International Review of Financial Analysis 72 (2020) 101562
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International Review of Financial Analysis
journal homepage: www.elsevier.com/locate/irfa
Identifying the comovement of price between China's and international
crude oil futures: A time-frequency perspective
Xiaohong Huanga,b, Shupei Huanga,b,
a
b
T
⁎
School of Economics and Management, China University of Geosciences, Beijing 100083, China
Key Laboratory of Carrying Capacity Assessment for Resource and Environment, Ministry of Natural Resources, Beijing 100083, China
ARTICLE INFO
ABSTRACT
Keywords:
Comovement
Crude oil futures prices
Wavelet
Complex network
Identifying the comovement of price between China's and international crude oil futures can help different
market players gain a deeper understanding of the world crude oil market. This paper uses the wavelet (wavelet
coherence and phase) methods to study the comovement characteristics at different time scales from three
aspects (the strength of comovement, the direction of comovement and the lead-lag relationship of price fluc­
tuation) and uses the complex network method to explore the evolutionary characteristics of the comovement
with time. We use the daily closing prices of WTI, Brent and China's crude oil futures (INE) as sample data. The
results show that the comovement between INE and international crude oil futures is extremely different from
that between other international crude oil futures, and the comovement at different time scales is also different.
Compared with the comovement between WTI and Brent crude oil futures, the comovement strength between
INE and international crude oil futures is weak and the comovement direction is unstable. China's crude oil
futures price fluctuation also tends to lag behind that of international crude oil futures. Compared with the longterm, the short-term comovement strength is weaker, the comovement states are more diverse and the transition
between comovement states is more complex. Moreover, during the evolution of time, some comovement states
have a higher probability of occurrence and they are also more stable than others. These findings are helpful for
policy makers to design policies and for investors to make investment decisions.
1. Introduction
China's status in the crude oil market has never been as significant
as it is now, because it is not only the world's largest importer of crude
oil but it also recently launched China's first crude oil futures market,
INE (China's first open futures, settled in RMB). So far, the daily trading
volume of China's crude oil futures has surpassed that of Dubai's crude
oil futures, and it has become the most traded crude oil futures contract
in Asia. Meanwhile, research into China's crude oil futures, such as
volatility characteristics (Ji & Zhang, 2019; Wang, Ye, & Wu, 2019;
Yang & Zhou, 2020), has also attracted the attention of more and more
scholars, investors and regulators.
However, one of the important ways to help different market
players understand the world crude oil market, the price comovement
between China's and international crude oil futures has not been stu­
died by scholars. The comovement between other crude oil markets has
been identified by many scholars, such as those between Brent, OPEC
and WTI (Beritzen, 2007; Coronado, Fullerton, & Rojas, 2018; Gulen,
1999; Kang & Yoon, 2013). As an important part of the crude oil futures
⁎
market, what are the comovement characteristics between China's and
international crude oil futures? What are the evolutionary character­
istics of comovement? We still do not know. Answering these questions
can help people better understand the world crude oil market and
provide some inspiration for different market players.
For regulators, by identifying the comovement characteristics and
its evolutionary features of China's and international crude oil futures,
it is helpful to reveal the impact of foreign crude oil markets on China's
crude oil market and the response of China's crude oil market to ex­
ternal shocks. It can also provide some inspiration for policymakers to
plan the future development of the crude oil futures market. For in­
vestors, since the crude oil futures market has the functions of arbitrage
and hedging (Chan & Woo, 2016), the answer to this question is useful
to make investment decisions. Therefore, it is necessary to study the
comovement between China's crude oil futures and international crude
oil futures.
For the identification of comovement features, many methods can
be used, such as Pearson correlation, econometric model (Wang et al.,
2020), and wavelet. However, based on the following reasons, this
Corresponding author at: School of Economics and Management, China University of Geosciences, Beijing 100083, China.
E-mail address: hspburn@163.com (S. Huang).
https://doi.org/10.1016/j.irfa.2020.101562
Received 29 July 2019; Received in revised form 25 June 2020; Accepted 20 July 2020
Available online 12 September 2020
1057-5219/ © 2020 Elsevier Inc. All rights reserved.
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
paper studies the comovement by the wavelet method. First, the focus
of different entities in the crude oil futures market is different. For
example, regulators may be more concerned about the long-term sta­
bility of the market and investors may care more about short-term
market price fluctuations (Huang, An, Huang, & Jia, 2018). In addition,
the correlation between the two time series may vary at different time
scales (Ji, Bouri, Roubaud, & Kristoufek, 2019; Pal & Mitra, 2019;
Zivkov, Balaban, & Djuraskovic, 2018). Pearson correlation and
econometric methods can only obtain information from the perspective
of time domain and cannot obtain it from the frequency domain. The
wavelet method, an improvement of the Fourier method, can analyze
the relationship between two time series from the perspective of the
time-frequency joint domain. It clearly reveal the variation character­
istics of time series at different time scales (Pal & Mitra, 2019). Second,
the econometric method requires the time series be a stationary se­
quence, but financial time series are mostly nonstationary. The wavelet
method does not require the stability of the time series. This method
was originally used primarily in noneconomic areas but has been
widely used in economic and financial fields in recent years (Huang,
An, Gao, & Sun, 2017; Huang, An, Gao, Wen, & Hao, 2017; Tweneboah
& Alagidede, 2018; Xu & Kinkyo, 2019). Therefore, based on the many
advantages of the wavelet, this paper mainly uses this method to
identify the comovement. Price comovement occurs when one's price
changes coincidentally with another's price. Therefore, we identify the
comovement characteristics in three aspects: the strength of comove­
ment, the direction of comovement, and the lead-lag relationship of
price fluctuation. We use wavelet coherence to analyze the strength of
comovement and wavelet phase to analyze the direction of comove­
ment and the lead-lag relationship of price fluctuation.
Through the wavelet method, the comovement characteristics at
different time scales can be identified. However, this is not enough to
help different market participants truly understand the comovement
between China's and international crude oil futures market. The
comovement between two prices is not static on each day (Ji, Bouri, &
Roubaud, 2018), so what are the characteristics of the evolution of
comovement over time at different time scales? This is also worth ex­
ploring. Studying it can deeply identify the comovement features and
help investors and regulators understand the development of China's
crude oil futures. Therefore, in order to explore the evolutionary
characteristics of comovement, we should introduce an additional
method.
At a certain time scale, the comovement state may change moment
by moment, and this changing comovement state constitutes a complex
network. Therefore, we introduce the complex network approach to
study the evolution of comovement. The complex network method has
been widely used in recent years to analyze the evolution of systems
from different perspectives, such as the integration of the financial
market (Ji et al., 2018), the volatility patterns of time series (Liu et al.,
2018), the EU Carbon Trade System (Liu, Gao, & Guo, 2018) and the
time-varying causality of multivariate time series (Jiang, Gao, An, Li, &
Sun, 2017). Through the analysis of network topology features, some
important nodes (Wang et al., 2019; Wang, Gao, An, Tang, & Sun,
2020) and relationships in the network can be identified. The appli­
cation of complex network methods by predecessors provide a more
important reference for this paper.
A complex network consists of nodes and edges. To extract the daily
comovement features as a node, we use different symbols to represent
different comovement features by the coarse graining method, and es­
tablish the comovement matrix. Then, we take the matrix as the node
and the mutual transformation of nodes as edges to establish the net­
work. The coarse graining method can omit the details and highlight
the features, which is very helpful for exploring the evolutionary fea­
tures of comovement (Wackerbauer, Witt, Altmanspacher, Kurths, &
Scheingraber, 1994). This method has been used by many scholars to
explore the evolution of systems (An, Gao, Fang, Huang, & Ding, 2014;
Fang, Gao, Huang, Jiang, & Liu, 2018).
In short, this paper mainly identify the comovement characteristics
at different time scales by the wavelet method, and explore the evolu­
tionary characteristics of comovement with time by the complex net­
work method. The rest of this paper has three sections. Section 2 briefly
introduces the data and methods (wavelet method and complex net­
work method). Section 3 is the results and discussion. The conclusions
based on the analysis are presented in Section 4.
2. Data and methods
2.1. Data and sample statistics
International crude oil futures, represented by WTI and Brent crude
oil futures, are recognized as the international crude oil benchmark
(Scheitrum, Carter, & Revoredo-Giha, 2018). Since China's crude oil
futures were officially listed on March 26, 2018, we use the daily
closing prices of continuous futures contracts1 of the INE, WTI and
Brent crude oil futures markets from March 26, 2018, to June 21, 2019
as sample data. We obtained the data from the Wind database. Because
Wind is the market leader in financial information services industry and
has been hailed as a major provider of Chinese financial information in
China and abroad. Wind's data has been cited in many academic papers
(Gao, Huang, Sun, Hao, & An, 2018; Liu et al., 2019; Liu, Gao, Fang,
et al., 2018). As the trading days of the crude oil futures are not uni­
form, in order to facilitate the analysis, we retained the price in­
formation of only the public trading days, a total of 295 trading days of
data, as shown in Fig. 1.
Table 1 is the basic statistical information of the data. First, the
standard deviation of INE is much higher than that of the international
crude oil futures, while the standard deviations of WTI and Brent crude
oil futures are almost the same. This implies that the price fluctuation of
INE is more violent than that of the international crude oil futures,
while the two international crude oil futures tend to show similar price
fluctuation. This also shows that compared with the international crude
oil futures market, the stability of China's crude oil futures market
needs to be further improved. Second, the value of Jarque-Bera in­
dicates that the price series of WTI, Brent and INE crude oil futures all
do not obey the normal distribution.
Furthermore, we initially identify the comovement characteristics of
China's and international crude oil futures at different time scales. First,
we decompose the three time series (INE, WTI and Brent crude oil fu­
tures price series) on multiple time scales through wavelet transform
method. Considering maximal overlap discrete wavelet transform
method has no special requirement on the length of the time series and
it is widely used by scholars, we use this method to decompose the
series (Li, Qi, Li, & Liu, 2019; Sui, Li, Feng, Liu, & Jiang, 2018). Taking
the small sample size into consideration, we decompose each time
series into seven detail levels and a trend level. Seven detail levels are
respectively associated with the changes of the time series at the time
scales of 2–4, 4–8, 8–16, 16–32, 32–64, 64–128 and 128–256 days. And
the trend level indicates the average behavior of the time series in the
long run, which indicates its time scale is more than 256 days
(Jammazi, 2012). Second, we calculate the Pearson correlation coeffi­
cients between three time series at different time scales to identify the
comovement strength and comovement direction, as shown in Fig. 2. It
can be seen from the figure that except for the time scale of 2–4 days,
the comovement directions of INE, WTI and Brent are the same at other
scales. In addition, the comovement strength of INE-WTI, INE-Brent,
and WTI-Brent increase with increasing time scale, and the comove­
ment strength of INE-WTI and INE-Brent is always smaller than that of
WTI-Brent at any time scale.
1
Continuous contract refers to the contract which is closest to the delivery
month among the contract of the current transaction. Continuous contract will
change with the delivery month.
2
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 1. Daily closing prices of INE, WTI and Brent crude oil futures.
Table 1
Summary statistics for the daily closing prices of INE, WTI and Brent crude oil futures.
INE
WTI
Brent
⁎⁎
⁎⁎⁎
Mean
Median
Max
Min
Std. dev.
Skewness
Kurtosis
Jarque-Bera
468.8115
62.46271
70.42193
468.2
64.38
71.86
590.6
74.67
84.86
342
42.68
50.49
45.0035
7.5455
6.9658
0.3875
−0.457
−0.476
3.123
2.121
2.477
7.5682⁎⁎
19.7669⁎⁎⁎
14.4932⁎⁎⁎
Significance at the 5% level.
Significance at the 1% level.
Fig. 2. The Pearson correlation coefficients of INE-WTI, INE-Brent and WTI-Brent at different time scales.
Third, we calculate the cross-correlation coefficient between three
time series at different time scales to identify the lead-lag relationship
of price fluctuation, as shown in Fig. 3. Cross-correlation coefficient is
considered as a good tool to explain the lead-lag relationship between
two markets (Jammazi, 2012). The cross-correlation coefficient rk of
two time series represents the correlation coefficient between two time
series when the first series leads or lags the second series the k periods.
The type of the lead-lag relationship between two time series is de­
termined by the sign of the k for which the rk is the largest. When k is
positive (negative), it means that the first series leads (lags) the second
series the k periods. The k equals zero, which means two series are
synchronized (Geng et al., 2016; Kydland & Prescott, 1990; Oladosu,
2009). It can be seen from Fig. 3 that no matter what time scale, WTI
and Brent crude oil futures prices are almost synchronous. For INE and
international crude oil futures prices, their original time series and the
long-term trend are synchronous, and there are lead-lag relationships
between them at the level of details and INE crude oil futures price
fluctuation leads WTI and Brent crude oil futures price fluctuations.
The above analysis is the preliminary identification of the comove­
ment between INE, WTI and Brent crude oil futures markets and it only
analyze the comovement characteristics at different time scales over a
period of time. Due to the cross-correlation and Pearson correlation
coefficients cannot identify the comovement characteristics at different
time points, we use wavelet coherence and phase angle to identify the
comovement characteristics at different time points and different time
scales. The following section will briefly introduce the wavelet co­
herence, the phase angle and the method of studying the evolutionary
characteristics of comovement with time (complex network method).
2.2. Methods
The paper discussed the comovement between three time series INE,
3
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 3. The cross-correlation coefficient rk of INE-WTI, INE-Brent and WTI-Brent at different time scales. The abscissa represents the first time series leads or lags the
second time series the k periods, and the ordinate is the value of cross-correlation coefficient.
WTI and Brent crude oil futures. The reason why we researched the
comovement between WTI and Brent crude oil futures is to use this as a
reference to better observe the difference between China's crude oil
futures and international crude oil futures. In this paper, first, we use
the wavelet coherence to analyze the comovement strength of INE-WTI,
INE-Brent and WTI-Brent at different time scales. Then, the wavelet
phase is used to describe the lead-lag relationship of price fluctuation
and the comovement direction between INE, WTI and Brent crude oil
futures. Finally, we use the complex network method to explore the
evolutionary characteristics of comovement over time at different time
scales. We calculate and visualize the values of wavelet coherence and
phase angle by MATLAB, and we draw the complex network diagram
and calculate the topological index of network by Gephi.
The definition of the square of the wavelet coherence coefficient is
the square of the absolute value of the smooth cross wavelet spectrum
normalized by the smooth wavelet power spectrum. The wavelet co­
herence in this paper refers to the wavelet square coherence. The for­
mula is expressed as:
2
R xy
( , s) =
=
1
|s|
t
s
,
s,
R, s
0
(1)
where s is the scaling parameter. By controlling its size, the width of the
wavelet can be amplified or reduced to obtain the information of the
signal in different frequency domains. And τ is the translation para­
meter, which refers to the position of the wavelet moving in time,
through which we can get the information of the signal in different time
domains.
For a time series x(t), its continuous wavelet transform is:
Wx; ( , s ) =
x (t )
1
|s|
t
s
dt
| s 1Wxy; ( , s ) |2
1
s
|Wx; ( , s )|2 s
1
|Wy; ( , s )|2
(4)
where ⟨.⟩ denotes smoothing in time domain and frequency domain
and s−1 is used to convert to energy density. The value range of
Rxy2(τ, s) is [0,1]. More details can be seen in Aguiar-Conraria and
Soares, (2014) and Torrence and Webster, (1999). The wavelet square
coherence represents the correlation coefficient of the two sequences at
the corresponding time and frequency. The larger the value is, the
stronger the comovement between the two time series is (Rua & Nunes,
2009). We divided the value of the wavelet square coherence into dif­
ferent intervals and each interval is represented by different symbols to
characterize the degree of coherence between two time series in dif­
ferent time and scales. The interval is divided as follows:
2.2.1. Wavelet coherence
The basis of the wavelet transform is the wavelet function Ψτ, s(t),
which is obtained by the scaling and the translation of the mother
wavelet function Ψ(t):
, s (t )
(3)
Wxy; ( , s ) = Wx; ( , s ) W y; ( , s )
2
L, Rxy
( , s)
2
Lm , Rxy
( , s)
2
R xy
(
, s) =
M,
2
Rxy
(
, s)
2
Hm , R xy
( , s)
H,
2
R xy
(
, s)
[0,0.2
[0.2,0.4
[0.4,0.6
[0.6,0.8
[0.8,1]
(5)
2.2.2. Phase angle
The lead-lag relationship of price fluctuation and the comovement
direction between two time series can be obtained from the phase
angle.
The phase angle is expressed as:
(2)
where * represents complex conjugate, and by mapping the time series
x(t) into a function of τ and s, the joint distribution information of time
series x(t) in different time domains and frequency domains after wa­
velet transform can be obtained.
For given two time series x(t) and y(t), we can define a cross-wa­
velet spectrum by their continuous wavelet transform:
( , s ) = tan
4
1
I { s 1Wxy; ( , s ) }
R { s 1Wxy; ( , s ) }
(6)
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 4. Detailed process of establishing comovement network. The process of establishing the INE-WTI network at the time scale of 2 days is used as an example.
where ℑ is the smoothed imaginary part and ℜ is the smoothed real
part. More details can be found in Torrence & Webster, (1999). The size
of the phase angle is generally expressed in radians and its value ranges
from −π to π. Its information can be represented by arrows in the
wavelet coherence diagram. When the range of phase angle is different,
the direction of the phase arrow, both the comovement direction and
the lead-lag relationship of fluctuations in time series x(t) and y(t) are
different. For example, when the phase angle belongs to (0, π/2), the
series x(t) and y(t) are in-phase and the fluctuation of series x(t) leads
the fluctuation of series y(t). At this time, the phase arrow is (↗). Inphase indicates that series x(t) and y(t) are positively correlated, and
the comovement direction between x(t) and y(t) is in the same direc­
tion. Anti-phase indicates that series x(t) and y(t) are negatively cor­
related, and the comovement direction between x(t) and y(t) is oppo­
site. The different information displayed by the phase angle in different
ranges is as follows, and we represent them with different symbols.
More details can be seen in Jiang, Nie, and Monginsidi (2017), Das,
Kannadhasan, Al-Yahyaee, & Yoon, (2018), and Vacha & Barunik,
(2012).
S1;
( , s ) = 0; in
phase; x (t ) and y (t ) move together;
phase arrow is
S2;
( , s)
(0,
S3;
( , s)
(
R1;
( , s) =
2
2
); in
, 0); in
phase; x (t ) is leading y (t ); phase arrow is
phase; y (t ) is leading x (t ); phase arrow is
or ; anti
phase; x (t ) and y (t ) move together;
phase arrow is
( , s) =
R2;
( , s)
(
,
2
); anti
phase; x (t ) is leading y (t );
phase arrow is
R3;
( , s)
( ,
2
); anti
phase; y (t ) is leading x (t );
phase arrow is
N1;
( , s) =
2
; x (t ) is leading y (t );
phase arrow is
N2;
( , s) =
2
; y (t ) is leading x (t ); phase arrow is
(7)
2.2.3. Establishing comovement matrices based on wavelet coherence and
phase symbols
By calculating the wavelet coherence and phase, we can get the
5
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 5. Wavelet coherence of INE-WTI, INE-Brent and WTI-Brent. The horizontal and vertical axes respectively represent time and time scale (period or frequency).
Each wavelet coherence value corresponds to a specific time scale and time point.
wavelet coherence matrix and the phase matrix respectively. The row of
the matrix represents the wavelet coherence value or phase value at
each time point in a certain time scale (frequency). The column of the
matrix represents the wavelet coherence value or phase value at each
time scale in a certain time point. In other words, each wavelet co­
herence value and phase value have the corresponding time point and
time scale. Based on the matrices of wavelet coherence and phase, we
define the matrix, which contains wavelet coherence information and
phase information, as comovement matrix:
Lt = [Rt t ]
3. Results and discussion
3.1. Identification of the comovement characteristics at different time scales
3.1.1. Identification of the comovement strength
Through wavelet coherence, small differences of comovement
strength between two crude oil futures prices at different time points
and time scales can be found (Huang et al., 2018). We visually present
the wavelet coherence matrices of three pairs of time series in Fig. 5.
The value of wavelet coherence ranges from 0 to 1. The larger the value
is, the redder the color is.
Comparing the wavelet coherence graphs of three pairs of time
series, we find that Fig. 5(a) and (b) are very similar, while Fig. 5(a) and
(b) show great differences from Fig. 5(c). This indicates that there is a
close connection between international crude oil futures (WTI and
Brent crude oil futures), while the link between INE and the interna­
tional crude oil benchmark is not. We also find that the INE-WTI and
INE-Brent comovement strength is always far less than the WTI-Brent
comovement strength at any time scale. This finding is consistent with
our findings of identifying the overall comovement strength. The dif­
ferences in comovement strength may be due to the differences in the
number of international investors. The comovement strength between
WTI and Brent crude oil futures is extremely strong, probably because
there are many international investors participating in the WTI and
Brent crude oil futures markets due to they have already become the
world's recognized crude oil benchmarks. The comovement strength
between China's and international crude oil futures is relatively low,
possibly due to the small number of international investors in China's
crude oil futures market. There are two main reasons why there are a
few international investors in China's crude oil futures market. The first
reason is that the listing time of China's crude oil futures is too short
compared with international crude oil futures, which has led many
(8)
Rtandϕtrespectively denote the wavelet coherence information and
phase information at time t in a certain time scale and they are re­
spectively represented by the defined symbol. The comovement ma­
trices of INE-WTI, INE-Brent and WTI-Brent are LIW, LIB and LWB re­
spectively.
2.2.4. Establishing a complex network based on comovement matrices
To analyze the evolutionary characteristics of the comovement be­
tween China's and international crude oil futures price, we introduced a
complex network method. We take the comovement matrix as the node,
the mutual transformation of nodes as edges and the number of times of
transformation between nodes as weights to establish a directed
weighted network. In this way, we can explore the evolutionary char­
acteristics of comovement over time at a certain time scale. We have
295 days of daily data, so the number of total weights is 294. Moreover,
since the comovement matrix at time point t may be consistent with the
comovement matrix at time point t + 1, the number of nodes will be
less than or equal to 295. The network construction process is shown in
Fig. 4.
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International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 6.
As depicted in Fig. 6, the INE-WTI and INE-Brent comovement di­
rections are not stable on a time scale of 2 to 8 days, and they often
show reverse comovement during the whole sample period. This im­
plies that the fluctuation period of comovement directions between INE
and international crude oil futures is probably one week. In a short
cycle of the week, the price change of China's crude oil futures is af­
fected by not only the price changes in WTI and Brent crude oil futures
but also by the exchange rate, domestic policies and investor behavior,
leading to reverse comovement between China's and international
crude oil futures. The INE-WTI and INE-Brent comovement directions
gradually stabilize and are always in the same direction with the time
scale from 8 days to more than 64 days. The above finding is almost
consistent with our previous analysis of the overall comovement di­
rection.
We can also see from Fig. 6 that the WTI-Brent comovement di­
rection is more stable than the INE-WTI and INE-Brent comovement
directions at any time scale, and their prices always tend to move in the
same direction during the sample period. This once again proves that
the relationship between INE and international crude oil futures is not
as close as that between international crude oil futures. Besides, the
three pairs of time series tend to move in the same direction. The reason
is they all represent the prices of the same type of goods (future crude
oil price) and the prices of the same type of goods tend to move in the
same direction.
international investors to adopt a wait-and-see attitude to avoid risks.
The second reason is INE's trading process is so complicated that pre­
vent international investors from entering the market. For example,
when international investors want to trade, they must exchange for
RMB. Furthermore, different financial environments and financial sys­
tems may also lead to different comovement strength. WTI and Brent
crude oil futures are respectively listed on the New York Mercantile
Exchange and London Intercontinental Exchange, and the United States
and the United Kingdom are among the earliest developed countries.
Compared with their financial market, China's financial environment
and financial system may be not mature enough. Therefore, from a
macroscopic point of view, this also leads to weak comovement
strength between China's and international crude oil futures.
The INE-WTI and INE-Brent comovement strengths fluctuate vio­
lently in the high-frequency range from 2 to 32 days during the entire
sample period, while the WTI-Brent comovement strength fluctuates
violently from 2 to 16 days. As the time scale increases, the fluctuations
of comovement strength become less dramatic. This means that the
fluctuation period of the comovement strength between INE and in­
ternational crude oil futures is within one month, while that between
international crude oil futures is half a month. And when the time scale
is longer than one month and half a month, respectively, the comove­
ment strength will gradually stabilize with the increase of the time
scale. Why would such a phenomenon happen? We infer that in the
short term, the price of crude oil futures is prone to change due to
different factors, such as political factors and speculation (Hamilton,
2009), and due to the gradual global integration, price fluctuation in
one market will soon be transmitted to another market through the
trading behavior of investors (Huang et al., 2018). Therefore, the
comovement strength will fluctuate with the fluctuation of market
prices and the impact of trading behaviors in the short term. In the long
run, according to the theory of economics, the price is ultimately de­
termined by the supply and demand of the market, and there is almost
no violent fluctuation in the supply and demand of crude oil in various
regions of the world (Huang et al., 2018). Therefore, the price of crude
oil futures will not have violent fluctuations in the long run and the
comovement strength gradually stabilizes as time scale increases.
Moreover, we can see that the WTI-Brent comovement strength is more
stable than the INE-WTI and INE-Brent comovement strength at any
time scale. This may be because whether in the short term or long term
(as opposed to short term), the WTI-Brent comovement strength is not
easily affected by various factors, while the comovement strength be­
tween China's and international crude oil futures is likely to change due
to various factors.
In addition, we find that as the time scale increases, the comove­
ment strength gradually increases. This finding is consistent with our
previous analysis of the overall comovement strength. The reason why
the comovement strength will increase with the increase of time scale is
that the longer the time scale, the higher the degree of market in­
tegration, and the more connections between the crude oil futures
markets (Pereira, Ferreira, Silva, Miranda, & Pereira, 2019).
In summary, the comovement strength between China's and inter­
national crude oil futures is different from the strength between other
international crude oil futures markets. Compared the comovement
strength between other international crude oil futures markets, the
comovement strength between China's and international crude oil fu­
tures markets is relatively weak and unstable. Besides, the comovement
strength at different time scales is also different. As the time scale in­
creases, the comovement strength gradually increases and stabilizes.
3.1.3. Identification of the lead-lag relationship of price fluctuation
Based on the phase matrices, we also identify the lead-lag re­
lationship of the INE-WTI, INE-Brent and WTI-Brent price fluctuation at
different time points and time scales, and the results are shown in
Fig. 7. There is no case where the phase angle is exactly equal to 0, −π
or π. Therefore, the price fluctuation of each pair of sequences has a
lead-lag relationship.
From Fig. 7, we can see that when the time scale ranges from 2 days
to 32 days, the lead-lag relationship among the INE, WTI and Brent
crude oil futures prices changes frequently. In addition, when the time
scale increases from 32 days to more than 64 days, the lead-lag re­
lationship gradually stabilizes and the INE price fluctuation usually lags
behind the WTI and Brent crude oil futures price fluctuations. This
implies that the period of the volatility of the lead-lag relationship is
about one month. There are two possible reasons to explain why the
INE price fluctuation usually lags behind the WTI and Brent crude oil
futures price fluctuations. The first reason is that WTI and Brent crude
oil futures are both international benchmarks with great influence.
When the prices of WTI and Brent crude oil futures are changed by
various factors, it will cause the INE price change. However, INE is not
an international benchmark for crude oil futures and the time to market
is much shorter than WTI and Brent crude oil futures. Therefore, INE
price changes tend not to affect the WTI and Brent crude oil futures
prices. The second reason is that China is the largest importer of crude
oil, which made China's crude oil futures prices largely affected by the
price of international crude oil benchmarks Wang, Ye, & Wu, 2019.
Moreover, we found that the price volatility of WTI crude oil futures
usually precedes that of Brent crude oil futures. There are also two
possible reasons to explain this phenomenon. First, from the perspective
of demand, due to the shale oil revolution, crude oil imports by the US
have decreased, weakening the impact of the external environment on
the WTI crude futures prices. Second, from the perspective of supply,
the increase in US crude oil exports has increased the impact on the
external environment.
The above finding is completely inconsistent with what we have
obtained from our overall analysis of the lead-lag relationship. This is
because the cross-correlation coefficient only identifies the lead-lag
relationship between two time series over a period of time, while the
phase angle can identify and find small differences of the lead-lag re­
lationship at different time points.
3.1.2. Identification of the comovement direction
In this paper, we identify the comovement direction at different
time points and time scales by phase matrices. At different time points
and time scales, the comovement directions of three pairs of time series
are all either in the same direction or the reverse direction. We visually
show the INE-WTI, INE-Brent and WTI-Brent comovement directions in
7
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 6. INE-WTI, INE-Brent and WTI-Brent comovement directions at different time scales. Yellow indicates that the comovement direction is the same, and blue
represents the reverse direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
performed in the form of 2n and the maximum scale of continuous
wavelet is 101 due to the small amount of sample data, we respectively
analyzed the evolution of comovement characteristics over time at the
time scales of 2, 4, 8, 16, 32, 64, and 101 days. In Fig. 8, we show part
of the network diagram. The rest of which we will put in the appendix
(Fig. 11). It can be seen that at any time scale, the network is not a fully
connected network. This means that the mutual transformation be­
tween nodes is not completely random but has some preference. To
better analyze the evolutionary characteristics of price comovement, we
analyzed the various indicators of the network, including the number of
nodes, the out-degree, the weighted degree and the weight of edge.
3.2. Exploration of the evolution of comovement characteristics at different
time scales
In the above, we analyzed the characteristics of comovement from
the three aspects. At different time points and time scales, the
comovement characteristics are constantly changing. However, the
above methods mainly analyze the comovement characteristics from
different time scales rather than different time points. This is a vertical
analysis rather than a horizontal analysis. To research the evolution of
comovement characteristics over time at different time scales, we map
the comovement features at different time points to a network at a
certain time scale.
The calculation of wavelet coherence and phase is based on a con­
tinuous wavelet, which means its scale is continuous. However, a dis­
crete wavelet can extract the characteristics of the time series better
than a continuous wavelet and they have a corresponding relationship
on the scale (Huang, An, Gao, Wen, & Jia, 2016; Wang, Liu, Huang, &
Lucey, 2019). Since the scale transformation of discrete wavelets is
3.2.1. Identification of the diversity of comovement states and the
complexity of linkage of comovement states
We counted the total number of nodes, the total out-degree and the
average out-degree of these 21 networks, as shown in Fig. 9.
The total number of nodes represents the diversity of comovement
states. In the networks of INE-WTI, INE-Brent and WTI-Brent, as the
Fig. 7. Lead-lag relationships of the INE-WTI, INE-Brent and WTI-Brent price fluctuations at different time scales.
8
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
3.2.2. Identification of the comovement states with high probabilities of
occurrence
The network established in this paper is a weighted network. The
weight of an edge represents the number of times that two nodes
convert between each other. The weighted degree of a node is the sum
of weights of all edges of this node. This indicator not only measures the
degree of the node but also measures the weight of the edge between
the node and its neighbors. Through this indicator, we can identify the
comovement matrices which have high probability of occurrence in the
network. The formula for calculating the weighted degree is
time scale increases from 2 days to 101 days, the number of nodes
shows a downward trend and it is only one at the time scale of
101 days. This also shows that as the time scale increases, the
comovement characteristics of the three pairs of time series gradually
stabilized. Excluding the time scale of 64 days, the number of nodes in
the WTI-Brent network is smaller than that in the INE-WTI and INEBrent networks at the same time scale. This also indicates that the
comovement between WTI and Brent crude oil futures is more stable
than that between China's and international crude oil futures.
The out-degree of a node is the number of edges from this node to
other nodes and here reflects how many other states that this
comovement state can transform. It measures the complexity of the
linkage of comovement states. The greater the total out-degrees and the
average out-degree are, the more complex the linkage of the comove­
ment states in the network are.
From Fig. 9, as the time scale increases, both the total out-degrees
and the average out-degree of the INE-WTI, INE-Brent and WTI-Brent
networks have a decreasing trend, which indicates the linkage of
comovement states in the network becomes simpler with the expansion
of the time scale. This means that in the short term, the transition be­
tween comovement states is diverse, while it is relatively fixed in the
long term (as opposed to short-term). This also implies that the market
volatility is more dramatic and the investment risk is greater in the
short-term than those in the long-term.
WDi =
Wij
(9)
where j represents the set of all neighboring nodes of node i and Wij
represents the number of times that node i and node j are converted
between each other.
In Fig. 8, the greater the weighted degree of the node is, the larger
the node volume is. We find that in each network, there are some nodes
that are significantly different in size from other nodes. This shows that
there are some nodes in the network whose probability of occurrence is
significantly larger than that of other nodes during the sample period.
To verify our conclusions, we calculated the cumulative weighted de­
gree distribution of nodes in the network. We sorted the weighted de­
gree of nodes in descending order. By taking the proportion of the cu­
mulative number of nodes to the total number of nodes as the abscissa
Fig. 8. The INE-WTI, INE-Brent and WTI-Brent networks at the time scales of 2, 4 and 8 days. The remaining networks at the time scales of 16, 32, 64 and 101 days
are placed in the appendix (Fig. 11).
9
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 9. The total number of nodes, total out-degree and average out-degree of the networks.
and the proportion of the cumulative weighted degree to the total
weighted degree as the ordinate, we depicted the distribution as shown
in Fig. 10.
We find that in each network, approximately 20% of the total nodes
occupied more than 40% of the total weighted degree. In addition, the
slope decreases with the increase of time scale in each distribution.
These phenomena indicate that some nodes do have a higher prob­
ability of occurrence than others during the sample period.
To find the characteristics of these nodes with high probabilities of
occurrence, we list the top 50% of the nodes based on the weighted
degree, as shown in Table 2.
In all WTI-Brent networks, the symbol of the top 50% nodes is
usually H and S. This also proves that the price comovement between
the two international crude oil benchmarks is strong and that their
prices always move in the same direction. It shows that the price trend
of international crude oil futures is extremely similar at any time. When
one's price rises or falls, the other's price will adjust quickly.
In addition, at the time scale of 2, 4, 8, 16 and 32 days, the co­
herence symbol of the top 50% nodes in the INE-WTI and INE-Brent
networks is always Hm, M or H, and the phase symbol gradually
changes from R to S with the increase of time scale. These also indicate
that the INE-WTI and INE-Brent comovement strengths are weaker than
the WTI-Brent comovement strength and that the prices of INE and
international crude oil benchmarks tend to rise and fall together in the
long term.
(Guo, Song, Li, Liu, & Guo, 2019). The correlation coefficient ranges
from −1 to 1. Table 3 shows the Pearson correlation coefficients for
each network. The missing part is because there is only one node in that
network.
The correlation coefficients are all positive, indicating that two
nodes with high probabilities of occurrence tend to link. However, in
the time scale of 2, 4, 16, or 32 days, the value is too small, meaning
that this tendency is not strong in a short cycle of a month and that
changes in the comovement states may exceed people's expectations.
Short-term investment in crude oil futures may have large risk, and riskaverse investors can choose other assets with less risk.
3.2.4. Identification of stable linkage between two comovement states
The weight of the edge reflects the frequency of conversion between
two nodes during the sample period. The larger the edge weight, the
more stable the link between two nodes is. In the Fig. 8, the weight of
the edge is proportional to the width of the edge. It can be seen from
Fig. 8 that some edges in the network are obviously wider than others.
To identify these stable links, we list the top four edge weights of some
networks in Table 4, with the edges of the remaining networks placed in
the appendix (Table 5).
The networks we build are directed networks, and according to the
construction principle of the edge, the source node and the target node
may be the same. We found that in each network, the source nodes and
target nodes of the top four edges are almost identical. This means that
these comovement states are usually stable and will not change in the
short term. More importantly, we found that these stable comovement
states usually have high probabilities of occurrence during the sample
period. Investors can use this feature to develop appropriate investment
strategies.
3.2.3. Whether the comovement states with high probabilities of occurrence
tend to connect
If two nodes with high probability of occurrence tend to link, then
the change in the comovement state is usually in line with people's
expectations. Conversely, if a node with a high probability of occur­
rence tends to connect to another node with a low probability of oc­
currence, then a comovement state may change into an unexpected
state in the evolution of time. We observed in the Fig. 8 that two nodes
with large weighted degree seem to tend to link. To verify this con­
jecture, we calculate the Pearson correlation coefficient of weighted
degree about the source node and target node. The formula is:
r=
n
i=1
n
i=1
(Si
(Si
S )2
S )(Ti
n
i=1
4. Conclusion
In this paper, first, we identified the comovement between China's
and international crude oil futures at different time scales from three
aspects: the strength of the comovement, the direction of the comove­
ment and the lead-lag relationship of the price fluctuation. Then, we
explored the evolutionary characteristics of the comovement at dif­
ferent time scales. Our research has the following main findings. First,
the comovement between China's and international crude oil futures is
very different from the comovement between other international crude
oil futures. Compared with the comovement between WTI and Brent
crude oil futures, its strength is weak and its direction is unstable. And
T)
(Ti
T )2
(10)
where Si and Ti respectively represent the weighted degree of the source
node and target node of edge i, and where S and T respectively re­
present the average weighted degree of the source node and target node
10
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Fig. 10. Cumulative weighted degree distribution of nodes in the networks. To simplify, we only show the distribution of the network that has more than one node.
Table 2
The top 50% nodes ordered by weighted degree. IW, IB and WB respectively
represent INE-WTI, INE-Brent and WTI-Brent. Although the ranking condition
of the INE-Brent network at the time scale of 64 days is not met (only one node
in this network), we still list it in the table.
Rank
Time scale = 2
IW
1
2
3
4
5
6
7
8
9
10
Total
Rank
1
2
3
4
Total
IB
Time scale = 4
WB
HmR2
MR2
LR3
HR2
LmR2
HmR3
MS3
MR3
LmS2
HmR2
HS3
MR2
HS2
HR2
HmS2
LmR2
HmS3
LS2
MS2
LR3
MS3
LmR3
MR2
LmS2
LmS3
MR3
LR2
0.73
0.73
0.95
Times scale = 16
IW
IB
WB
HS3
HS2
HmS2
HS3
HS2
HS3
HmS3
HmS2
HS2
HmS3
0.72
0.71
0.95
2-INE-WTI
Time scale = 8
IW
IB
WB
IW
IB
WB
HmS3
MS3
HmR2
LS2
LR3
LmS2
LmS3
MR2
HmS3
MS3
HmR2
HS3
LmS3
LR3
HmS2
LmR2
LmS2
HS3
HS2
HmS2
HmS3
MS3
MS2
MR2
HS3
HmS3
LmR2
LmS3
MS3
LmS2
HmS2
HmS3
HS3
MS3
MR2
LmS2
LR3
HS2
HS3
HmS2
HmS3
0.84
0.80
0.94
Time scale = 32
IW
IB
WB
HmS3
HmS3
HS2
MS2
MS2
LmS2
LmS2
0.81
0.86
0.94
Time scale = 64
IW
IB
WB
HS3
HS3
HS2
HmS2
HmS3
HS3
0.78
0.86
0.71
Table 4
The top four edges of the INE-WTI network at the time scales of 2, 4, 8, 16, 32
and 64 days. The edges of the remaining networks are placed in the appendix
(Table 5).
0.63
1
Source
HmR2
H m R2
HR2
HR2
MR2
MR2
LR3
LR3
16-INE-WTI
Source Target
HmS2
HmS2
HS3
HS3
HmS3
HmS3
HS2
HS2
Table 3
The Pearson correlation coefficient of weighted degree about the source node
and target node in the networks.
INE-WTI
INE-Brent
WTI-Brent
4
8
16
32
64
101
0.2805
0.3514
0.4916
0.3187
0.3222
0.4810
0.6360
0.4440
0.6616
0.6750
0.4625
0.1177
0.3308
0.2437
0
0.6717
Null
0.7577
Null
Null
Null
Weight
Source
23
18
14
13
HmS3
HmS3
MS3
MS3
HmR2
HmR2
LS2
LS2
32-INE-WTI
Source Target
HmS3
HmS3
MS2
MS2
LmS2
LmS2
HS3
HS3
Weight
70
52
42
35
Target
8-INE-WTI
Weight
Source
61
33
26
20
HS3
HS3
HmS3
HmS3
LmR2
LmR2
LmS3
LmS3
64-INE-WTI
Source Target
HS3
HS3
HmS2
HmS2
HmR2
HmR2
HmR3
HmR3
Weight
112
58
57
29
Target
Weight
90
34
26
22
Weight
222
29
24
16
investors. Second, the comovement characteristics between China's
crude oil futures and international crude oil futures at different time
scales are also very different. The comovement characteristics (the
comovement strength, the comovement direction and the lead-lag re­
lationship of price fluctuation) are more stable in the long-term than in
the short-term. And compared with the long-term, the short-term
comovement strength is weaker, the comovement states are more di­
verse and the transition between comovement states is more complex.
This implies that in the short-term, the relationship between crude oil
markets is more complicated and the investment risks are greater than
those in the long-term. Investors should pay more attention to pre­
venting short-term risks. Third, at each time scale, during the evolution
of time, some comovement states have a higher probability of occur­
rence and they are also more stable than others. This conclusion can
provide more support for investors to make long-term and short-term
investment strategies.
0.80
2
Target
4-INE-WTI
China's crude oil futures price fluctuation tends to lag behind that of
international crude oil futures. This implies that there may be a gap
between China's crude oil futures and international crude oil bench­
marks. In the future, regulators of China's crude oil futures should pay
attention to improving the financial environment and simplifying the
trading procedures to facilitate the participation of the international
Acknowledgements
This work is supported by National Natural Science Foundation of
China (Grant No. 41801106, No. 71991481 and No.71991480) and
Scientific Research Program funded by Shaanxi Provincial Education
Department, China (Program No.17JZ039).
11
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
Appendix A
Fig. 11. The INE-WTI, INE-Brent and WTI-Brent networks at the time scales of 16, 32, 64 and 101 days.
Table 5
The top four edges of the INE-Brent and WTI-Brent networks at the time scales of 2, 4, 8, 16, 32, 64 and 101 days, and that of INE-WTI network at the time scale of
101 days.
2-INE-Brent
4-INE- Brent
8-INE- Brent
Source
Target
Weight
Source
Target
Weight
Source
Target
Weight
HR2
HmR2
LmR2
MR2
16-INE- Brent
Source
HS3
HS2
HmS2
HmS3
2-WTI-Brent
Source
HS3
HS2
HS3
HS2
16-WTI- Brent
Source
HS2
HS3
HmS3
HS3
101-INE- WTI
Source
HS3
HR2
HmR2
LmR2
MR2
24
20
16
10
HmS3
HmR2
HS3
MS3
61
23
20
19
87
62
36
18
Weight
89
47
31
29
Target
HmS3
MS2
HmS2
LmS2
Weight
123
44
39
39
HmS3
HS3
MS3
MR2
64-INE- Brent
Source
HS3
HmS3
HS3
MS3
MR2
Target
HS3
HS2
HmS2
HmS3
Target
HS3
Weight
294
Target
HS3
HS2
HS2
HS3
Weight
107
75
14
12
Target
HS3
HS2
HmS2
HmS3
Weight
97
88
22
17
Target
HS2
HS3
HmS2
HmS3
Weight
106
60
46
32
Target
HS2
HS3
HmS3
HS2
Weight
154
114
7
4
Target
HS2
HS3
HS2
HS3
Weight
185
106
2
1
Target
HS2
HmS3
HS3
MR3
Weight
128
57
49
29
Target
HS3
Weight
294
HmS3
H m R2
HS3
MS3
32-INE- Brent
Source
HmS3
MS2
HmS2
LmS2
4-WTI- Brent
Source
HS3
HS2
HmS2
HmS3
32-WTI- Brent
Source
HS2
HS3
HS3
HS2
101-INE- Brent
Source
HS3
Target
HS3
Weight
294
Target
HS2
Weight
294
12
8-WTI- Brent
Source
HS2
HS3
HmS2
HmS3
64-WTI- Brent
Source
HS2
HmS3
HS3
MR3
101-WTI- Bren
Source
HS2
International Review of Financial Analysis 72 (2020) 101562
X. Huang and S. Huang
References
and Brent crude oil markets. Environmental and Resource Economics Review, 22,
671–689.
Kydland, F. E., & Prescott, E. C. (1990). Business cycles: Real facts and a monetary myth.
Quarterly Review, 14(2), 3–18.
Li, H., Qi, Y., Li, C., & Liu, X. (2019). Routes and clustering features of PM2.5 spillover
within the Jing-Jin-Ji region at multiple timescales identified using complex networkbased methods. Journal of Cleaner Production, 209, 1195–1205.
Liu, S., Fang, W., Gao, X., An, F., Jiang, M., & Li, Y. (2019). Long-term memory dynamics
of crude oil price spread in non-dollar countries under the influence of exchange
rates. Energy, 182, 753–764.
Liu, S., Gao, X., Fang, W., Sun, Q., Feng, S., Liu, X., & Guo, S. (2018). Modeling the
complex network of multidimensional InformationTime series to characterize the
volatility pattern evolution. IEEE Access, 6, 29088–29097.
Liu, Y., Gao, X., & Guo, J. (2018). Network features of the EU carbon trade system: An
evolutionary perspective. Energies, 11, 1501.
Oladosu, G. (2009). Identifying the oil price-macroeconomy relationship: An empirical
mode decomposition analysis of US data. Energy Policy, 37, 5417–5426.
Pal, D., & Mitra, S. K. (2019). Oil price and automobile stock return co-movement: A
wavelet coherence analysis. Economic Modelling, 76, 172–181.
Pereira, E. J.d. A. L., Ferreira, P. J. S., Silva, M. F.d., Miranda, J. G. V., & Pereira, H. B. B.
(2019). Multiscale network for 20 stock markets using DCCA. Physica A, 529, 121542.
Rua, A., & Nunes, L. C. (2009). International comovement of stock market returns: A
wavelet analysis. Journal of Empirical Finance, 16, 632–639.
Scheitrum, D. P., Carter, C. A., & Revoredo-Giha, C. (2018). WTI and Brent futures pricing
structure. Energy Economics, 72, 462–469.
Sui, G., Li, H., Feng, S., Liu, X., & Jiang, M. (2018). Correlations of stock price fluctuations
under multi-scale and multi-threshold scenarios. Physica A-Statistical Mechanics and
Its Applications, 490, 1501–1512.
Torrence, C., & Webster, P. J. (1999). Interdecadal changes in the ENSO-monsoon system.
Journal of Climate, 12, 2679–2690.
Tweneboah, G., & Alagidede, P. (2018). Interdependence structure of precious metal
prices: A multi-scale perspective. Resources Policy, 59, 427–434.
Vacha, L., & Barunik, J. (2012). Co-movement of energy commodities revisited: Evidence
from wavelet coherence analysis. Energy Economics, 34, 241–247.
Wackerbauer, R., Witt, A., Altmanspacher, H., Kurths, J., & Scheingraber, H. (1994). A
comparative classification of complexity-measures. Chaos, Solitons & Fractals, 4,
133–173.
Wang, Z., Gao, X., An, H., Tang, R., & Sun, Q. (2020). Identifying influential energy stocks
based on spillover network. International Review of Financial Analysis, 68, 101277.
Wang, Z., Gao, X., Tang, R., Liu, X., Sun, Q., & Chen, Z. (2019). Identifying influential
nodes based on fluctuation conduction network model. Physica A-Statistical Mechanics
and Its Applications, 514, 355–369.
Wang, F., Ye, X., & Wu, C. (2019). Multifractal characteristics analysis of crude oil futures
prices fluctuation in China. Physica A-Statistical Mechanics and Its Applications, 533,
122021.
Wang, X., Liu, H., Huang, S., & Lucey, B. (2019). Identifying the multiscale financial
contagion in precious metal markets. International Review of Financial Analysis, 63,
209–219.
Xu, L., & Kinkyo, T. (2019). Changing patterns of Asian currencies' co-movement with the
US dollar and the Chinese renminbi: Evidence from a wavelet multiresolution ana­
lysis. Applied Economics Letters, 26, 465–472.
Yang, J., & Zhou, Y. (2020). Return and volatility transmission between China's and in­
ternational crude oil futures markets: A first look. Journal of Futures Markets, 40,
860–884.
Zivkov, D., Balaban, S., & Djuraskovic, J. (2018). What multiscale approach can tell about
the Nexus between exchange rate and stocks in the major emerging markets? Finance
a Uver-Czech Journal of Economics and Finance, 68, 491–512.
Aguiar-Conraria, L., & Soares, M. J. (2014). The continuous wavelet transform: Moving
beyond uni- and bivariate analysis. Journal of Economic Surveys, 28, 344–375.
An, H. Z., Gao, X. Y., Fang, W., Huang, X., & Ding, Y. H. (2014). The role of fluctuating
modes of autocorrelation in crude oil prices. Physica A-Statistical Mechanics and Its
Applications, 393, 382–390.
Beritzen, J. (2007). Does OPEC influence crude oil prices? Testing for co-movements and
causality between regional crude oil prices. Applied Economics, 39, 1375–1385.
Chan, H. L., & Woo, K.-Y. (2016). An investigation into the dynamic relationship between
international and China's crude oil prices. Applied Economics, 48, 2215–2224.
Coronado, S., Fullerton, T. M., & Rojas, O. (2018). A nonlinear empirical analysis of oil
price co-movements. International Journal of Energy Economics and Policy, 8, 290–294.
Das, D., Kannadhasan, M., Al-Yahyaee, K. H., & Yoon, S. M. (2018). A wavelet analysis of
co-movements in Asian gold markets. Physica A-Statistical Mechanics and Its
Applications, 492, 192–206.
Fang, W., Gao, X., Huang, S., Jiang, M., & Liu, S. (2018). Reconstructing time series into a
complex network to assess the evolution dynamics of the correlations among energy
prices. Open Physics, 16, 346–354.
Gao, X., Huang, S., Sun, X., Hao, X., & An, F. (2018). Modelling cointegration and granger
causality network to detect long-term equilibrium and diffusion paths in the financial
system. Open Science, 5, 172092.
Geng, J.-B., Ji, Q., & Fan, Y. (2016). The behaviour mechanism analysis of regional
natural gas prices: A multi-scale perspective. Energy, 101, 266–277.
Gulen, S. G. (1999). Regionalization in the world crude oil market: Further evidence.
Energy Journal, 20, 125–139.
Guo, C., Song, Y., Li, H., Liu, N., & Guo, S. (2019). Evolutionary characteristics of M&A
involving parties of Chinese listed companies based on two-mode network. Physica AStatistical Mechanics and Its Applications, 532, Article 121870.
Hamilton, J. D. (2009). Understanding crude oil prices. Energy Journal, 30, 179–206.
Huang, S., An, H., Gao, X., & Sun, X. (2017). Do oil price asymmetric effects on the stock
market persist in multiple time horizons? Applied Energy, 185, 1799–1808.
Huang, S., An, H., Gao, X., Wen, S., & Hao, X. (2017). The multiscale impact of exchange
rates on the oil-stock nexus: Evidence from China and Russia. Applied Energy, 194,
667–678.
Huang, S., An, H., Gao, X., Wen, S., & Jia, X. (2016). The global interdependence among
oil-equity nexuses. Energy (Oxford), 48, 259–271.
Huang, S., An, H., Huang, X., & Jia, X. (2018). Co-movement of coherence between oil
prices and the stock market from the joint time-frequency perspective. Applied Energy,
221, 122–130.
Jammazi, R. (2012). Cross dynamics of oil-stock interactions: A redundant wavelet ana­
lysis. Energy, 44, 750–777.
Ji, Q., Bouri, E., & Roubaud, D. (2018). Dynamic network of implied volatility trans­
mission among US equities, strategic commodities, and BRICS equities. International
Review of Financial Analysis, 57, 1–12.
Ji, Q., Bouri, E., Roubaud, D., & Kristoufek, L. (2019). Information interdependence
among energy, cryptocurrency and major commodity markets. Energy Economics, 81,
1042–1055.
Ji, Q., & Zhang, D. (2019). China's crude oil futures: Introduction and some stylized facts.
Finance Research Letters, 28, 376–380.
Jiang, M., Gao, X., An, H., Li, H., & Sun, B. (2017). Reconstructing complex network for
characterizing the time-varying causality evolution behavior of multivariate time
series. Scientific Reports, 7, 10486.
Jiang, Y., Nie, H., & Monginsidi, J. Y. (2017). Co-movement of ASEAN stock markets: New
evidence from wavelet and VMD-based copula tests. Economic Modelling, 64,
384–398.
Kang, S. H., & Yoon, S.-M. (2013). Information transmission of volatility between WTI
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