019 0.021 0.025 0.023 0.012 0.011 0.008 0.041 0.021 0.019 0.030 0.034 0.026 0.027 0.030 0.080 0.000 0.025 0.083 0.533 0.159

0.002 0.002 0.004 0.001 0.001 0.002 0.002 0.004 0.002 0.003 0.002 0.002 0.001 0.001 0.002 0.004 0.005 0.000 0.004 0.043 0.099

0.014 0.014 0.021 0.018 0.007 0.010 0.009 0.031 0.014 0.015 0.023 0.027 0.019 0.021 0.023 0.056 0.062 0.017 0.000 0.400 0.031

0.201 0.248 0.280 0.249 0.108 0.143 0.094 0.336 0.233 0.253 0.334 0.356 0.314 0.315 0.331 0.375 0.374 0.141 0.369 Total 0.266

Note: We use abbreviations of the first three letters of each commodity to represent it. Please refer to Table 1 for the full name. From is the aggregation of each row and To is the aggregation of each column showing the dependency effects from the system and on the system, respectively, whereas Net is the simple difference between these two.

International Review of Economics and Finance 67 (2020) 118–132

Cof

123

Coc Cof Cor Cot Lea Liv Ora Soy Sug Whe Alu Cop Gol Nic Sil Cru Hea Nat Gas To Net

Coc

Table 2 Dependency matrix in the commodity market system over the full sample.

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5. Empirical results We take two steps for both the dependency network approach and the centrality network approach: The first step is to investigate the full sample using a static analysis, and then we extend to the time-varying dynamics of the network. To enable the transformation from static to dynamic analysis, we adopt a rolling-window approach (such as Diebold & Yilmaz, 2014). The window length is 130 observations, approximately corresponding to a half-year window size.

5.1. Dependency network 5.1.1. Static analysis of the full sample Considering all the 19 commodity futures as a portfolio of assets, we estimate pairwise dependency across all pairs (excluding selfexplanation) and fit them into a matrix, the dependency matrix, which is comparable to the connectedness matrix in Diebold and Yilmaz (2014). Similar to Diebold and Yilmaz (2014), though with different meanings, we construct three additional measures. They are the From measure, To measure and Net measure. While From is an aggregation across a row and To is an aggregation over a column, their difference is defined as Net. These measures, especially Net, convey important information with regard to which variable plays the most important role in affecting others. The commodity with the highest contribution to the system plays the most important mediating role in the system and thus should be considered the most important component. Table 2 reports the results of the dependency matrix, with the three measures shown in the last column and two bottom rows. In Table 2, we also report the total dependency measure (Total), which is equivalent to the one in Diebold and Yilmaz (2014) and measures the level of aggregate systemic linkage. The total level of dependency is 0.266. Among all commodities, crude oil has notably higher dependency effects than all other commodities. This can also be clearly seen in Fig. 2 which plots the three measures discussed above for all commodities. The top three contributors are sequentially crude oil, heating oil, and copper, corresponding to the To value of 0.597, 0.533 and 0.494, respectively. These three commodities are also ranked as the top three (in the same order) in terms of the Net values. Following the recent literature, we also plot pairwise directed dependency across all variables into a network graph. Given that the dependency effects are asymmetric, variable i may be more dependent on variable j than from the opposite direction. If that is the case, we can then draw an arrow pointing from j to i, to show that variable i is net dependent on variable j, or in other words variable j is exerting net influence on variable i. This can be applied to all pairs to portray a general network and show how the system is interconnected. Fig. 3 plots the directed pairwise dependency across all 19 commodities. The size of each node is determined by the number of outward arrows the node has. A larger node indicates more outgoing arrows and therefore reflects the node’s relatively stronger influence in the dependency network. Not surprisingly, results based on node sizes in Fig. 3 are consistent with the results of aggregation measures mentioned above (To and Net): crude oil is the most important commodity among all and has larger dependency effects than all the other commodities which are also net contributors. To give a clearer picture, we proceed to simplify the complete network by restricting the number of edges. Only the top 20% connections ranked by the net pairwise dependency values are shown in this new graph, known as the core directed dependency network (see Fig. 4). As the new core network retains only the most valuable information, it is easier to capture the most important pairwise relationships. The majority of commodities are net receivers in the system. Crude oil has the greatest net positive dependency effects on other commodities, with 14 outward arrows. Heating oil is the second and then copper, both having 6 outward arrows in the core directed dependency network. In general, the message sent by the core network graph is very clear: crude oil price plays a pivotal

Fig. 2. System dependency measures for the full sample. 124

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Fig. 3. Complete dependency network (directed pairwise net dependency).

role in the dependency network, or in other words, the price of crude oil futures contains useful information that can affect correlations among other commodity prices. 5.1.2. Dynamic analysis with rolling windows As mentioned earlier, the whole sample period covers a long time span during which a number of systemic events occurred, such as the 2008 global financial crisis. It should be reasonably expected that the correlation and dependency structures within the system of commodity futures have changed substantially over time. It is therefore necessary to progress from static analysis to a dynamic setup. A rolling-window approach is therefore adopted with the window size roughly equaling half of a year. We first report the rolling-window version of total dependency degrees in Fig. 5. Although the average value of the total dependency is 0.282, which is very close to the static full sample result (0.266), Fig. 5 shows substantial variations of the overall dependency over time. The value can go as high as 0.9 during the 2008 global financial crisis period and drops dramatically to lower than 0.1 between 2014 and 2015. It is intuitively not surprising to find that dependency increases during the financial crisis period as a result of the higher level of systemic risk in general markets, which is consistent with Zhang and Broadstock (2018), among others, arguing that systemic risk in the global commodity markets increased due to the global financial crisis. The authors also find that the trend of systemic risk has been decreasing in the recent years, which can provide plausible explanations to another non-negligible trend shown in our result: the interdependency in the system drops considerably during certain recent periods, such as 2014–2015 and 2017–2018. This declining dependency may be attributed to the dominating role of crude oil in this system. Crude oil prices have recently experienced dramatic changes, propelled by reasons that are quite different from those in the 2008 global financial crisis episode. We can also link these phenomena to the global business cycle. From Fig. 5, we can see a clear cycling pattern of the total network dependency over time. It is expected to observe a countercyclical dependency in commodity markets that moves in an opposite direction to the overall state of the economy. As a result, the interdependency should increase during times of crises and recession of the global economic condition, while decreasing in calm and expansion times. Kilian and Zhou (2018) document that global real economic activity has experienced sharp decreases during the 2008 global financial crisis, 2012–2013 and 2015–2016, which correspond to the periods of higher levels of total dependency in the commodity market shown in Fig. 5. When the global real economic condition recovers, the level 125

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Fig. 4. Core dependency network (top 20% pairwise net dependency).

Fig. 5. Total dependency degree of the time-varying directed dependency network. 126

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of dependency falls. Further results on individual commodities using rolling windows are reported in Table 3. It shows the average value and the standard deviation of From, To and Net measures for each commodity. The general rankings are the same as in the static analysis. On average, crude oil, heating oil and copper still remain to be the top three To and Net contributors in the system. While the overall dependency in the system is found to vary over time, it is expected to see the time varying patterns also in each commodity’s case. To be concise, we only plot the three leading commodities and discuss their time varying Net dependency (see Fig. 6). Seen in Fig. 6, all the three series experience substantial variations over time. Their patterns of fluctuation during the sample period are very similar, although crude oil clearly plays a leading role in most of the rolling windows, staying above the other two for most of the time. Heating oil follows a similar trend to crude oil, though exhibiting generally lower impact on the dependency network. As the third important commodity, copper generally makes less net contribution to the system relative to the other two. It, however, occasionally takes the lead. For example, copper makes obviously higher net contribution to the dependency network between mid-2012 and late 2013, during which the copper market is crashing. 5.2. Centrality network To test the robustness of our findings based on the dependency network analysis, we then use standard correlation estimation to construct a centrality network. Following the MST method introduced in section 3.2, we obtain the following results. 5.2.1. Static analysis of the full sample The full sample static results are shown in Fig. 7 using the MST approach. The graph clearly illustrates several interesting patterns of the network structure. First, commodities within the same categories (namely, energy, metal and agricultural commodities) tend to position together and form clusters in the network. This clustering effect in the network implies that commodities of the same categories exhibit stronger tendency of price synchronization and co-movements. Second, copper, heating oil and corn are identified as the three local core commodities for each commodity category, and are marked in blue. This finding is partly consistent with the results from the dependency network analysis that copper, heating oil and crude oil are the top three contributors of net dependency. Third, crude oil is shown to be a most special commodity, as it plays a critical role of connecting together two main clusters, energy and metal. This result is robust, as crude oil is also recognized as the largest node in the dependency network, indicating its role as the most important dependency contributor. For individual commodities, crude oil acts as a bridge to link together the two most connected commodities (copper and heating oil). It is thus reasonable to argue that crude oil is the most important component enabling the formation of this whole network. This especially important linking role of crude oil in the network is in line with the general belief that crude oil is the most important commodity in the global commodity markets, thus providing reliable evidence to the justification of the highest weight set for crude oil in the CRB index. 5.2.2. Dynamic analysis with rolling windows Similar to the dependency network analysis, we also adopt a rolling-window approach so as to evaluate the time varying nature of the MST. Using the rolling normalized tree length, we are able to construct the system integration index, which is essentially the inverse ofLðtÞ. It is bounded between 0 and 1 and plotted in Fig. 8. The system integration index based on the time-varying MST and the total Table 3 Summary statistics of the rolling-window dependency network. From

Cocoa Coffee Corn Cotton Lean Hogs Live Cattle Orange Juice Soybean Sugar Wheat Aluminum Copper Gold Nickel Silver Crude Oil Heating Oil Natural Gas Gasoline

To

Net

Mean

Std.dev

Mean

Std.dev

Mean

Std.dev

0.224 0.273 0.299 0.264 0.114 0.171 0.105 0.355 0.242 0.271 0.356 0.374 0.314 0.332 0.340 0.394 0.391 0.158 0.385

0.201 0.241 0.247 0.218 0.112 0.204 0.137 0.249 0.222 0.243 0.245 0.254 0.216 0.235 0.228 0.259 0.260 0.180 0.259

0.127 0.233 0.292 0.196 0.041 0.102 0.031 0.426 0.174 0.224 0.382 0.509 0.283 0.303 0.428 0.602 0.538 0.056 0.416

0.136 0.234 0.271 0.200 0.048 0.156 0.058 0.334 0.239 0.245 0.281 0.377 0.268 0.247 0.289 0.401 0.381 0.103 0.359

0.098 0.040 0.007 0.068 0.073 0.069 0.073 0.071 0.068 0.047 0.026 0.134 0.030 0.029 0.088 0.208 0.147 0.102 0.031

0.077 0.061 0.057 0.062 0.075 0.077 0.090 0.135 0.093 0.068 0.092 0.151 0.143 0.085 0.115 0.157 0.141 0.091 0.133

Note: The windows size is set to be 130 daily observations (approximately half year). 127

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Fig. 6. Time-varying Net dependency for the top three contributors.

dependency degree for the dependency network are not exactly the same by definition or value, but they are both informative indicators reflecting the overall level of connectedness within the system. Generally, the patterns revealed in Fig. 8 are very similar to those in Fig. 5 using the total dependency degree indicator. Remarkably higher levels of co-movement and dependency are observed during the 2008 global financial crisis period. The level of co-movement starts falling from 2012. A temporary increase in the index is then seen during 2015 and 2016. Again, we can connect the dynamics of system integration (the overall level of co-movement) in commodity markets with the global real economic conditions and trends (such as Kilian & Zhou, 2018) to find possible explanations.

Fig. 7. Static centrality network in the global commodity markets. 128

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Fig. 8. System integration index of the centrality network over time.

Another technique we use in the analysis of time-varying centrality network is to calculate a multi-step survival ratio (as given by equation (4)), which is essentially a measure of MST stability. Shown in Fig. 9, we report the survival ratios of the estimated MSTs computed under 2-, 6-, 12-, and 24-step. Average survival ratios decrease with the increase in the number of steps, suggesting a diminishing stability of the network with an extended time span. The general information here supports a reasonably stable system where links survive for a relatively long period. Further information about the time-varying systemic importance of the commodity series in centralities networks can be seen in Table 4. It is worth mentioning again that the evaluation of one variable’s systemic importance is multi-dimensional, based on three distinctive centrality measures. Degree, closeness and betweenness have different meanings and do not necessarily measure the same dimension of centrality in a network. In general, a systemically important variable is expected to have higher degree centrality, lower closeness centrality2 and higher betweenness centrality. Table 4 reports the mean and standard deviation of these centrality measures for each of the 19 commodities during the sample period. Shown in Table 4, the top three commodities ranked by degree centrality from high to low are copper, soybean, and heating oil. This ranking is roughly consistent with the most central nodes found in Fig. 7. The average degree centrality for crude oil is 2.627, slightly lower than the top three and ranked the fourth among all. For closeness centrality, the top three commodities (from low to high in magnitudes) are copper, crude oil and soybean, while the top three commodities by betweenness centrality are sequentially copper, soybean and crude oil. The importance of crude oil and copper is reconfirmed by these centrality measures. It is, however, necessary to account for why soybean is identified as one of the core commodities here. First, soybean is indeed one of the most important commodities as shown in Fig. 3 and Table 3. It plays a critical role in the static dependency network shown in Fig. 3, and is found to be the only net positive contributor to the dependency network within the agricultural category in Table 3. Second, the construction of the centrality network is based on correlations. We have seen that commodities of the same category are more likely to display price co-movements. Given that our sample includes more agricultural commodities (10 out of 19) than other types, it is not surprising to find that the most important agricultural commodity also plays a leading role in the overall centrality network. Nonetheless, results here only show simple averages of all windows. We can still confidently claim that crude oil is one of the most systemically important commodities in the system.

5.3. Aggregation over commodity categories Motivated by the presence of a clustering effect discussed above, this section further explores the dependency patterns for each main category of commodities. We divide the 19 commodities into three categories accordingly, which are energy (four commodities), metal (five commodities) and agricultural (ten commodities) products. For brevity, we only report the time-varying Net dependency measures of each category for the rolling-window dependency networks in Fig. 10.3 In Fig. 10, the time-varying pattern of the Net measure for each category of commodities can be easily spotted, and there are significant variations in the earlier part of the sample period, after which Net dependency of all three categories is shown to be smooth. Another key graphic indication is that agricultural products in general form a net receiver category. Despite strong variations over time, the aggregate Net value of this category remains negative in almost all windows. The opposite contribution is made by energy

2 3

In practice, people also use the inverse of equation (6) to measure closeness, which has essentially the same meaning as our definition. Other dependency measures are not reported here for brevity but are available upon request. 129

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Fig. 9. Multi-step survival ratios of the time-varying centrality networks.

commodities as a whole, whose net contribution to the system is consistently positive over time. Metal commodities make positive net contributions for most of the time, albeit lower in magnitudes than energy commodities. The exception is between 2013 and 2014, when the energy products as a whole no longer exhibits positive net dependency. 6. Conclusions This paper applies a newly developed dependency network approach to studying the interdependency across the returns of 19 commodity futures, which are the key components of the CRB index. Some interesting and dynamic results are found. The main results from our dependency network analysis are very intuitive. From both static and dynamic analyses, three commodities, crude oil, heating oil and copper, are found to play critical roles in the commodity network. These findings confirm the conclusions in several existing studies, for example, Zhang and Broadstock (2018) who also claim that crude oil is important in the global commodity markets and its price movements have significant spillover effects on other main commodity prices. Albeit these common findings, our results convey quite different messages from the extant studies, in that the pivotal role of crude oil in our dependency network does not necessarily suggest that crude oil price movements can trigger co-movements or be used to explain the movements of other commodities. Instead, we argue that it merely indicates the key connecting role of crude oil price in linking together the components in the general commodity market to form an interconnected network. Our main results are further confirmed using graph theory. An MST is constructed to reveal the centrality of the commodity market. Table 4 Summary statistics of time-varying centralities. Degree centrality

Cocoa Coffee Corn Cotton Lean Hogs Live Cattle Orange Juice Soybean Sugar Wheat Aluminum Copper Gold Nickel Silver Crude Oil Heating Oil Natural Gas Gasoline

Closeness centrality

Betweenness centrality

Mean

Std.dev

Mean

Std.dev

Mean

Std.dev

1.193 1.933 2.299 1.535 1.318 1.707 1.062 2.793 1.516 1.688 1.786 3.488 1.680 1.670 2.423 2.627 2.732 1.090 1.460

0.415 0.978 0.783 0.730 0.527 0.623 0.247 1.176 0.764 0.768 0.881 1.176 0.963 0.906 0.878 1.132 0.925 0.292 0.690

90.278 84.642 78.300 84.820 98.889 93.514 96.891 65.456 87.859 86.126 77.589 63.466 78.989 78.427 70.356 64.611 67.836 94.250 78.069

18.724 18.693 16.319 18.881 19.126 21.260 21.070 17.962 18.878 16.486 16.695 15.307 17.139 15.867 14.702 16.594 15.261 20.343 15.349

0.032 0.133 0.212 0.090 0.035 0.081 0.009 0.378 0.089 0.122 0.127 0.416 0.136 0.127 0.245 0.353 0.287 0.012 0.088

0.080 0.156 0.159 0.134 0.062 0.082 0.043 0.190 0.140 0.149 0.159 0.187 0.201 0.185 0.186 0.211 0.184 0.043 0.150

Note: The window size is set to be 130 daily observations. 130

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Fig. 10. Time varying Net dependency for three commodity categories.

The centrality network in the form of a static MST clearly illustrates a clustering effect of commodities, meaning that commodities in the same categories are more likely to co-move with each other. While heating oil, copper and corn are found to be the centers of their own clusters, crude oil’s pivotal role in connecting together the clusters and forming the network still remains robust. Some further conclusions can be drawn from the dynamic analysis on network dependency, summarized as follows. First, we do confirm a strong time-varying pattern existing in the dependency network of the global commodity markets. Second, the level of the total dependency in the global commodity markets is closely linked with the global business cycle (for example, Kilian & Zhou, 2018), demonstrating a countercyclical pattern. Third, crude oil, or more generally energy commodities as a whole, keeps playing a net contributor role, making positive net contributions to the commodity network, whereas agricultural products are shown to be the net receiver for most of the sample period. Furthermore, given that correlation is the one of the main proxies relevant for portfolio construction and investment decision making, our results based on this approach generate useful practical implications to market participants. Savvy investors should be aware that in the presence of strengthening interdependence across commodity markets, it becomes increasingly difficult to effectively reduce portfolio risks by holding diversified commodity categories, consistent with the argument by Pindyck and Rotemberg (1990) that the prices of raw commodities have a persistent tendency to move together. Similar to the tight cross-market dependency, commodities within a category also present a high degree of co-movement. Holding diversified commodities within one category thus offers little help for portfolio hedging. Investors seeking for potential investment opportunities in the commodity markets should watch closely the dynamics of the dependence across these markets and adjust their investment strategies and asset allocations accordingly. Alternatively, diversified assets such as cryptocurrencies can be incorporated in the portfolio to improve the possibility and efficiency of risk hedging. This study well depicts the dynamic and complex relationships among commodity returns. Notwithstanding, there are a few questions worth further investigation. One possible direction for future research is to study the risk spillover relationship among commodity markets under extreme conditions. With increasing uncertainties of the macroeconomy and market environment, risk has become a main feature of the commodity market system in the post-financial crisis era. Investigating risk and volatility spillovers across commodity markets should benefit potential users and provide investors with further practical guidance. CRediT authorship contribution statement Fei Wu: Methodology, Investigation, Writing - original draft. Wan-Li Zhao: Data curation, Software, Visualization. Qiang Ji: Conceptualization, Methodology, Writing - review & editing. Dayong Zhang: Supervision, Validation, Writing - review & editing. Acknowledgements Supports from the National Natural Science Foundation of China under Grant No. 71974159, No. 71974181, No. 71774152, Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant: Y7X0231505), 111 Project B16040 are acknowledged. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.iref.2020.01.004.

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