Test for volatility spillover effects in Japan’s oil futures markets by a realized variance approach

Tadahiro Nakajima (Faculty of Economics, Kobe University, Kobe, Japan and The Kansai Electric Power Company, Incorporated, Osaka, Japan)

Studies in Economics and Finance

ISSN: 1086-7376

Article publication date: 2 November 2018

Issue publication date: 21 June 2019

Abstract

Purpose

The purpose of this paper is twofold. First, the paper examines the risk transmission between crude oil and petroleum product prices of Japan’s oil futures market. Second, it compares the performance of two tests for Granger causality using realized variance (RV) and the exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model.

Design/methodology/approach

The author measures the daily RV of crude oil, kerosene and gasoline futures listed on the Tokyo Commodity Exchange using high-frequency data, and he examines the Granger causality in variance between these variables using the vector autoregression model. Further, the author estimates the EGARCH model based on daily data and test for Granger causality in variance between commodity futures using Hong’s (2001) approach.

Findings

The results of the RV approach reveal that the hypothesis on the existence of a mutual volatility spillover between crude oil and petroleum product markets is accepted. However, the results of the conventional approach indicate that all the hypotheses on Granger causalities in variance are rejected. The methodology based on intraday high-frequency data exhibits higher power than the conventional approach based on daily data.

Originality/value

This is the first paper to investigate Japan’s oil market using RV. The authors conclude that the approach based on RV is universally adoptable when testing for Granger causality in variance.

Keywords

Citation

Nakajima, T. (2019), "Test for volatility spillover effects in Japan’s oil futures markets by a realized variance approach", Studies in Economics and Finance, Vol. 36 No. 2, pp. 224-239. https://doi.org/10.1108/SEF-01-2017-0011

Publisher

:

Emerald Publishing Limited

Copyright © 2018, Emerald Publishing Limited


1. Introduction

For a long time now, crude oil and oil products have been listed on commodity exchanges. These listings have made a large contribution to their efficient price formation. In 1983, the North American West Texas Intermediate (WTI) crude oil was listed on the New York Mercantile Exchange (NYMEX); in 1988, the European Brent crude oil was listed on the International Petroleum Exchange; and in 2001, the Middle East Dubai-Oman crude oil was listed on the Tokyo Commodity Exchange (TOCOM). Furthermore, gasoline and kerosene were listed on TOCOM in 1999. However, the world oil market is overwhelmed with uncertainty factors, including geopolitical risks in oil-producing countries, fluctuations in demand of newly emerging countries and movement of speculative money. In addition, the oil market is fraught with issues of greenhouse gas and resource nationalism. No company or country can avoid these risk factors, although states with huge power may hope to control such issues. We have taken some measures to stabilize the world oil market. First, the International Energy Agency and the Organization of the Petroleum Exporting Countries have committed to provide and share fair and objective information about the oil market. Secondly, we have had allowances that can flexibly respond to rapid fluctuations in the supply-demand balance. The main oil supplying countries have surplus productive capacity. On the other hand, the Organization for Economic Co-operation and Development countries can adjust the oil consumption in response to price fluctuations by increasing the use of oil substitute energy sources. In addition, these countries not only develop oil stockpiling bases but also hold a system to interchange the stockpiling of crude oil in the event of emergency supply disruptions. Finally, there is the enhancement of commodity exchange. Commodity exchanges have greatly contributed to the price formation efficiency. We have increased and diversified the market participants to ensure sufficient liquidity as well as to reflect a variety of information. Moreover, we have committed to disclose the trading information to ensure market transparency. However, the volatility tends to increase even with these measures.

This study’s contribution is twofold. First, this is the first study to investigate the volatility spillover effects between crude oil and oil product prices on Japan’s oil futures markets. Numerous studies have examined the volatility of oil prices in the context of the relationship between different economic variables in various countries. Ng and Pirrong (1996) examine the spot and futures price dynamics of gasoline and heating oil on NYMEX. They argue that there are volatility spillovers from futures to spot markets. Thuraisamy et al. (2013) and Li and Yang (2013) examine the volatility relationship between crude oil prices and economic variables other than oil product prices. However, there is no literature that analyzes the volatility transmission between crude oil and oil product markets in Japan. Figure 1 traces the first differences in the Dubai-Oman crude oil prices from the beginning of 2002 to the end of 2014. The volatility has become gradually higher and has been remarkable especially since 2007. On the other hand, petroleum and its products are essential for both daily public life and production activities. Moreover, crude oil has particularly been gaining importance as an alternative investment asset because the crude oil market was believed to have a low relationship with the traditional financial markets. In recent years, some articles[1] investigate the relationship between energy and stock prices to discuss how the financialization of energy commodities may affect potential hedging activities. Thus, it is important for practical as well as academic purposes to analyze the price volatilities of petroleum and its products, particularly in view of the strong demand for a risk hedge strategy against price fluctuations, such as option pricing and portfolio construction.

Second, we examine volatility spillover effects between markets by using both the realized variance (RV) model and the exponential generalized autoregressive conditional heteroscedasticity (EGARCH) model. Wei (2012) compares the volatility forecasting performance of three models in the Chinese fuel oil futures market, the generalized autoregressive conditional heteroscedasticity (GARCH) model, the stochastic volatility (SV) model and the RV model. Souček and Todorova (2014, 2013) estimate oil price volatility using the RV to investigate whether volatility affects other economic variables. However, there is no study that compares how the test for Granger causality in variance perform under the autoregressive conditional heteroscedasticity (ARCH) type model and the RV model.

Volatility is the second moment of asset returns and one of the determinants of value-at-risk (VaR)[2]. Furthermore, it is an essential variable in derivatives pricing. During the Black and Scholes (1973) era, volatility was assumed constant for a certain period and the sample standard deviation of returns was regarded as estimation of volatility. However, since several decades, it has been generally believed that volatility fluctuates stochastically. Enlge (1982) proposed ARCH as a time series model to formulate this stochastic fluctuation. Many models extending the ARCH model have been proposed. Bollerslev (1986); Glosten et al. (1993) and Nelson (1991) designed the GARCH model, the Glosten–Jagannathan–Runkle (GJR) model and the EGARCH model, respectively. These models assume that volatility is a latent variable that cannot be observed and estimate parameters from their returns. On the other hand, Andersen and Bollerslev (1998) proposed the realized volatility as an estimator that does not depend on theoretical models. Realized volatility is the square root of the RV, that is, the sum of squares of returns in a specific period. Realized volatility is a consistent estimator of the volatility in a period. Because the daily volatility can be easily estimated from high-frequency intraday data with the use of data processing technology based on a computer, a number of papers adopt the RV to examine the volatility of various economic variables.

The remainder of this paper is organized as follows. Section 2 explains the methodology applied; Section 3 describes the empirical results and the analyzed data, and Section 4 provides a summary and concludes the paper.

2. Methodology

This paper analyzes the volatility of crude oil, kerosene and gasoline futures prices listed on TOCOM. First, we calculate the RV for each commodity using high-frequency data and test the volatility spillover between the three markets using the Granger causality test based on the vector autoregression (VAR) model.

Second, we test Granger causality from the returns to volatility and apply the ARCH-type model to these futures prices. We then estimate the EGARCH model for each commodity futures price and test the Granger causality in variance between these markets using the Hong (2001) model, which uses the cross-correlation function (CCF) approach.

Finally, we compare the RV-VAR and EGARCH-CCF tests.

2.1 Realized variance measurement

To measure the variance of asset prices, Andersen and Bollerslev (1998) proposed the RV, which does not depend on a theoretical model. Moreover, the RV does not need to estimate any parameter, unlike the ARCH-type and SV models.

Given the n intraday returns {rt, rt+1/n, rt+2/n,⋯, rt+(n−1)/n} at date t, the RV at date t is as follows:

(1) RVt=i=0n-1rt+i/n2

Suppose that asset price ln P(s) at time s follows a diffusion process of the following equation:

(2) dlnPs=μsds+σsdWs
where μ(s) is the instantaneous drift; σ(s) is the instantaneous volatility; and W(s) is the Wiener process. In this case, the true variance at date t, σt2, can be written as follows:
(3) σt2=tt+1σs2ds

Under the condition of n → ∞ in equation (1), the condition of RVtσt2 holds. Therefore, we can estimate the precise volatility using the RV.

Two factors need to be considered when using the RV. First, microstructure noise increases as n increases. Zhang et al. (2005); Zhang (2006) and Bandi and Russell (2008) propose the optimal sampling methods. On the other hand, Andersen et al. (2001) suggest that a 5 min sampling interval is optimal for a liquid market. The same sampling frequency was chosen by Liu and Wan (2012); Haugom et al. (2014) and Sévi (2015). This paper follows these studies in that we use a 5 min sampling frequency.

Second, while calculating the RV, we need to pay attention on and/or make some adjustments with regard to the closing time of the trading market. In other words, we define the volatility at date t as that for the period from the previous trading at date t − 1 to the previous trading at date t, and therefore, while using the RV, we need to consider the period from the previous trading at date t − 1 to the first trading at date t. Similarly, when a market has a lunch break, as does TOCOM, the same consideration is needed.

This study adopts the Hansen and Lunde (2005) adjustment method. Hansen and Lunde (2005) propose the correction value, RVhlt, as follows:

(4) RVhlt=t=1TRt-R¯2t=1TRVtRVt
where Rt is the daily return; R¯ is the sample average of the daily returns; and RVt is calculated excluding nights and lunch break.

2.2 Granger causality test using vector autoregression

Granger (1969) developed a technique to examine the causality between variables systematically. The VAR model consisting of two stationary time series, xt and yt, can be written as follows:

(5) xt=a10+k=1pa11kxt-k+k=1pa12kyt-k+u1t
(6) yt=a20+k=1pa21kxt-k+k=1pa22kyt-k+u2t
where u1t and u2t are disturbance terms. This paper determines the appropriate lag length p on the Schwarz information criterion (SIC).

To examine the Granger causality from xt to yt, one needs to test the following hypothesis:

H0:a211=a212==a21p=0
(7) HA:a2110 or a2120 or or a21p0

If the null hypothesis is rejected, some coefficients are statistically significant, implying that xt Granger causes yt. On the other hands, if the null hypothesis is accepted, no coefficient is statistically significant, implying no Granger causality from xt to yt.

2.3 Granger causality test using the cross-correlation function

Cheung and Ng (1996) propose a two-stage procedure to test for causality in variance based on the residual CCF approach. This paper adopts the Hong (2001) test, an extension of the Cheung and Ng’s (1996) technique.

Here, we consider two stationary time series, Xt and Yt. At the first stage, we estimate the EGARCH model for each economic variable. At the second stage, we construct the squared standardized residuals by conditional variances and use the CCF of squared standardized residuals to test the null hypothesis of no causality in variance. Cheung and Ng’s (1996) statistic to test the null hypothesis of no causality in variance from lag 1 to k can be written as follows:

(8) S=Ti=1kcuv2icuu0cvv0
where cuv(i) is the i-th lag sample cross-covariance of squared standardized residuals, cuu(0) and cvv(0) are the sample variances of squared standardized residuals, and T is the sample size. If the test statistic is larger than the critical value of the chi-square distribution, the null hypothesis is rejected. To avoid severe size distortion, Hong (2001) suggested the following statistic:
(9) M=S-k2k

If this test statistic is larger than the critical standard normal distribution, the null hypothesis of no causality in variance is rejected.

3. Empirical results

3.1 Data and realized variance measurement

Our sample data cover the period from 1 January, 2013 to 30 December, 2014. This paper uses the intraday high frequency data in TOCOM’s oil market. Trading hours are separated between day session and night session. The day session opens at 9:00 and closes at 15:15, and the night session opens at 16:30 and closes at 4:00. The data used in this paper consist of daily and 5 min variables of the Dubai crude oil, kerosene and gasoline futures prices. These prices are expressed in yen per kiloliter. Figure 2 traces the natural logarithms of crude oil prices and its returns. Figures 3 and 4 provide the time series plots of kerosene and gasoline, respectively, and are based on daily data. As expected, these prices show similar movement, as the market trends of crude oil and petroleum product are displayed. Furthermore, the trends of these returns look exactly the same, including the returns’ spiking dates. Therefore, we believe that the hypothesis of volatility spillover between these markets is accepted by the Granger causality in variance tests using daily returns data.

We compute the RV for each variable using the widely used 5 min sampling interval. Because most of the microstructure noise effects can be removed, we can use as many observations as possible. The RV of crude oil, kerosene and gasoline, as defined by equation (4), are plotted in Figures 5, 6 and 7, respectively. Although the RVs of these three economic variables represent approximately the same scale and trends, the difference in spikes and persistence can be recognized with our detailed observations. Thus, it is doubtful whether the hypothesis of risk transmission between these markets can be accepted by Granger causality tests using the RV model.

The conditions for the Granger causality test based on VAR model is that all the variables are stationary. Accordingly, before we proceed with each causality test, we test for the stationarity status of the RV series of crude oil, kerosene and gasoline. Additionally, we test for the stationarity status of the daily returns series of crude oil, kerosene and gasoline futures prices. Various kinds of unit root tests have been proposed; however, we cannot determine which unit root test technique is the most appropriate. Therefore, we adopt the PP unit root test developed by Phillips and Perron (1988) and the KPSS unit root test developed by Kwiatkowski et al. (1992). The PP test has a null hypothesis of a unit root as against the alternative hypothesis of a stationarity. In contrast, the KPSS test has a null hypothesis of a stationarity as against the alternative hypothesis of a unit root. The results obtained through PP test applied to the RV series and the first differences of natural logarithm of these prices are reported in Table I. The results reject the unit root hypothesis in all cases. Table II reports the results obtained through the KPSS test applied to all the variables. The results indicate an acceptance of the stationarity hypothesis except for three cases. After considering the results of the PP and KPSS unit root tests, we might conclude that all the variables are stationary. Therefore, we can use the VAR model to test the Granger causality between the RV series of three commodity futures prices. Moreover, we can use the VAR model to test the Granger causality between the RV series and the first difference series of natural logarithm of price for each commodity.

3.2 Test for granger causality in variance using the realized variance and value-at-risk models

Let COt, KEt and GSt be the crude oil RV series, kerosene RV series and gasoline RV series, respectively. We determine the appropriate lag orders by minimizing the SIC to estimate the VAR model as follows:

(10) COt=2.79×10-5+5.88×10-1COt-1+2.13×10-1KEt-1+1.08×10-1GSt-1
(11) KEt=2.66×10-5+3.37×10-1COt-1+1.64×10-1KEt-1+2.67×10-1GSt-1
(12) GSt=2.19×10-5+2.08×10-1COt-1+8.13×10-2KEt-1+4.73×10-1GSt-1

We then conduct the Granger causality test in variance with equations (10), (11) and (12). The results are reported in Table III. We find that crude oil price Grange-causes both kerosene and gasoline prices in variance. The risk of crude oil price, which is the cost of petroleum product, should be the risk of the petroleum product’s prices. We can show that both kerosene and gasoline prices Granger-cause the crude oil price in variance. This indicates that the market risk of petroleum product affects its cost market risk. Moreover, Granger causality in variance exists between kerosene and gasoline prices. We can assume that the risk of these two markets transmits between each other through the raw material market because there are no substitute goods between kerosene and gasoline.

3.3 Test for granger causality in variance using the EGARCH model and Hong’s (2001) approach

First, we test the Granger causality from each time series return to variance by each VAR model. Let rCOt, rKEt and rGSt be the daily returns series of crude oil, kerosene and gasoline, respectively. We determine the appropriate lag orders by minimizing the SIC to estimate the VAR model as follows:

(13) rCOt=-4.25×10-4+4.26×10-3rCOt-1+1.29×10-2rCOt-2+2.74COt-1-3.39COt-2
(14) COt=2.88×10-5-2.46×10-3rCOt-1-1.26×10-3rCOt-2+0.623COt-1+0.208COt-2
(15) rKEt=-6.61×10-4+3.97×10-2rKEt-1-4.48×10-4rKEt-2-3.18KEt-1+4.81KEt-2
(16) KEt=2.84×10-5-2.12×10-3rKEt-1-8.03×10-4rKEt-2+0.512KEt-1+0.287KEt-2
(17) rGSt=-4.69×10-4+5.28×10-2rGSt-1+9.62×10-3rGSt-2+1.40GSt-1-0.864GSt-2
(18) GSt=2.31×10-5-2.21×10-3rGSt-1-8.39×10-4rGSt-2+0.502GSt-1+0.319GSt-2

Table IV reports the Granger causality test results. The hypothesis that each of the returns Granger causes each variance is accepted, whereas the hypothesis that each of the variances Granger causes each return is rejected. These results indicate that the application of the ARCH-model to these commodity futures prices is appropriate.

Second, we estimate the crude oil, kerosene and gasoline price dynamics using the EGARCH model developed by Nelson (1991). The conditional mean (xt) and variance (ht) are expressed respectively as follows:

(19) xt=cm+i=1kωixt-i+εt, εt=htut,
(20) loght=cv+i=1lαi|εt-iht-i|+βiεt-iht-i+i=1mγiloght-i,
where εt is the prediction error series; ut has a generalized error distribution (GED) which can be expressed as:
(21) ut=νλ21+1/νΓ(1/ν)exp(12|zλ|ν),λ=22/νΓ(1/ν)Γ(3/ν),
where Γ(·) is the Gamma function, and ν is the shape parameter, satisfying the condition 0 <ν ≤ ∞. All parameters (cm, ωi, cv, αi, βi, γi and ν) are estimated by using the maximum-likelihood method and selecting the lag lengths of each equation based on SIC. There is no non-negativity requirement on these parameters because logarithmic transformation is taken. The estimates are reported in Table V.

We select the AR(1)-EGARCH(1, 1) model for all series. As shown in Table V, the coefficients of the ARCH term (α), asymmetric parameter (β) and GARCH term (γ) are statistically significant at the 5 per cent level for all variables, except for the asymmetric parameter (β) coefficient for gasoline prices. The GED parameter (ν) is also statistically significant for all variables. As each is estimated to be less than two, the error terms are fat-tailed. In addition, the Granger causality test in variance using the CCF approach needs no autocorrelation of the squared standardized residuals. Therefore, we check the models’ specification using the Ljung–Box test. Table V also indicates the Ljung–Box test statistics at lag 10, Q2(10), which is the test statistics for the null hypothesis of no autocorrelation up to order 10 for squared standardized residuals. The null hypothesis is accepted for all variables.

Table VI reports the Granger causality test in variance results from the CCF approach. As no test statistics are larger than the critical 5 per cent significance level, the tests based on the EGARCH and CCF models do not reject the null hypothesis of no causality in variance between the three variables whereas those based on the RV and VAR models accept all of the Granger causality in variance between the variables.

4. Concluding remarks

This study is the first to examine Japan’s oil markets with the RV calculated using high-frequency data. We use the intraday high-frequency data to calculate the daily RV of crude oil, kerosene and gasoline futures prices listed on TOCOM. We investigate the volatility spillover effects between these markets using the Granger causality test method and the VAR model composed of the daily RV of these commodity futures. Statistically significant at the 5 per cent level, a mutual Granger causality in variance exists between the three variables, and, therefore, we establish the following hypothesis: The cost and price risks of petroleum product interact, whereas the price risk of petroleum product spillover each other thorough the cost risk.

In 2015, the annual trading volume of crude oil futures was 4 million contracts of 50 kiloliters on TOCOM, while it amounted to 202 million contracts of 1,000 barrels on NYMEX and 184 million contracts of 1,000 barrels on the Intercontinental Exchange (ICE). Numerous studies have tested the Granger causality in variance among crude oil prices and other economic variables, however there are no studies that investigate the volatility transmission between crude oil and oil product markets. As an extension to this study, we hope to confirm this hypothesis for NYMEX or ICE.

Furthermore, we estimate the EGARCH model with daily data and examine the volatility spillover between commodity markets using the Hong’s (2001) approach. First, we build the VAR model for each commodity using the daily returns and the RV to test the Granger causality from the returns to variance. This study also confirms that the application of these three economic variables to the ARCH-type model is valid because the hypothesis of no Granger causality is rejected for each of the futures. The Hong (2001) test results do not show any Granger causality in variance between these variables. Although a glance through Figures 2-4, which trace the daily data of these prices and returns, may indicate that the hypothesis of volatility spillover between these markets is accepted by the Granger causality in variance tests using the daily return data, these empirical results do not reject the hypothesis of no Granger causality in variance. In contrast, Figures 5-6, which display the daily RV, do not predict the risk transmission between these markets although the Granger causality test using the RV-VAR model accepts the hypothesis of volatility spillover effects. The results that betray our easy intuition are interesting. This indicates that the Granger causality in variance test using the RV-VAR model is more powerful than that using the EGARCH-CCF model. Because of the increasing convenience of accessing high-frequency intraday data and the advancement in computer processing power, for research on volatility, we recommend the use of the RV approach, which can be considered the actually measured value of variance.

Additionally, this paper examines the relationship between these futures prices. We test the Granger causality in mean between these futures markets by Hong’s (2001) and Granger’s (1969) approaches using these daily returns. The results do not reject the hypothesis of no Granger causality in mean in all cases. However, we cannot deny the mutual effect between the three futures prices because at the 5 per cent significance level, both the correlation between two out of the three futures returns and the correlation between the standardized residuals of each EGARCH model exist. As these markets have high liquidity, we believe that the smooth and quick arbitrage between these markets can efficiently form these prices in less than 24 h. The returns from one out of three do not help in the prediction of others’ returns so long as the daily data are used. In this context, we recommend the use of intraday high-frequency data.

Once the volatility rises in one market, market players take the investment behavior in other markets for their actual loss measures and risk management. In other words, they sell other assets with unrealized capital gains to fill up the loss and buy or sell other assets to rebuild their portfolio composition ratio. These investment behaviors cause the volatility transmission between markets. The investors’ behavior to volatility fluctuations is rarely decided mechanically because there are many cases that require human beings’ judgment based on the market value of their portfolio, calculation of their VaR and research on other markets. Therefore, the volatility transmits between markets over more than a day, even if markets have high liquidity. On the other hand, the return transmission between markets is mainly caused by arbitrage in which algorithmic trading is used. Thus, if we can ensure market liquidity, the return transmits between markets in less than a day at the latest. In this case, we can accept the efficient price formation hypothesis when analyzing the daily data. This study reveals that the above hypotheses can be accepted in TOCOM.

Figures

First differences in Dubai-Oman futures prices

Figure 1.

First differences in Dubai-Oman futures prices

Time series plots of the crude oil prices and returns

Figure 2.

Time series plots of the crude oil prices and returns

Time series plots of the kerosene prices and returns

Figure 3.

Time series plots of the kerosene prices and returns

Time series plots of the gasoline prices and returns

Figure 4.

Time series plots of the gasoline prices and returns

Time series plots of the crude oil RV

Figure 5.

Time series plots of the crude oil RV

Time series plots of the kerosene RV

Figure 6.

Time series plots of the kerosene RV

Time series plots of the gasoline RV

Figure 7.

Time series plots of the gasoline RV

The PP unit root test results

Variables Adj. t-Statistics
None Constant Constant + trend
Crude oil
RV –4.193* –6.102* –6.182*
Return –21.882* –21.890* –22.194*
Kerosene
RV –4.948* –7.901* –7.895*
Return –20.916* –20.919* –21.188*
Gasoline
RV –4.659* –6.761* –6.750*
Return –20.832* –20.836* –20.983*
Critical value (1% level) –2.570 –3.443 –3.977
Note:
*

Indicates that the unit root hypothesis is rejected at the 1% significance level

The KPSS unit root test results

Variables LM-Statistics
Constant Constant + trend
Crude oil
RV 0.364 0.368*
Return 0.682 0.135
Kerosene
RV 0.378 0.383*
Return 0.647 0.136
Gasoline
RV 0.365 0.355*
Return 0.422 0.119
Critical value (1% level) 0.739 0.216
Note:
*

Indicates that the stationary hypothesis is rejected at the 1% significance level

Granger causality in variance test results based on the RV and the VAR model

Null hypothesis F statistics Probabilities
KE does not Granger cause CO 13.1679 0.00*
CO does not Granger cause KE 48.4025 0.00*
GS does not Granger cause CO 8.75141 0.00*
CO does not Granger cause GS 15.2824 0.00*
GS does not Granger cause KE 31.4083 0.00*
KE does not Granger cause GS 5.37568 0.02*
Note:
*

Means significant at the 5% level

Granger-causality between the return and variance

Null hypothesis F statistics Probabilities
rCO does not Granger cause CO 32.1967 0.00*
CO does not Granger cause rCO 0.15306 0.86
rKE does not Granger cause KE 18.6804 0.00*
KE does not Granger cause rKE 0.40405 0.67
rGS does not Granger cause GS 24.8307 0.00*
GS does not Granger cause rGS 0.02715 0.97
Note:
*

Means significant at the 5% level

Estimation results of the AR-EGARCH model

Estimated coefficients Crude oil Kerosene Gasoline
Mean
cm –0.000278 (0.50) 0.000138 (0.71) 0.000187 (0.61)
ω –0.031753 (0.49) 0.007452 (0.87) 0.048044 (0.31)
Variance
cv –0.279060 (0.03)* –0.378540 (0.01)* –0.386600 (0.00)*
α 0.175325 (0.00)* 0.217120 (0.00)* 0.227288 (0.00)*
β –0.066639 (0.01)* –0.066958 (0.03)* –0.038025 (0.19)
γ –0.983223 (0.00)* 0.976285 (0.00)* 0.976559 (0.00)*
GED
ν 1.336151 (0.00)* 1.321078 (0.00)* 1.416286 (0.00)*
Diagnostic
Q2(10) 1.9221 (1.00) 0.9374 (1.00) 1.0708 (1.00)
Notes:

The values in parentheses indicate the p-values. Q2(10) is a test statistic for the null hypothesis; it indicates that no autocorrelation exists up to order 10 for standardized residuals squared;

*

means significant at the 5% level

Granger causality in variance test results based on the EGARCH model and the Hong’s (2001) approach

Causality direction Hong’s statistic (lag length = 1) Hong’s statistic (lag length = 5)
Kerosene to crude oil –0.70074 (0.24) –1.35291 (0.09)
Crude oil to kerosene –0.67859 (0.25) –1.27933 (0.10)
Gasoline to crude oil –0.70289 (0.24) –1.25389 (0.11)
Crude oil to gasoline –0.63066 (0.26) –1.13610 (0.13)
Kerosene to gasoline –0.69300 (0.24) –1.46529 (0.07)
Gasoline to kerosene –0.70633 (0.24) –1.32288 (0.09)
Note:

The values in parentheses indicate the p-values

Notes

1.

Representative articles include Batten et al. (2017).

2.

Huang and Tseng (2009); Rossignolo et al. (2013) and Kinateder and Wagner (2014) are some of the typical studies that derive VaR from estimated volatility.

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Corresponding author

Tadahiro Nakajima can be contacted at: nakajima.tadahiro@a4.kepco.co.jp