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The lead-lag relationships between spot and futures prices of natural gas
Physica A 490 (2018) 203–211
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Physica A journal homepage: www.elsevier.com/locate/physa
The lead-lag relationships between spot and futures prices of natural gas Yahui Zhang a , Li Liu b, * a b
School of Economics and Management, Nanjing University of Science and Technology, China School of Finance, Nanjing Audit University, China
highlights • • • • •
We employ both linear and nonlinear causality tests. We find linear causality from futures to spot prices. We find bidirectional causality between spot and futures prices. Volatility spillover can partly explain the nonlinear causality behavior. The cross-correlations are also analyzed.
article
info
Article history: Received 1 April 2017 Received in revised form 29 June 2017
Keywords: Natural gas Spot and futures Causality analysis Nonlinearity
a b s t r a c t The lead-lag relationships between spot and futures markets are of great interest for academics. Previous studies neglect the possibility of nonlinear behaviors which may be caused by asymmetry or persistence. To fill this gap, this paper uses the MF-DCCA method and the linear and nonlinear causality tests to explore the causal relationships between natural gas spot and futures prices in the New York Mercantile Exchange. We find that spot and futures prices are positive cross-correlated, the natural gas futures can linearly Granger cause spot price, and there are bidirectional nonlinear causality relationships between natural gas spot and futures prices. Further, we explore the sources of nonlinear causality relationships, and find that the volatility spillover can partly explain the nonlinear causality and affect their cross-correlations. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Natural gas is the cleanest fossil fuel, its carbon emissions are much lower than those of coal and oil, and other pollutants are emitted less. With the increasing awareness of energy conservation, the study of natural gas price is gradually important. As a market form of high-class, the natural gas futures market has been widely adopted by the developed countries due to its price discovery and hedging function. It is important to investigate the characteristic of natural gas market and the lead-lag relationship between natural gas prices and spot prices. Research on the lead-lag relationship can efficiently help investors better predict the prices and adjust investment plans to avoid risks. However, previous investigations on the leadlag relationships between spot and futures prices are mainly focused on stock index futures, bulk commodities and energy products and there are fewer works to explore the regular of natural gas. For example, Hasbrouck [1] analyzes the standard & Poor’s 100 and the NASDAQ 500 index of the day price formation. The author shows that the corresponding mini futures play
*
Correspondence to: 86 West Yushan Road, Pukou District, Nanjing 211815, China. E-mail address: [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.physa.2017.08.018 0378-4371/© 2017 Elsevier B.V. All rights reserved.
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Y. Zhang, L. Liu / Physica A 490 (2018) 203–211
a leading role in price discovery. Zhang [2] uses correlation analysis and Granger causality test to analyze the relationship of spot and futures in CSI 300 Index. Their result is that the correlation of spot and futures in CSI 300 Index is very strong, but they do form a mutually determined relationship. Srinivasan and Ibrahim [3] examine Gold futures and spot markets of NCDEX by VECM and ECM–EGARCH models. The result indicates that the spot market of Gold plays a dominant role and serves as effective price discovery vehicle. Shakeel and Purankar [4] find bidirectional causality relationships between spot and futures series of castor seed and soybean, suggesting that both the spot and futures markets of the selected agricultural commodity play the leading role through price discovery process in India. Mehrara and Hamldar [5] investigate the relationships between spot and futures prices in Brent crude oil market and find that the coefficient of the ECT and lagged explanatory variables are significant in both equations which indicates that long-run as well as short-run bidirectional causalities between log of spot and futures prices. Liu et al. [6] research the characteristic electric power futures in Nordic electricity market and American PJM electricity market and the dynamic relationship between electrical futures and spot prices. The results show that the linear combination of futures price and spot price has balance convergence trends in the long-term. There is a long-term equilibrium relationship between futures price and spot price. Futures price and spot price have interaction and mutual influence, but futures price is in the dominant position. Previous studies have neglected some of the problems. First of all, there are fewer studies exploring the causal relationships between the natural gas spot market and futures market, and as the market of natural gas becoming more and more important, the research to natural gas market is increasingly needed. Secondly, past studies for spot and futures only consider the linear Granger causality and ignore the actual nonlinear behaviors, but as shown in many researches, the nonlinear structures of energy prices have been widely considered. For example, Davis and Rodriguez-Yam [7] use the Structural Break Estimation for Nonstationary Time Series Models and Savolainen [8] use the Markov regime switching models to describe the behavior of energy futures returns on a commodity level; Bekiros et al. [9] emphasize the role of economic and firm-level uncertainty measures in predicting volatility of stock returns, and presage against using linear models which are likely to suffer from misspecification in the presence of parameter instability and nonlinear spillover effects. Therefore, this paper is based on the previous studies, using MF-DCCA method and the linear and nonlinear causality tests to examine the causal relationships between spot and futures markets for natural gas. This paper uses the linear and nonlinear causality tests to explore the causal relationships between natural gas spot and futures prices in the New York Mercantile Exchange. The results show that the natural gas futures are the linear Granger reasons in spot price, and there are bidirectional nonlinear causality relationships between natural gas spot and futures prices. Further, it explores the reasons of nonlinear causality, and finds that the volatility spillover can partly explain the nonlinear causality between spot and futures prices. The remainder of the paper is organized as follow. Section 2 outlines the empirical method. Section 3 explains the data and their preliminary features. Section 4 reports the empirical results and Section 5 concludes. 2. Econometric methodology This section gives a brief description on the econometric methodology of linear and nonlinear causality tests for the causal relationships. The linear causality test is the standard Granger [10] method built on the predictive regression. The nonlinear causality test is the Diks and Panchenko [11] method, a modified version of Hiemstra and Jones [12] method. 2.1. Linear Granger causality test For the null hypothesis that the variable Xt cannot Granger cause Yt . The test equation of traditional linear Granger causality can be written as,
∆Yt = γ +
p ∑ i=1
αi ∆Yt −i +
q ∑
βj ∆Xt −j + εt
(1)
j=1
where, γ is a constant, p and q are the lag lengths and the disturbance term εt is assumed to be a white noise. ∆Xt and ∆Yt denote the first differences of Xt and Yt , respectively. The null hypothesis of no Granger causality was described as the equation: β1 = β2 = · · · = βq = 0. The standard Wald F -statistic is used to detect the Granger causality relationship as,
(RSS (p) − RSS (p, q)) /q ∼ F (p, T − p − q − 1) (2) RSS (p, q) /(T − q − p − 1) where, RSS (p, q) is the sum of squared residuals of Eq. (1), RSS (p) is the sum of squared residuals of a univariate autoregression for ∆Yt with the lag order p, and T is the number of observations. F =
Engle and Granger [13] argued that, if two series are cointegrated, the causality testing should be performed based on a vector error correction (VECM) specification rather than an unrestricted VAR. The VECM has the following form:
∆Yt = −p (Yt −1 − λXt −1 ) +
p ∑ i=1
αi ∆Yt −1 +
q ∑
βi ∆Xt −i
j=1
where, [1 − λ] is the cointegration vector and λ is the cointegration coefficient.
(3)
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2.2. A nonlinear Granger causality test It has been well documented in the literature that due to the multiplicity of influencing factors, the financial time series present complex nonlinear structures. The linear causality test is likely to lead to incorrect result when the tested series display nonlinear property. Baek and Brock [14] first developed the nonlinear causality test for the interaction between stationary sequences. On this basis, Hiemstra and Jones [12] relax the requirement of Baek and Brock method [14] that the stationary series are independent and identically distributed by allowing the existence of a short auto-correlation, which improves the original nonlinear causality test. Furthermore, Diks and Panchenko [11] pointed out that the Hiemstra and Jones test [12] is not strictly consistent with the definition of Granger causality test, and cannot be used to do causality analysis. In order to overcome this drawback, Diks and Panchenko [11] construct a new nonparametric test which can be used to analyze the nonlinear Granger causality relationship between two stationary time series. This test can be described as follows: Assume that {Xt , Yt , t ≥ 1} are two strictly stationary time series, in Granger’s sense [10], {Xt } is a strictly Granger cause of {Yt } if past and current values of {Xt } contain additional information on future value of {Yt } that is not contained only in the past and current {Yt } values. ( ) l l Assume the delay vectors Xt x = (Xt −l+1 , . . . , Xt ) and Yt x = (Yt −l+1 , . . . , Yt ) , lx , ly ≥ 1 , the null hypothesis that past l observations of Xt x contain no useful information of Yt +1 can be described using the following equation:
⏐( ⏐
ly
l
H0 : Yt +1 ⏐ Xt x ; Yt
)
⏐ ⏐ l ∼ Yt +1 ⏐ Yt y
(4)
where, ‘∼’ denotes(the equivalence in distribution. For two strict stationary time series, Eq. (4) actually considers the ) distribution of the lx + ly + 1 dimensional vector Wt = (Xt , Yt , Zt ) where Zt = Yt +1 . Under the null hypothesis, the distribution of Wt is invariant. We drop the time index and follow Bekiros and Diks [15] assume lx = ly = 1. Thus, under the null hypothesis, the conditional distribution of Z given by (X , Y ) = (x, y) is equivalent to that of Z given by Y = y. In Eq. (2), the joint probability density distribution fX ,Y ,Z (x, y, z ) and its marginal should satisfy the following equation: fX ,Y ,Z (x, y, z ) fY (y)
=
fX ,Y (x, y) fY (y)
·
fY ,Z (y, z ) fY (y)
.
(5)
The above equation implies that X and Z are in dependent conditionally on Y = y for each fixed value of y. Diks and Panchenko [11] show that this formulated null hypothesis implies the following equation: q ≡ E fX ,Y ,Z (x, y, z ) fY (y) − fX ,Y (x, y) fY ,Z (y, z ) = 0.
]
[
(6)
Let fˆW (Wi ) denote a local density estimator of a dw -variate random vector W at Wi as the following equation: fˆw (Wi ) = (2εn )−dw (n − 1)−1
∑
IijW ,
(7)
j,j̸ =i
where, IijW = I Wi − Wj < εn , I(·) is the indicator function and εt is the bandwidth, depending on the sample size n. Given this estimator, the test statistic is a scaled sample version of q in Eq. (6):
(
Tn (εn ) =
n−1
∑(
n (n − 2)
)
)
fˆX ,Y ,Z (Xi , Yi , Zi ) fˆY (Yi ) − fˆX ,Y (Xi , Yi ) fˆY ,Z (Yi , Zi ) .
(8)
i
For lx = ly = 1, if εn = Cn−β (C > 0, 1/4 < β < 1/3), Diks and Panchenko [11] prove under strong mixing that the statistic in Eq. (8) follows the asymptotic distribution:
√ (Tn (εn ) − q) n
Sn
D
−→ N (0, 1)
(9)
D
where, −→ denotes convergence in distribution and Sn is the asymptotic variance of Tn (·). Following Diks and Panchenko’s suggestion [11], we use a right-tailed test.1 3. Data and preliminary analysis We choose daily price data of natural gas spot and futures traded in NYMEX. Our sample covers the period from January 7, 1997 to February 27, 2016, containing 4790 observations for each series. The data is obtained from the website of the US Energy Information Administration (EIA).2 We use price data of natural gas spot and futures with four different maturity contracts. The futures price is quoted for delivering a specified quantity of a commodity at a specified time and place in the future. Contract 1 denotes a futures contract with the earliest delivery date. Contracts 2–4 denote the successive delivery 1 Here, the author greatly thanks Dr. Diks for providing the programming code of this nonlinear causality test. 2 Website: http://www.eia.gov.
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Fig. 1. Spot and futures prices of natural gas.
Table 1 Descriptive statistics of natural gas spot and futures returns.
Mean (%) Maximum (%) Minimum (%) Std. dev.(%) Skewness Kurtosis Jarque–Bera Q(10)
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
−0.016
−0.014
−0.011
−6.65E−03
−3.66E−03
32.44
23.24 −22.64 3.1833 0.320 7.280 3733.35*** 29.950***
57.67 −56.82 44.79 0.616 23.757 86183.67*** 237.66***
−19.90 3.5059 0.523 8.293 5802.76*** 38.811***
21.52
18.27
−31.12
−19.37
2.8647 0.086 8.928 7010.31*** 21.619**
2.5878 0.059 7.366 3803.22*** 30.042***
Note: The Jarque–Bera statistic tests for the null hypothesis of Gaussian distribution. Q(10) is the Ljung–Box statistic of the return series for up to the 10th order serial correlation. The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively.
months following Contract 1. For natural gas, each contract expires on the third business day prior to the 25th calendar day of the month preceded the delivery month. If the 25th calendar day of the month is a non-business day, trading ceases on the third business day prior to the business day preceded the 25th calendar day. After a contract expires, Contract 1 for the remainder of that calendar month is the second following month. Fig. 1 gives the illustrations of evolutions of natural gas spot and futures prices. We can find that the dynamic changes of prices of different futures are very similar. In July 2001, natural gas prices reached the local peak. The plausible explanations are two aspects. First, the oil crisis during that period led to increases in the price of its substitution: natural gas. Second, more than 10 natural gas exporting countries including Russia and Iran agreed to set up a ‘‘gas OPEC’’ to control the production of natural gas. Both events caused higher natural gas prices. The largest crash occurred during the period of mid 2008–2009, which can be related to the effect of the global financial crisis. After the financial crisis, the gas prices evolved at a relatively low level. In the following part of this paper, we will focus on the natural logarithmic forms of natural gas prices. We use SP, FU1, FU2, FU3 and FU4 to denote spot price and futures prices of four contracts maturing in one, two, three and four months, respectively. Table 1 shows the descriptive statistics of the natural gas spot and futures price returns, defined by the first-order differences of logarithmic prices. The mean values of the five return series are relatively small and close to zero. The standard deviation and the range of spot return (maximum–minimum) is greater than futures returns, implying that spot market is more volatile. The standard deviations decrease with the increase of futures maturity, consistent with the well-known Samuelson effect. All five series are right-skewed and leptokurtic, evidenced by positive skewness and kurtosis higher than 3. The Jarque–Bera statistics show rejections of the null hypothesis of Gaussian distribution at 1% significance level for each series, indicating the fat-tailed distribution. The Q (10) statistics reject the null hypothesis at 5% confidence level, implying that natural gas spot and futures returns have strong correlated behaviors.
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Table 2 Result of unit root test. SP Intercept only ADF −2.572* PP −2.683* KPSS 1.636** Trend and intercept ADF −2.507 PP −2.628 KPSS 1.470**
FU1
FU2
FU3
FU4
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
−2.076 −2.100
−1.889 −1.930
−1.790 −1.839
−1.678 −1.686
−58.390** −66.427**
−74.287*** −74.339***
−76.653** −73.678**
−72.149** −72.137**
−72.817** −72.882**
1.690**
1.768***
1.867**
1.967**
0.050
0.113
0.143
0.191
0.260
**
***
**
**
−1.977 −1.995
−1.760 −1.794
−1.609 −1.648
−1.432 −1.434
−58.389 −66.423**
−74.288 −74.338***
−73.659 −73.689**
−72.162 −72.155**
−72.840** −72.910**
1.538**
1.606***
1.675**
1.734**
0.020
0.033
0.033
0.030
0.032
Note: This table reports the results of testing for unit root in logarithms of natural gas prices and their first order differences. The null hypothesis of ADF and PP tests is unit root and that of KPSS test is stationarity. The optimal lag lengths of ADF test are chosen based on Schwarz information criterion (SIC) and the optimal bandwidths of PP and KPSS are determined based on Newey–West criterion. * Rejections at 10% significance level. ** ***
Rejections at 5% significance level. Rejections at 1% significance level. Table 3 Correlation matrix between spot and futures returns.
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
∆SP ∆FU1 ∆FU2 ∆FU3 ∆FU4
1 0.3293 0.2420 0.2302 0.1992
1 0.9128 0.8141 0.7319
1 0.9289 0.8371
1 0.9263
1
PII ∆SP ∆FU1 ∆FU2 ∆FU3 ∆FU4
1.0000 0.1442 0.1332 0.1431 0.1479
1.0000 0.9484 0.9051 0.8719
1.0000 0.9652 0.9111
1.0000 0.9665
1.0000
PI
4. Empirical results To avoid the spurious regression results caused by the nonstationarity, the linear Granger causality analysis should be performed on stationary time series. Thus, prior to causality analysis, we should employ unit root tests to analyze the stationarity properties of the natural gas prices series. The results of unit root tests are presented in Table 2. We determine the order of integration of each series using Augmented Dickey–Fuller (ADF) test [16], Phillips–Perron (PP) test [17] and KPSS test [18]. The information criterion, like Schwartz Information Criterion (SIC), can be used to decide the lag length of the ADF method, the optimal bandwidths of PP and KPSS methods are determined by the Newey–West criterion. Based on each of these three methods, the unitroot tests are performed by taking into account two different models (with constant and with constant and trend). The null hypothesis of ADF and PP tests is a unit root while that of KPSS test is a stationary time series. For each futures price series, the ADF and PP statistics cannot reject the unit root null hypothesis while KPSS statistics significantly reject the null hypothesis of stationarity, implying a nonstationary process. For natural gas spot price series, although ADF and PP statistics show rejections of unit root null hypothesis at 10% significance level with intercept only, the KPSS statistic reject the null hypothesis of stationarity at 1% significance level. Thus, we can conclude that natural gas spot price series is nonstationary at 1% significance level. For the first differences of spot and futures prices series, all three types of statistics consistently indicate that they are stationary. In summary, we can conclude that each natural gas price series contain a unit root while the first order differences (i.e., the returns) are stationary. To analyze the changes in the lead-lag relationships between spot and futures over time, the period is divided into two segments. PI represents the period from January 7, 1997 to December 31, 2006, and PII represents the period from January 1, 2007 to February 27, 2016. Table 3 reports the correlation matrix among five return series in two periods. We can find in Table 3 that spot and futures returns are positively correlated. The correlations between two futures returns are much larger than those between returns spot and futures. With the contact time increasing, the correlations of the futures prices decrease. It is worth mentioning that the correlated behaviors between spot and futures returns are stronger in PI than PII. We also investigate the cross-correlations between natural gas spot and futures prices. It is widely accepted that financial time series are multifractal [19], we estimate the cross-correlation exponents with the MF-DCCA method following Wang et al. [20]. I will not go into details about MF-DCCA method here, please refer to Wang’s paper for the details. According Wang et al. [20], if scaling exponent is bigger than 0.5, the cross-correlations between the kinds off fluctuations related to q of two series are persistent (positive). An increase of one price is likely to be followed by an increase of the other price. If
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Table 4 Scaling exponents for return series with q varying from −10 to 10. q
PI
−10 −6 −2 2 6 10
PII
SP-FU1
SP-FU2
SP-FU3
SP-FU4
SP-FU1
SP-FU2
SP-FU3
SP-FU4
0.8394 0.8132 0.7839 0.7688 0.7329 0.6996
0.8750 0.8434 0.7994 0.7805 0.7582 0.7325
0.9871 0.859 0.8064 0.7775 0.7516 0.7256
0.9043 0.8698 0.8698 0.7996 0.7668 0.7370
0.8660 0.8324 0.8020 0.8095 0.7971 0.7669
0.8728 0.8355 0.7981 0.805 0.7968 0.7690
0.8524 0.8218 0.7958 0.8103 0.8047 0.7759
0.8504 0.8277 0.8074 0.8281 0.8228 0.7911
Table 5 Results of cointegration test between natural gas spot and futures prices. No.of cointegration equations
PI
PII
Trace statistic
None At most 1
Max-eigen statistic
Trace statistic
Max-eigen statistic
SPFU1
SPFU2
SPFU3
SPFU4
SPFU1
SPFU2
SPFU3
SPFU4
SPFU1
SPFU2
SPFU3
SPFU4
SPFU1
SPFU2
SPFU3
SPFU4
87.67* 2.87
70.23* 2.40
49.23* 2.19
38.60* 1.76
84.80* 2.87
67.83* 2.40
47.04* 2.19
36.84* 1.76
176.86* 1.30
86.18* 1.20
40.65* 0.71
29.81* 0.46
175.56* 1.30
84.98* 1.20
39.95* 0.71
29.35* 0.46
Note: This table shows the results of Johansen cointegration test. * Rejections at 5% significance level.
Table 6 Results of linear causality analysis. Variable
PI
PII
Raw data X
Y
SP SP SP SP
FU1 FU2 FU3 FU4
X →Y
Y →X *** *** *** ***
ECM filtered series
Raw data
X →Y
X →Y
Y →X
ECM filtered series Y →X
X →Y
Y →X
*** *** *** ***
Note: This table reports the results of linear causality test between natural gas spot and futures prices. The causality analysis for the cointegrated pairs is performed based on VEC specifications. The optimal lag lengths are determined based on Schwarz Information Criterion (SIC). ‘X → Y ’ denotes the null hypothesis that X does not Granger cause Y . The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively.
scaling exponent is smaller than 0.5, the cross-correlations between the kinds of fluctuations related to q of two series are anti-persistent (negative). An increase of one price is likely to be followed by a decrease of the other price. If scaling exponent is equal with 0.5, one series is not cross-correlated with the other, and the change of one price cannot affect the behavior the other price. As we can see in Table 4, all the scaling exponents in both PI and PII is bigger than 0.5, it means the relationships between natural gas spot and futures are persistent cross-correlated. And as the q changed, the scaling exponents vary with the change of q, which verifies the multiple characteristics of natural gas price series. When two time series are cointegrated, it is better for causality test to use a vector error correction model (VECM) rather than an unrestricted vector autoregressive model (VAR) [13]. Thus, we analyze the cointegration relationships between spot and futures before the causality analysis. Table 5 shows the results of the two periods of cointegration analysis based on Johansen’s maximum eigenvalues and trace test [21–23]. The results show that in the two periods, none of the maximum eigenvalue statistics or trace statistics can reject the null hypothesis of no cointegration between spot and futures. For more details, both of trace statistics and maximum eigenvalue statistics reject null hypothesis of no cointegration relationship between spot and futures prices at 5% significance level but cannot reject the null hypothesis of at most one cointegration equation. That is, natural gas spot and futures prices are cointegrated. Therefore, the linear Granger causality analysis between spot and futures prices in both two periods will be performed based on VECM specification. According to the results of previous cointegration tests, the VECM models of spot and futures price series are established by Schwarz information criterion (SIC). For convenience, we use the ‘‘X →Y ’’ to represent the null hypothesis: X does not Granger cause Y . As can be seen in Table 6, the results in PI indicate that natural gas futures prices can linearly Granger cause spot price. This evidence is consistent with the majority of the literatures on the price discovery of futures markets. Table 5 also reports the results of linear Granger causality tests on VECM filtered residuals. We can find that the linear causality behaviors in PI between natural gas prices disappear after the procedure of VECM filtering. And the result of PII is the same as PI. In existing studies, the nonlinear structures of energy prices have been well confirmed [24–27]. Moreover, nonlinear linkages between two energy prices have been found in recent studies [28,29]. Thus, it is necessary to examine the existence of nonlinear Granger causality. For this consideration, we detect the nonlinear structure of our sample data and then test the nonlinear causality between natural gas spot and futures returns using a nonparametric method proposed by Diks and Panchenko [11].
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Table 7 Results of BDS test. Length in S.D. (σ 2 )
Embedding dimension
PI
PII
Z statistic
0.7 0.7 0.7 0.7 0.7
2 3 4 5 6
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
13.91*** 17.03*** 19.21*** 20.98*** 23.13***
4.48*** 6.76*** 8.05*** 8.96*** 10.59***
2.71*** 4.71*** 5.91*** 6.48*** 7.66***
1.92** 3.54*** 4.78*** 5.22*** 6.16***
3.34*** 4.65*** 5.76*** 5.93*** 6.76***
10.42*** 13.10*** 14.16*** 15.59*** 17.40***
4.95*** 8.35*** 9.74*** 10.79*** 11.91***
2.59*** 5.64*** 6.62*** 7.54*** 8.33***
1.25* 3.54*** 4.78*** 5.22*** 6.16***
3.34* 4.65*** 5.76*** 5.93*** 6.76***
Note: This table reports the results of BDS test. * Rejections at 10% significance level. ** ***
Rejections at 5% significance level. Rejections at 1% significance level.
Table 8 Result of nonlinear causality analysis. Variable
PI
PII
Raw data
ECM filtered series
Raw data
ECM filtered series
X
Y
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
SP SP SP SP
FU1 FU2 FU3 FU4
*** *** *** **
*** *** *** ***
*** *** *** ***
*** *** *** ***
*** *** *** ***
*** *** *** ***
*** *** ** **
*** *** *** ***
Note: This table reports the results of nonlinear causality test between energy prices and exchange rates. ‘X →Y ’ denotes the null hypothesis that X does not Granger cause Y . We set the lag length lx = ly = 1. The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively.
We choose the BDS test to detect the nonlinear structures of five return series in both two periods and show the results in Table 7. Let the embedding dimension (m) vary from 2 to 6. As we can see, the BDS statistics of five return series at most of embedding dimensions in both two periods show rejections of the null hypothesis of independent identity distributions at 1% significance level, in favor of nonlinear structures. Even though in PI, the BDS statistics of futures contract with the maturities of three months returns at m = 2 reject the null hypothesis at 5% significance level and in PII the BDS statistics of futures contract with the maturities of three and four months returns at m = 2 reject the null hypothesis at 10% significance level, the result is still valid. Thus, we conclude that the spot and futures returns display significant nonlinear properties, confirming the results in existing literatures [24–27]. Table 8 reports the results of Diks and Panchenko’s nonparametric Granger causality test [11]. When time series present nonlinear structures, the traditional Granger causality test would be invalid. Therefore, we use the nonparametric test method [11] to explore the nonlinear relationships between the spot price and futures prices of the natural gas market in the New York Stock Exchange. According Bekiros and Diks [15], we discuss the results for lags lx = ly = 1, the constant C for the bandwidth εn is set to be 7.5, which is suggested by Diks and Panchenko [11]. With the optimal value of β = 2/7 given by Diks and Panchenko [11] and about 2400 observations in each period, the selected bandwidth of εn is approximately one standard deviation of time series. In order to investigate whether there are any remaining causal relationships which are strictly nonlinear, we also report the nonlinear causality test for error correction model residuals. In both PI and PII, we can find significant bidirectional causality relationships existing between spot and four futures prices. Even for the residuals of the VECM model, the nonlinear causality relationships are still significant. Why do the results of linear and nonlinear Granger tests have the significant difference? What is the source of nonlinear Granger causality? Bekiros and Diks [15] show that volatility spillover can partly explain nonlinear causality. Based on this consideration, we employ a GARCH–BEKK model [30] to study the effects of volatility spillover (second-order co-movement) on nonlinear causality in our case. The BEKK–GARCH (1, 1) specification is defined as: Ht = C ′ C +
q ∑
A′jk εt −j εt′ −j Ajk +
j=1
p ∑
1/2
G′jk Ht −j Gjk , εt = Ht
vt .
(10)
j=1
Here, C⏐, Ajk and Gjk are (N ×N) matrices and C is an upper triangular matrix. Ajk is the conditional covariance matrix of εt with εt ⏐Φt −1 ∼ (0, Ht ) and Φt −1 is the available information set at time t − 1. The standardized residuals vt are obtained by 1/2
whitening the residuals εt /Ht . Table 9 reports the results of nonlinear causality analysis before and after GARCH–BEKK filtering in two periods. We can find that after BEKK–GARCH (1, 1) filtered, nonlinear causality behaviors are weaker in both of two series. For example, the lead-lag relationship between spot and futures with the maturities of four mouths disappear and spot price can nonlinearly
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Table 9 Result of nonlinear causality test for GARCH–BEKK filtered series. Variable
PI
PII
Raw data
BEKK–GARCH filtered series
Raw data
BEKK–GARCH filtered series
X
Y
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
SP SP SP SP
FU1 FU2 FU3 FU4
*** *** *** **
*** *** *** ***
* ***
*** *** ***
*** *** *** ***
*** *** *** ***
*
*** **
Note: This table reports the results of nonlinear causality test for GARCH–BEKK filtered residuals. ‘X → Y ’ denotes the null hypothesis that X does not Granger cause Y . We set the lag length lx = ly = 1. The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively. Table 10 Scaling exponents for GARCH–BEKK filtered series with q varying from −10 to 10. q
−10 −6 −2 2 6 10
PI
PII
SP-FU1
SP-FU2
SP-FU3
SP-FU4
SP-FU1
SP-FU2
SP-FU3
SP-FU4
0.8372 0.7880 0.6942 0.5112 0.3052 0.2229
0.8001 0.7605 0.6748 0.5262 0.3835 0.3167
0.7806 0.7387 0.6585 0.5225 0.4037 0.3488
0.7853 0.7378 0.6555 0.5333 0.4331 0.3847
0.7456 0.7240 0.6770 0.5625 0.4343 0.3807
0.7204 0.7016 0.6633 0.5688 0.4596 0.4094
0.7059 0.6847 0.6461 0.5633 0.4726 0.4293
0.6934 0.6717 0.6331 0.5560 0.4775 0.4402
Granger cause FU1 at 10% significance level only. Thus, volatility spillover can partly explain the nonlinearity in causality relationships, and from the results of nonlinear causality test for GARCH–BEKK filtered series, the weaker nonlinear causality behaviors caused by the volatility spillover in PII is stronger than which is in PI, it means the volatility spillover is stronger and stronger over time. We also explored the impacts of volatility spillover on cross correlations. Table 10 report the scaling exponents for GARCH–BEKK filtered series with q varying from −10 to 10. Compared with the scaling exponents in Table 4, the scaling exponents for GARCH–BEKK filtered series are less than the scaling exponents for returns series, which indicates that volatility spillover also affect the cross correlations between natural gas spot and futures prices. For further discuss, the scaling exponents vary with the change of q, when q > 0, the scaling exponents are bigger than 0.5, it means the GARCH– BEKK filtered series have the positive cross-correlations, and when q > 0, the cross-correlations are negative. The results are the same in PI and PII. Furthermore, the scaling exponents in PII are much close to 0.5, indicating the weak cross-correlations, which confirm the previous conclusion that the volatility spillover is stronger and stronger over time and can partly explain the lead-lag relationships between natural gas spot and futures. 5. Conclusions We explore the lead-lag relationships by the MF-DCCA method, the results indicate that the lead-lag relationships are existed and positive. Using the linear Granger causality test, we find that during both two periods, futures prices can Granger cause spot prices and the futures returns have a dominated role in the price discovery process. From nonlinear Granger causality test results, we can see that there are bidirectional Granger causality relationships between futures and spot returns. After the vector error correction model (ECM) filtering, the cross-correlation and bidirectional Granger causality still existed and significant. In addition, we consider the volatility spillover effects on the lead-lag relationships between spot and futures. The GARCH–BEKK model is used to capture the spillover effect. We find that volatility spillover can partly explain the nonlinear causality and the nonlinear cross-correlation. Acknowledgments We would like to show our sincere gratitude to anonymous reviewers whose comments and suggestions greatly improved the quality of the manuscript. This work is supported by the National Science Foundation of China under the grant Nos. 71401077 and 71771124. Li Liu’s work is also sponsored by Qing Lan Project in Jiangsu province, the financial support from open project of Jiangsu key laboratory of financial engineering (NSK2015-10) and the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (PPZY2015B104). References [1] J. Hasbrouck, Intraday price formation in U.S. equity index markets, J. Finance 58 (2003) 2375–2399. [2] Y. Zhang, The empirical research of the relationship of spot and futures: based on the data of CSI300 index future, Value Eng. 33 (2011). [3] P. Srinivasan, P. Ibrahim, Price discovery and asymmetric volatility spillovers in Indian spot-futures gold markets, Int. J. Econ. Sci. Appl. Res. 5 (2012) 65–80.
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Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
The lead-lag relationships between spot and futures prices of natural gas Yahui Zhang a , Li Liu b, * a b
School of Economics and Management, Nanjing University of Science and Technology, China School of Finance, Nanjing Audit University, China
highlights • • • • •
We employ both linear and nonlinear causality tests. We find linear causality from futures to spot prices. We find bidirectional causality between spot and futures prices. Volatility spillover can partly explain the nonlinear causality behavior. The cross-correlations are also analyzed.
article
info
Article history: Received 1 April 2017 Received in revised form 29 June 2017
Keywords: Natural gas Spot and futures Causality analysis Nonlinearity
a b s t r a c t The lead-lag relationships between spot and futures markets are of great interest for academics. Previous studies neglect the possibility of nonlinear behaviors which may be caused by asymmetry or persistence. To fill this gap, this paper uses the MF-DCCA method and the linear and nonlinear causality tests to explore the causal relationships between natural gas spot and futures prices in the New York Mercantile Exchange. We find that spot and futures prices are positive cross-correlated, the natural gas futures can linearly Granger cause spot price, and there are bidirectional nonlinear causality relationships between natural gas spot and futures prices. Further, we explore the sources of nonlinear causality relationships, and find that the volatility spillover can partly explain the nonlinear causality and affect their cross-correlations. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Natural gas is the cleanest fossil fuel, its carbon emissions are much lower than those of coal and oil, and other pollutants are emitted less. With the increasing awareness of energy conservation, the study of natural gas price is gradually important. As a market form of high-class, the natural gas futures market has been widely adopted by the developed countries due to its price discovery and hedging function. It is important to investigate the characteristic of natural gas market and the lead-lag relationship between natural gas prices and spot prices. Research on the lead-lag relationship can efficiently help investors better predict the prices and adjust investment plans to avoid risks. However, previous investigations on the leadlag relationships between spot and futures prices are mainly focused on stock index futures, bulk commodities and energy products and there are fewer works to explore the regular of natural gas. For example, Hasbrouck [1] analyzes the standard & Poor’s 100 and the NASDAQ 500 index of the day price formation. The author shows that the corresponding mini futures play
*
Correspondence to: 86 West Yushan Road, Pukou District, Nanjing 211815, China. E-mail address: [email protected] (L. Liu).
http://dx.doi.org/10.1016/j.physa.2017.08.018 0378-4371/© 2017 Elsevier B.V. All rights reserved.
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Y. Zhang, L. Liu / Physica A 490 (2018) 203–211
a leading role in price discovery. Zhang [2] uses correlation analysis and Granger causality test to analyze the relationship of spot and futures in CSI 300 Index. Their result is that the correlation of spot and futures in CSI 300 Index is very strong, but they do form a mutually determined relationship. Srinivasan and Ibrahim [3] examine Gold futures and spot markets of NCDEX by VECM and ECM–EGARCH models. The result indicates that the spot market of Gold plays a dominant role and serves as effective price discovery vehicle. Shakeel and Purankar [4] find bidirectional causality relationships between spot and futures series of castor seed and soybean, suggesting that both the spot and futures markets of the selected agricultural commodity play the leading role through price discovery process in India. Mehrara and Hamldar [5] investigate the relationships between spot and futures prices in Brent crude oil market and find that the coefficient of the ECT and lagged explanatory variables are significant in both equations which indicates that long-run as well as short-run bidirectional causalities between log of spot and futures prices. Liu et al. [6] research the characteristic electric power futures in Nordic electricity market and American PJM electricity market and the dynamic relationship between electrical futures and spot prices. The results show that the linear combination of futures price and spot price has balance convergence trends in the long-term. There is a long-term equilibrium relationship between futures price and spot price. Futures price and spot price have interaction and mutual influence, but futures price is in the dominant position. Previous studies have neglected some of the problems. First of all, there are fewer studies exploring the causal relationships between the natural gas spot market and futures market, and as the market of natural gas becoming more and more important, the research to natural gas market is increasingly needed. Secondly, past studies for spot and futures only consider the linear Granger causality and ignore the actual nonlinear behaviors, but as shown in many researches, the nonlinear structures of energy prices have been widely considered. For example, Davis and Rodriguez-Yam [7] use the Structural Break Estimation for Nonstationary Time Series Models and Savolainen [8] use the Markov regime switching models to describe the behavior of energy futures returns on a commodity level; Bekiros et al. [9] emphasize the role of economic and firm-level uncertainty measures in predicting volatility of stock returns, and presage against using linear models which are likely to suffer from misspecification in the presence of parameter instability and nonlinear spillover effects. Therefore, this paper is based on the previous studies, using MF-DCCA method and the linear and nonlinear causality tests to examine the causal relationships between spot and futures markets for natural gas. This paper uses the linear and nonlinear causality tests to explore the causal relationships between natural gas spot and futures prices in the New York Mercantile Exchange. The results show that the natural gas futures are the linear Granger reasons in spot price, and there are bidirectional nonlinear causality relationships between natural gas spot and futures prices. Further, it explores the reasons of nonlinear causality, and finds that the volatility spillover can partly explain the nonlinear causality between spot and futures prices. The remainder of the paper is organized as follow. Section 2 outlines the empirical method. Section 3 explains the data and their preliminary features. Section 4 reports the empirical results and Section 5 concludes. 2. Econometric methodology This section gives a brief description on the econometric methodology of linear and nonlinear causality tests for the causal relationships. The linear causality test is the standard Granger [10] method built on the predictive regression. The nonlinear causality test is the Diks and Panchenko [11] method, a modified version of Hiemstra and Jones [12] method. 2.1. Linear Granger causality test For the null hypothesis that the variable Xt cannot Granger cause Yt . The test equation of traditional linear Granger causality can be written as,
∆Yt = γ +
p ∑ i=1
αi ∆Yt −i +
q ∑
βj ∆Xt −j + εt
(1)
j=1
where, γ is a constant, p and q are the lag lengths and the disturbance term εt is assumed to be a white noise. ∆Xt and ∆Yt denote the first differences of Xt and Yt , respectively. The null hypothesis of no Granger causality was described as the equation: β1 = β2 = · · · = βq = 0. The standard Wald F -statistic is used to detect the Granger causality relationship as,
(RSS (p) − RSS (p, q)) /q ∼ F (p, T − p − q − 1) (2) RSS (p, q) /(T − q − p − 1) where, RSS (p, q) is the sum of squared residuals of Eq. (1), RSS (p) is the sum of squared residuals of a univariate autoregression for ∆Yt with the lag order p, and T is the number of observations. F =
Engle and Granger [13] argued that, if two series are cointegrated, the causality testing should be performed based on a vector error correction (VECM) specification rather than an unrestricted VAR. The VECM has the following form:
∆Yt = −p (Yt −1 − λXt −1 ) +
p ∑ i=1
αi ∆Yt −1 +
q ∑
βi ∆Xt −i
j=1
where, [1 − λ] is the cointegration vector and λ is the cointegration coefficient.
(3)
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2.2. A nonlinear Granger causality test It has been well documented in the literature that due to the multiplicity of influencing factors, the financial time series present complex nonlinear structures. The linear causality test is likely to lead to incorrect result when the tested series display nonlinear property. Baek and Brock [14] first developed the nonlinear causality test for the interaction between stationary sequences. On this basis, Hiemstra and Jones [12] relax the requirement of Baek and Brock method [14] that the stationary series are independent and identically distributed by allowing the existence of a short auto-correlation, which improves the original nonlinear causality test. Furthermore, Diks and Panchenko [11] pointed out that the Hiemstra and Jones test [12] is not strictly consistent with the definition of Granger causality test, and cannot be used to do causality analysis. In order to overcome this drawback, Diks and Panchenko [11] construct a new nonparametric test which can be used to analyze the nonlinear Granger causality relationship between two stationary time series. This test can be described as follows: Assume that {Xt , Yt , t ≥ 1} are two strictly stationary time series, in Granger’s sense [10], {Xt } is a strictly Granger cause of {Yt } if past and current values of {Xt } contain additional information on future value of {Yt } that is not contained only in the past and current {Yt } values. ( ) l l Assume the delay vectors Xt x = (Xt −l+1 , . . . , Xt ) and Yt x = (Yt −l+1 , . . . , Yt ) , lx , ly ≥ 1 , the null hypothesis that past l observations of Xt x contain no useful information of Yt +1 can be described using the following equation:
⏐( ⏐
ly
l
H0 : Yt +1 ⏐ Xt x ; Yt
)
⏐ ⏐ l ∼ Yt +1 ⏐ Yt y
(4)
where, ‘∼’ denotes(the equivalence in distribution. For two strict stationary time series, Eq. (4) actually considers the ) distribution of the lx + ly + 1 dimensional vector Wt = (Xt , Yt , Zt ) where Zt = Yt +1 . Under the null hypothesis, the distribution of Wt is invariant. We drop the time index and follow Bekiros and Diks [15] assume lx = ly = 1. Thus, under the null hypothesis, the conditional distribution of Z given by (X , Y ) = (x, y) is equivalent to that of Z given by Y = y. In Eq. (2), the joint probability density distribution fX ,Y ,Z (x, y, z ) and its marginal should satisfy the following equation: fX ,Y ,Z (x, y, z ) fY (y)
=
fX ,Y (x, y) fY (y)
·
fY ,Z (y, z ) fY (y)
.
(5)
The above equation implies that X and Z are in dependent conditionally on Y = y for each fixed value of y. Diks and Panchenko [11] show that this formulated null hypothesis implies the following equation: q ≡ E fX ,Y ,Z (x, y, z ) fY (y) − fX ,Y (x, y) fY ,Z (y, z ) = 0.
]
[
(6)
Let fˆW (Wi ) denote a local density estimator of a dw -variate random vector W at Wi as the following equation: fˆw (Wi ) = (2εn )−dw (n − 1)−1
∑
IijW ,
(7)
j,j̸ =i
where, IijW = I Wi − Wj < εn , I(·) is the indicator function and εt is the bandwidth, depending on the sample size n. Given this estimator, the test statistic is a scaled sample version of q in Eq. (6):
(
Tn (εn ) =
n−1
∑(
n (n − 2)
)
)
fˆX ,Y ,Z (Xi , Yi , Zi ) fˆY (Yi ) − fˆX ,Y (Xi , Yi ) fˆY ,Z (Yi , Zi ) .
(8)
i
For lx = ly = 1, if εn = Cn−β (C > 0, 1/4 < β < 1/3), Diks and Panchenko [11] prove under strong mixing that the statistic in Eq. (8) follows the asymptotic distribution:
√ (Tn (εn ) − q) n
Sn
D
−→ N (0, 1)
(9)
D
where, −→ denotes convergence in distribution and Sn is the asymptotic variance of Tn (·). Following Diks and Panchenko’s suggestion [11], we use a right-tailed test.1 3. Data and preliminary analysis We choose daily price data of natural gas spot and futures traded in NYMEX. Our sample covers the period from January 7, 1997 to February 27, 2016, containing 4790 observations for each series. The data is obtained from the website of the US Energy Information Administration (EIA).2 We use price data of natural gas spot and futures with four different maturity contracts. The futures price is quoted for delivering a specified quantity of a commodity at a specified time and place in the future. Contract 1 denotes a futures contract with the earliest delivery date. Contracts 2–4 denote the successive delivery 1 Here, the author greatly thanks Dr. Diks for providing the programming code of this nonlinear causality test. 2 Website: http://www.eia.gov.
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Fig. 1. Spot and futures prices of natural gas.
Table 1 Descriptive statistics of natural gas spot and futures returns.
Mean (%) Maximum (%) Minimum (%) Std. dev.(%) Skewness Kurtosis Jarque–Bera Q(10)
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
−0.016
−0.014
−0.011
−6.65E−03
−3.66E−03
32.44
23.24 −22.64 3.1833 0.320 7.280 3733.35*** 29.950***
57.67 −56.82 44.79 0.616 23.757 86183.67*** 237.66***
−19.90 3.5059 0.523 8.293 5802.76*** 38.811***
21.52
18.27
−31.12
−19.37
2.8647 0.086 8.928 7010.31*** 21.619**
2.5878 0.059 7.366 3803.22*** 30.042***
Note: The Jarque–Bera statistic tests for the null hypothesis of Gaussian distribution. Q(10) is the Ljung–Box statistic of the return series for up to the 10th order serial correlation. The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively.
months following Contract 1. For natural gas, each contract expires on the third business day prior to the 25th calendar day of the month preceded the delivery month. If the 25th calendar day of the month is a non-business day, trading ceases on the third business day prior to the business day preceded the 25th calendar day. After a contract expires, Contract 1 for the remainder of that calendar month is the second following month. Fig. 1 gives the illustrations of evolutions of natural gas spot and futures prices. We can find that the dynamic changes of prices of different futures are very similar. In July 2001, natural gas prices reached the local peak. The plausible explanations are two aspects. First, the oil crisis during that period led to increases in the price of its substitution: natural gas. Second, more than 10 natural gas exporting countries including Russia and Iran agreed to set up a ‘‘gas OPEC’’ to control the production of natural gas. Both events caused higher natural gas prices. The largest crash occurred during the period of mid 2008–2009, which can be related to the effect of the global financial crisis. After the financial crisis, the gas prices evolved at a relatively low level. In the following part of this paper, we will focus on the natural logarithmic forms of natural gas prices. We use SP, FU1, FU2, FU3 and FU4 to denote spot price and futures prices of four contracts maturing in one, two, three and four months, respectively. Table 1 shows the descriptive statistics of the natural gas spot and futures price returns, defined by the first-order differences of logarithmic prices. The mean values of the five return series are relatively small and close to zero. The standard deviation and the range of spot return (maximum–minimum) is greater than futures returns, implying that spot market is more volatile. The standard deviations decrease with the increase of futures maturity, consistent with the well-known Samuelson effect. All five series are right-skewed and leptokurtic, evidenced by positive skewness and kurtosis higher than 3. The Jarque–Bera statistics show rejections of the null hypothesis of Gaussian distribution at 1% significance level for each series, indicating the fat-tailed distribution. The Q (10) statistics reject the null hypothesis at 5% confidence level, implying that natural gas spot and futures returns have strong correlated behaviors.
Y. Zhang, L. Liu / Physica A 490 (2018) 203–211
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Table 2 Result of unit root test. SP Intercept only ADF −2.572* PP −2.683* KPSS 1.636** Trend and intercept ADF −2.507 PP −2.628 KPSS 1.470**
FU1
FU2
FU3
FU4
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
−2.076 −2.100
−1.889 −1.930
−1.790 −1.839
−1.678 −1.686
−58.390** −66.427**
−74.287*** −74.339***
−76.653** −73.678**
−72.149** −72.137**
−72.817** −72.882**
1.690**
1.768***
1.867**
1.967**
0.050
0.113
0.143
0.191
0.260
**
***
**
**
−1.977 −1.995
−1.760 −1.794
−1.609 −1.648
−1.432 −1.434
−58.389 −66.423**
−74.288 −74.338***
−73.659 −73.689**
−72.162 −72.155**
−72.840** −72.910**
1.538**
1.606***
1.675**
1.734**
0.020
0.033
0.033
0.030
0.032
Note: This table reports the results of testing for unit root in logarithms of natural gas prices and their first order differences. The null hypothesis of ADF and PP tests is unit root and that of KPSS test is stationarity. The optimal lag lengths of ADF test are chosen based on Schwarz information criterion (SIC) and the optimal bandwidths of PP and KPSS are determined based on Newey–West criterion. * Rejections at 10% significance level. ** ***
Rejections at 5% significance level. Rejections at 1% significance level. Table 3 Correlation matrix between spot and futures returns.
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
∆SP ∆FU1 ∆FU2 ∆FU3 ∆FU4
1 0.3293 0.2420 0.2302 0.1992
1 0.9128 0.8141 0.7319
1 0.9289 0.8371
1 0.9263
1
PII ∆SP ∆FU1 ∆FU2 ∆FU3 ∆FU4
1.0000 0.1442 0.1332 0.1431 0.1479
1.0000 0.9484 0.9051 0.8719
1.0000 0.9652 0.9111
1.0000 0.9665
1.0000
PI
4. Empirical results To avoid the spurious regression results caused by the nonstationarity, the linear Granger causality analysis should be performed on stationary time series. Thus, prior to causality analysis, we should employ unit root tests to analyze the stationarity properties of the natural gas prices series. The results of unit root tests are presented in Table 2. We determine the order of integration of each series using Augmented Dickey–Fuller (ADF) test [16], Phillips–Perron (PP) test [17] and KPSS test [18]. The information criterion, like Schwartz Information Criterion (SIC), can be used to decide the lag length of the ADF method, the optimal bandwidths of PP and KPSS methods are determined by the Newey–West criterion. Based on each of these three methods, the unitroot tests are performed by taking into account two different models (with constant and with constant and trend). The null hypothesis of ADF and PP tests is a unit root while that of KPSS test is a stationary time series. For each futures price series, the ADF and PP statistics cannot reject the unit root null hypothesis while KPSS statistics significantly reject the null hypothesis of stationarity, implying a nonstationary process. For natural gas spot price series, although ADF and PP statistics show rejections of unit root null hypothesis at 10% significance level with intercept only, the KPSS statistic reject the null hypothesis of stationarity at 1% significance level. Thus, we can conclude that natural gas spot price series is nonstationary at 1% significance level. For the first differences of spot and futures prices series, all three types of statistics consistently indicate that they are stationary. In summary, we can conclude that each natural gas price series contain a unit root while the first order differences (i.e., the returns) are stationary. To analyze the changes in the lead-lag relationships between spot and futures over time, the period is divided into two segments. PI represents the period from January 7, 1997 to December 31, 2006, and PII represents the period from January 1, 2007 to February 27, 2016. Table 3 reports the correlation matrix among five return series in two periods. We can find in Table 3 that spot and futures returns are positively correlated. The correlations between two futures returns are much larger than those between returns spot and futures. With the contact time increasing, the correlations of the futures prices decrease. It is worth mentioning that the correlated behaviors between spot and futures returns are stronger in PI than PII. We also investigate the cross-correlations between natural gas spot and futures prices. It is widely accepted that financial time series are multifractal [19], we estimate the cross-correlation exponents with the MF-DCCA method following Wang et al. [20]. I will not go into details about MF-DCCA method here, please refer to Wang’s paper for the details. According Wang et al. [20], if scaling exponent is bigger than 0.5, the cross-correlations between the kinds off fluctuations related to q of two series are persistent (positive). An increase of one price is likely to be followed by an increase of the other price. If
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Table 4 Scaling exponents for return series with q varying from −10 to 10. q
PI
−10 −6 −2 2 6 10
PII
SP-FU1
SP-FU2
SP-FU3
SP-FU4
SP-FU1
SP-FU2
SP-FU3
SP-FU4
0.8394 0.8132 0.7839 0.7688 0.7329 0.6996
0.8750 0.8434 0.7994 0.7805 0.7582 0.7325
0.9871 0.859 0.8064 0.7775 0.7516 0.7256
0.9043 0.8698 0.8698 0.7996 0.7668 0.7370
0.8660 0.8324 0.8020 0.8095 0.7971 0.7669
0.8728 0.8355 0.7981 0.805 0.7968 0.7690
0.8524 0.8218 0.7958 0.8103 0.8047 0.7759
0.8504 0.8277 0.8074 0.8281 0.8228 0.7911
Table 5 Results of cointegration test between natural gas spot and futures prices. No.of cointegration equations
PI
PII
Trace statistic
None At most 1
Max-eigen statistic
Trace statistic
Max-eigen statistic
SPFU1
SPFU2
SPFU3
SPFU4
SPFU1
SPFU2
SPFU3
SPFU4
SPFU1
SPFU2
SPFU3
SPFU4
SPFU1
SPFU2
SPFU3
SPFU4
87.67* 2.87
70.23* 2.40
49.23* 2.19
38.60* 1.76
84.80* 2.87
67.83* 2.40
47.04* 2.19
36.84* 1.76
176.86* 1.30
86.18* 1.20
40.65* 0.71
29.81* 0.46
175.56* 1.30
84.98* 1.20
39.95* 0.71
29.35* 0.46
Note: This table shows the results of Johansen cointegration test. * Rejections at 5% significance level.
Table 6 Results of linear causality analysis. Variable
PI
PII
Raw data X
Y
SP SP SP SP
FU1 FU2 FU3 FU4
X →Y
Y →X *** *** *** ***
ECM filtered series
Raw data
X →Y
X →Y
Y →X
ECM filtered series Y →X
X →Y
Y →X
*** *** *** ***
Note: This table reports the results of linear causality test between natural gas spot and futures prices. The causality analysis for the cointegrated pairs is performed based on VEC specifications. The optimal lag lengths are determined based on Schwarz Information Criterion (SIC). ‘X → Y ’ denotes the null hypothesis that X does not Granger cause Y . The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively.
scaling exponent is smaller than 0.5, the cross-correlations between the kinds of fluctuations related to q of two series are anti-persistent (negative). An increase of one price is likely to be followed by a decrease of the other price. If scaling exponent is equal with 0.5, one series is not cross-correlated with the other, and the change of one price cannot affect the behavior the other price. As we can see in Table 4, all the scaling exponents in both PI and PII is bigger than 0.5, it means the relationships between natural gas spot and futures are persistent cross-correlated. And as the q changed, the scaling exponents vary with the change of q, which verifies the multiple characteristics of natural gas price series. When two time series are cointegrated, it is better for causality test to use a vector error correction model (VECM) rather than an unrestricted vector autoregressive model (VAR) [13]. Thus, we analyze the cointegration relationships between spot and futures before the causality analysis. Table 5 shows the results of the two periods of cointegration analysis based on Johansen’s maximum eigenvalues and trace test [21–23]. The results show that in the two periods, none of the maximum eigenvalue statistics or trace statistics can reject the null hypothesis of no cointegration between spot and futures. For more details, both of trace statistics and maximum eigenvalue statistics reject null hypothesis of no cointegration relationship between spot and futures prices at 5% significance level but cannot reject the null hypothesis of at most one cointegration equation. That is, natural gas spot and futures prices are cointegrated. Therefore, the linear Granger causality analysis between spot and futures prices in both two periods will be performed based on VECM specification. According to the results of previous cointegration tests, the VECM models of spot and futures price series are established by Schwarz information criterion (SIC). For convenience, we use the ‘‘X →Y ’’ to represent the null hypothesis: X does not Granger cause Y . As can be seen in Table 6, the results in PI indicate that natural gas futures prices can linearly Granger cause spot price. This evidence is consistent with the majority of the literatures on the price discovery of futures markets. Table 5 also reports the results of linear Granger causality tests on VECM filtered residuals. We can find that the linear causality behaviors in PI between natural gas prices disappear after the procedure of VECM filtering. And the result of PII is the same as PI. In existing studies, the nonlinear structures of energy prices have been well confirmed [24–27]. Moreover, nonlinear linkages between two energy prices have been found in recent studies [28,29]. Thus, it is necessary to examine the existence of nonlinear Granger causality. For this consideration, we detect the nonlinear structure of our sample data and then test the nonlinear causality between natural gas spot and futures returns using a nonparametric method proposed by Diks and Panchenko [11].
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Table 7 Results of BDS test. Length in S.D. (σ 2 )
Embedding dimension
PI
PII
Z statistic
0.7 0.7 0.7 0.7 0.7
2 3 4 5 6
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
∆SP
∆FU1
∆FU2
∆FU3
∆FU4
13.91*** 17.03*** 19.21*** 20.98*** 23.13***
4.48*** 6.76*** 8.05*** 8.96*** 10.59***
2.71*** 4.71*** 5.91*** 6.48*** 7.66***
1.92** 3.54*** 4.78*** 5.22*** 6.16***
3.34*** 4.65*** 5.76*** 5.93*** 6.76***
10.42*** 13.10*** 14.16*** 15.59*** 17.40***
4.95*** 8.35*** 9.74*** 10.79*** 11.91***
2.59*** 5.64*** 6.62*** 7.54*** 8.33***
1.25* 3.54*** 4.78*** 5.22*** 6.16***
3.34* 4.65*** 5.76*** 5.93*** 6.76***
Note: This table reports the results of BDS test. * Rejections at 10% significance level. ** ***
Rejections at 5% significance level. Rejections at 1% significance level.
Table 8 Result of nonlinear causality analysis. Variable
PI
PII
Raw data
ECM filtered series
Raw data
ECM filtered series
X
Y
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
SP SP SP SP
FU1 FU2 FU3 FU4
*** *** *** **
*** *** *** ***
*** *** *** ***
*** *** *** ***
*** *** *** ***
*** *** *** ***
*** *** ** **
*** *** *** ***
Note: This table reports the results of nonlinear causality test between energy prices and exchange rates. ‘X →Y ’ denotes the null hypothesis that X does not Granger cause Y . We set the lag length lx = ly = 1. The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively.
We choose the BDS test to detect the nonlinear structures of five return series in both two periods and show the results in Table 7. Let the embedding dimension (m) vary from 2 to 6. As we can see, the BDS statistics of five return series at most of embedding dimensions in both two periods show rejections of the null hypothesis of independent identity distributions at 1% significance level, in favor of nonlinear structures. Even though in PI, the BDS statistics of futures contract with the maturities of three months returns at m = 2 reject the null hypothesis at 5% significance level and in PII the BDS statistics of futures contract with the maturities of three and four months returns at m = 2 reject the null hypothesis at 10% significance level, the result is still valid. Thus, we conclude that the spot and futures returns display significant nonlinear properties, confirming the results in existing literatures [24–27]. Table 8 reports the results of Diks and Panchenko’s nonparametric Granger causality test [11]. When time series present nonlinear structures, the traditional Granger causality test would be invalid. Therefore, we use the nonparametric test method [11] to explore the nonlinear relationships between the spot price and futures prices of the natural gas market in the New York Stock Exchange. According Bekiros and Diks [15], we discuss the results for lags lx = ly = 1, the constant C for the bandwidth εn is set to be 7.5, which is suggested by Diks and Panchenko [11]. With the optimal value of β = 2/7 given by Diks and Panchenko [11] and about 2400 observations in each period, the selected bandwidth of εn is approximately one standard deviation of time series. In order to investigate whether there are any remaining causal relationships which are strictly nonlinear, we also report the nonlinear causality test for error correction model residuals. In both PI and PII, we can find significant bidirectional causality relationships existing between spot and four futures prices. Even for the residuals of the VECM model, the nonlinear causality relationships are still significant. Why do the results of linear and nonlinear Granger tests have the significant difference? What is the source of nonlinear Granger causality? Bekiros and Diks [15] show that volatility spillover can partly explain nonlinear causality. Based on this consideration, we employ a GARCH–BEKK model [30] to study the effects of volatility spillover (second-order co-movement) on nonlinear causality in our case. The BEKK–GARCH (1, 1) specification is defined as: Ht = C ′ C +
q ∑
A′jk εt −j εt′ −j Ajk +
j=1
p ∑
1/2
G′jk Ht −j Gjk , εt = Ht
vt .
(10)
j=1
Here, C⏐, Ajk and Gjk are (N ×N) matrices and C is an upper triangular matrix. Ajk is the conditional covariance matrix of εt with εt ⏐Φt −1 ∼ (0, Ht ) and Φt −1 is the available information set at time t − 1. The standardized residuals vt are obtained by 1/2
whitening the residuals εt /Ht . Table 9 reports the results of nonlinear causality analysis before and after GARCH–BEKK filtering in two periods. We can find that after BEKK–GARCH (1, 1) filtered, nonlinear causality behaviors are weaker in both of two series. For example, the lead-lag relationship between spot and futures with the maturities of four mouths disappear and spot price can nonlinearly
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Table 9 Result of nonlinear causality test for GARCH–BEKK filtered series. Variable
PI
PII
Raw data
BEKK–GARCH filtered series
Raw data
BEKK–GARCH filtered series
X
Y
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
X →Y
Y →X
SP SP SP SP
FU1 FU2 FU3 FU4
*** *** *** **
*** *** *** ***
* ***
*** *** ***
*** *** *** ***
*** *** *** ***
*
*** **
Note: This table reports the results of nonlinear causality test for GARCH–BEKK filtered residuals. ‘X → Y ’ denotes the null hypothesis that X does not Granger cause Y . We set the lag length lx = ly = 1. The asterisks *, ** and *** denote rejections at 10%, 5% and 1% significance levels, respectively. Table 10 Scaling exponents for GARCH–BEKK filtered series with q varying from −10 to 10. q
−10 −6 −2 2 6 10
PI
PII
SP-FU1
SP-FU2
SP-FU3
SP-FU4
SP-FU1
SP-FU2
SP-FU3
SP-FU4
0.8372 0.7880 0.6942 0.5112 0.3052 0.2229
0.8001 0.7605 0.6748 0.5262 0.3835 0.3167
0.7806 0.7387 0.6585 0.5225 0.4037 0.3488
0.7853 0.7378 0.6555 0.5333 0.4331 0.3847
0.7456 0.7240 0.6770 0.5625 0.4343 0.3807
0.7204 0.7016 0.6633 0.5688 0.4596 0.4094
0.7059 0.6847 0.6461 0.5633 0.4726 0.4293
0.6934 0.6717 0.6331 0.5560 0.4775 0.4402
Granger cause FU1 at 10% significance level only. Thus, volatility spillover can partly explain the nonlinearity in causality relationships, and from the results of nonlinear causality test for GARCH–BEKK filtered series, the weaker nonlinear causality behaviors caused by the volatility spillover in PII is stronger than which is in PI, it means the volatility spillover is stronger and stronger over time. We also explored the impacts of volatility spillover on cross correlations. Table 10 report the scaling exponents for GARCH–BEKK filtered series with q varying from −10 to 10. Compared with the scaling exponents in Table 4, the scaling exponents for GARCH–BEKK filtered series are less than the scaling exponents for returns series, which indicates that volatility spillover also affect the cross correlations between natural gas spot and futures prices. For further discuss, the scaling exponents vary with the change of q, when q > 0, the scaling exponents are bigger than 0.5, it means the GARCH– BEKK filtered series have the positive cross-correlations, and when q > 0, the cross-correlations are negative. The results are the same in PI and PII. Furthermore, the scaling exponents in PII are much close to 0.5, indicating the weak cross-correlations, which confirm the previous conclusion that the volatility spillover is stronger and stronger over time and can partly explain the lead-lag relationships between natural gas spot and futures. 5. Conclusions We explore the lead-lag relationships by the MF-DCCA method, the results indicate that the lead-lag relationships are existed and positive. Using the linear Granger causality test, we find that during both two periods, futures prices can Granger cause spot prices and the futures returns have a dominated role in the price discovery process. From nonlinear Granger causality test results, we can see that there are bidirectional Granger causality relationships between futures and spot returns. After the vector error correction model (ECM) filtering, the cross-correlation and bidirectional Granger causality still existed and significant. In addition, we consider the volatility spillover effects on the lead-lag relationships between spot and futures. The GARCH–BEKK model is used to capture the spillover effect. We find that volatility spillover can partly explain the nonlinear causality and the nonlinear cross-correlation. Acknowledgments We would like to show our sincere gratitude to anonymous reviewers whose comments and suggestions greatly improved the quality of the manuscript. This work is supported by the National Science Foundation of China under the grant Nos. 71401077 and 71771124. Li Liu’s work is also sponsored by Qing Lan Project in Jiangsu province, the financial support from open project of Jiangsu key laboratory of financial engineering (NSK2015-10) and the Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (PPZY2015B104). References [1] J. Hasbrouck, Intraday price formation in U.S. equity index markets, J. Finance 58 (2003) 2375–2399. [2] Y. Zhang, The empirical research of the relationship of spot and futures: based on the data of CSI300 index future, Value Eng. 33 (2011). [3] P. Srinivasan, P. Ibrahim, Price discovery and asymmetric volatility spillovers in Indian spot-futures gold markets, Int. J. Econ. Sci. Appl. Res. 5 (2012) 65–80.
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