متغیر زمان و وابستگی نامتقارن بین بازار نفت خام آتی و لحظه ای : شواهدی از مدل آرجی گارچ رابطه ای مبتنی بردوگانه ترکیب

This paper designs a Mixture copula-based ARJI–GARCH model to simultaneously investigate the dynamic process of crude oil spot and futures returns and the time-varying and asymmetric dependence between spot and futures returns. The individual behavior of each market is modeled by the ARJI–GARCH process. The time-varying and asymmetric dependence is captured by the Mixture copula which is composed of the Gumbel copula and Clayton copula. Empirical results show three important findings. First, jumping behavior is an important process for each market. Second, spot and futures returns do not have the same jump process. Third, the tail dependence between spot and futures markets is time-varying and asymmetric with the magnitude of upper tail dependence being slightly weaker than that of lower tail dependence.

A growing body of studies has investigated the impact of crude oil price on economic and financial developments and the findings indicate a negative effect. These reports have stimulated subsequent research that aims to better understand crude oil prices in spot and futures markets. Due to the unusually large changes that have recently occurred in these markets and their asymmetric relationship, the current paper proposes a Mixture copula-based ARJI–GARCH model to investigate both the interdependence of oil spot and futures prices and their individual dynamic behaviors. Previous studies in this area have provided limited results. Studies show that generalized autoregressive conditional heteroskedasticity (GARCH) model and its extended models are very popular approach in modeling and forecasting spot return and futures return. For exploring the linear relationship between spot and futures returns, the constant correlation coefficient (CCC) GARCH and the dynamic conditional correlation (DCC) GARCH have been adopted.1However, the aforementioned approaches are hampered in their ability to successfully capture the true dynamic relationship between spot and futures markets. The reason is that they have yet to provide a flexible and reasonable way to jointly investigate the marginal behavior of these markets and their time-varying and asymmetric relationship. Recent studies have indicated that the processes of crude oil spot and futures prices heavily rely on many components, including usual factors (e.g., routine energy consumptions and productions) and unusual factors (e.g., unexpected demand and supply shocks). The aforementioned GARCH models are only suitable for modeling the volatility clustering characteristics caused by the usual factors. However, when sudden and large price changes happen, the adequacy of the GARCH models patently cannot be guaranteed. To overcome this defect, Chan and Maheu (2002) propose the autoregressive conditional jump intensity (ARJI) model, which simultaneously considers the persistence in conditional variance and irregular price changes. This model seems to be a better specification in describing the crude oil price dynamics. For example, Lee et al. (2010) utilize the ARJI specification to investigate the time series properties of the West Texas Intermediate crude oil spot and futures markets and demonstrate that periods of sudden large changes in crude oil price can be appropriately identified through the jump intensity, which describes the possible number of jumps at each time point.2 Several empirical studies have documented the importance of the fat-tailed property in accurately forecasting the Value-at-Risk (VaR) of crude oil spot and futures prices. For example, Giot and Laurent (2003) demonstrate that the skewed student distribution is better in evaluating VaR forecasts than the normal distribution. Fan et al. (2008) find that the generalized error distribution (GED) has better performance in forecasting VaR than the normal distribution. On the other hand, Agnolucci (2009) shows that the accuracy of volatility forecasts hinges on the distribution specification and that the GED specification performs better at forecasting conditional variance than the normal specification. With respect to the ARJI–GARCH model, an advantage worth mentioning is that it can generate skewed and fat-tailed conditional density even if the assumption that error terms are normally distributed holds.3 Although the correlation coefficient, determined by the bivariate GARCH model with the elliptic distributed error terms (e.g., the normal distribution or Student's t distribution), is the most popular measure for describing the linear relationship between crude oil spot and futures returns, it cannot capture their nonlinear relation. This is a critical drawback, since the nonlinear relation between spot and futures returns has become an important issue in recent years. Another drawback is that the correlation measure may misestimate their linear relationship due to the incorrect specification of the bivariate distribution. The copula approach developed by Patton (2006) can remedy the above two weaknesses simultaneously. It can deal with the computational difficulty in estimating complex joint distribution, and measure the nonlinear interdependence through two measures: the tail dependence index and Kendall's tau index. The main reason for the popularity of the copula approach is that the Sklar (1959) theorem indicates that the joint density of two returns can be obtained if both the marginal density of each return and the copula function can be correctly specified. It is well-known that the high correlation coefficient between spot and futures markets could be the reason that the futures product is a good hedge instrument for the corresponding spot good: futures prices contain information that can help to forecast spot prices. However, this high linkage does not mean that the two markets, in most cases, move in the same direction, irrespective of an increase or decrease in price. Moreover, the high linkage does not guarantee that the two markets will have a symmetric correlation. Unlike the existing studies which use the correlation coefficient to measure the linkage between crude oil spot and futures markets, this paper focuses on the possible asymmetric tail dependence and the possible asymmetric comovement. Therefore, this paper develops a Mixture copula-based ARJI–GARCH model to discuss the nonlinear dependence between crude oil spot and futures markets. In essence, combining the ARJI–GARCH model with the mixture copula specification yields a new model. The main advantages of this new proposed model are briefly introduced as follows. First, in order to take the vital characteristics (e.g., volatility clustering, a sudden large variation in price, cointegration and the non-normal distribution) found in the extant literature into account, the ARJI–GARCH model with the long-term disequilibrium adjustment term (the so-called error correction term) is used to model the process for crude oil spot and futures returns.4 Moreover, to investigate whether they are affected by the same factors, the research allows the spot and futures returns to have different jump processes. Second, the mixed copula is composed of the Gumbel copula, which focuses only on the right-tail structure, and the Clayton copula, which concentrates attention only on the left-tail structure. Hence, the Mixture copula allows the asymmetric dependence, i.e., the comovement structure between spot and futures returns in the extreme right tail, to be different from that in the extreme left tail. Moreover, the nonlinear dependence is allowed to be time-varying. In comparison to the Gaussian copula, Student's t copula, Gumbel copula and Clayton copula,5 the Mixture copula has more flexibility in describing the joint dependence over time without assuming complex joint distributions. The remainder of this paper is organized as follows. Section 2 briefly introduces the copula theory and then describes the marginal distributions with the ARJI–GARCH process and four copula densities. Section 3 reports the empirical results and discusses the asymmetric dependency relation. Section 4 provides the conclusions.

A great many studies have investigated the linkage between crude oil spot and futures markets by utilizing the correlation coefficient. However, the information provided by the correlation coefficient is severely limited. What is the relationship between spot and futures markets when both are in a very good condition? What is their relationship when both markets are in a very bad condition? Finally, how do they differ in good and bad conditions? This paper designs a Mixture copula-based ARJI–GARCH model to answer the above questions. The model adopted here has at least two advantages. First, the ARJI–GARCH model can capture usual and unusual price changes for each market. Second, the Mixture copula, which is composed of the Gumbel copula and the Clayton copula, can measure the tail dependence between two markets and investigate whether there is a difference in right-tailed and left-tailed dependence. Some interesting results are observed. First, for each market, the ARJI–GARCH model has a better in-sample fit than the AR–GARCH model. Second, the spot and futures markets have very different jump processes. The common jump specification adopted by recent studies may distort the relationship between the two markets. Moreover, the disequilibrium term is a significant explanatory variable for the spot return but not for the futures return. Third, the concordance probability is remarkably lager than the disconcordance probability. Fourth, the Mixture copula shows evidence in favor of asymmetric tail dependence: the upper tail dependence is slighter lower than the lower tail dependence.