Modeling conditional correlations in Crude oil returns
Roengchai Tansuchat
Faculty of Economics, Maejo University, Thailand
Faculty of Economics, Chiang Mai University, Thailand
Chialin Chang
Department of Applied Economics
NationalChungHsingUniversity, Taichung, Taiwan
Michael McAleer
School of Economics and Commerce
University of Western Australia
Abstract
This paper estimates univariate and multivariate conditional volatility and conditional correlation models on spot, forward and futures returns from 3 major benchmarks of international crude oil markets: Brent, WTI and Dubai. Modelling the conditional correlation is conducted by using CCC model of Bollerslev (1990), VARMA-GARCH model of Ling and McAleer (2003), VARMA-AGARCH model of McAleer, et al. (2008) and DCC model of Engle (2002). The constant conditional correlations across the conditional volatilities of returns estimated by CCC-GARCH(1,1) are high. This paper also presents the ARCH and GARCH effect for returns and significant interdependences in the conditional volatilities across returns in each market.The estimates of volatility spillovers and asymmetric effects for negative and positive shocks on conditional variance suggest that VARMA-GARCH is superior to VARMA-AGARCH model. In addition to these results, DCC model gives the statistically significant estimates for the returns in each market. Obviously, this finding shows that the assumption of constant conditional correlation was not hold in empirical experiment.
Keywords : Constant conditional correlation; Dynamic conditional correlation; Multivariate GARCH model; Forward prices and returns; Futures prices and returns; WTI; Brent; Dubai
JEL : C22, C32, G17, G32
Email :
1. Introduction
Crude oil is arguably the most influential physical commodity in the world as it provides energy for all kinds of human activities in forms of refined energy products such as liquefied petroleum gases (LPGs), gasoline and diesel. As a result, crude oil is the world’s most dynamically traded commodity and affects many economies around the globe. For instance, Sadorsky (1999) found that oil price volatility shocks have asymmetric effects on the economy, namely, changes in oil prices affect economics activity, but changes in economic activities have little impact on oil price. Consequently, oil price fluctuations have large macroeconomic impacts. In addition, Guo and Kliesen (2005) explained changing in oil price affects the aggregate economic activity in two ways: (1) the change in the dollar price of crude oil (relative price change) and (2) the increase in uncertainty about future price.
Substantial research has been conducted on the volatility of spot, forward and futures price. The models of the crude oil price volatility are classified as univariate and multivariate conditional volatility. In case of univariate conditional volatility model, Fong and See (2002) examined the temporal behavior of daily returns on crude oil futures using a Markov switching model of the conditional volatility. Lanza et al. (2006), used the AR(1)-GARCH(1,1) and AR(1)-GJR(1,1) to estimate the conditional volatility based on the forward and futures returns. Manera et al. (2006), used the univariate ARCH and GARCH to estimates spot and forward returns, standard diagnostic tests also showed that the AR(1)-GARCH(1,1) and AR(1)-GJR(1,1) specification were statistically adequate for both the conditional mean and the conditional variance. Sadorsky (2006) investigated the forecast performance of a large number of models. The fitted model for heating oil and natural gas volatility is TGARCH, whereas for crude oil and unleaded gasoline volatility is GARCH model. Lee and Zyren (2007) calculated historical volatility and GARCH models to compare the historical price volatility behavior of crude oil, motor gasoline and heating oil in U.S. markets since 1990. They combined a shift variable in the GARCH/TARCH models to capture the response of price volatility to a change in OPEC’s pricing behavior. Narayan and Narayan (2007) model crude oil price volatility using daily data by EGARCH model to gauge two features of crude oil price volatility, namely asymmetry and persistence of shocks.
For the multivariate conditional volatility model, Lanza et al. (2006) estimated modeling dynamic conditional correlations (DCC) in WTI oil forward and future returns by CCC-MGARCH of Bollerslev (1990) and DCC of Engle (2002). They found that DCC can vary dramatically, being negative in four of ten cases and being close to zero in another five cases. Only in the case of the dynamic volatilities of the three-month and six-month future returns is the range of variation relatively narrow. Manera et al. (2006), estimated DCC in the returns on Tapis oil spot and one month forward prices using CCC-MGARCH, VARMA-GARCH of Ling and McAleer (2003), VARMA-AGARCHof McAleer et al. (2008) and DCC, and also tested and compared for volatility specification.
Trojani and Audrino (2005) purposed a new multivariate DCC-GARCH model (multivariate tree-structured DCC-GARCH) that extended existing approaches by admitting multivariate thresholds in conditional volatilities and conditional correlations. They found that in some Monte-Carlo simulations the model is able to fir correctly a GARCH-type dynamics and a complex threshold structure in conditional volatilities and correlations of simulated data. In empirical data of international equity markets, the estimated conditional volatilities are strongly influenced by GARCH-type and multivariate threshold effects. They concluded that conditional correlations are determined by simple threshold structures whereas no GARCH-type effect could be identified.
The purpose of this paper is to estimate the univariate and multivariate conditional volatility models for the returns on spot, forward and futures in Brent, WTI and Dubai markets. The paper is organized as follow, section 2 discusses briefly the univariate and multivariate GARCH models to be estimated. Section 3 explains the data, descriptive statistic and unit-root testing. Section 4 describes the empirical estimates and some diagnostic tests of the univariate and multivariate models. Section 5 provides some concluding remarks.
2.Econometric Models
2.1Univariate conditional volatility model
Following the seminal paper by Engle (1982), consider the time series and the associated error, , which is the expectation of the conditional mean on the information set at time . The generalized autoregressive conditional heteroscedastity (GARCH) (r,s) model of Bollerslev (1986) following from ARMA model is as follows:
, (1)
(2)
where , and are sufficient to ensure that the conditional variance . represents the ARCH effect, or the short-run persistence of the shock to returns and represent the GARCH effects. The sum of measures the persistence of the contribution of shocks to return i to expected long-run persistence.
Equation (2) assumes that the conditional variance is a function of the magnitudes of the lagged residuals and not their signs that means a positive shock has the same impact on the conditional variance as a negative shock . To accommodate differential impacts on the conditional variance between positive and negative shock, Glosten et al. (1992) proposed the asymmetric GARCH, or GJR model, is given by
(3)
where
(4)
is an indicator function to differentiate between positive and negative shocks on conditional volatility. When , the sufficient condition to ensure that the conditional variance is , , and . The short-run persistence of positive and negative shocks is given by and respectively. When the conditional shocks, , follow a symmetric distribution, the expected short-run persistence is , and the contribution of shocks to expected long-run persistence is .
In order to estimate parameters of model (1)-(3), the technique is known as maximum likelihood estimation using a joint normal distribution of . However, when the process of does not follow normal distribution or the conditional distribution is not perfectly known, the solution to maximizing the likelihood function is defined as the pseudo- or quasi-MLE (QMLE).
The basic properties of GARCH family, Bollerslev (1986) showed the necessary and sufficient condition for the second-order stationarity of GARCH model as: . For the GARCH(1,1) model, Nelson (1990) obtained the log-moment condition for the strict stationary and ergodicity as follow: , which is important in deriving the statistical properties of QMLE. For the GJR(1,1), Ling and McAleer (2002) presented the necessary and sufficient condition for is . McAleer et al. (2002) established the log-moment condition for GJR(1,1) as , and showed that it is sufficient for the consistence and asymptotic normality of the QMLE for GJR(1,1)
2.2Multivariate conditional volatility model
The typical specification underlying the multivariate conditional mean and conditional variance in returns are given as follow:
(5)
where , is a sequence of independently and identically distributed (i.i.d.) random vectors, is the past information available to time t, , m is number of returns, and . The constant conditional correlation (CCC) model of Bollerslev (1990) assumes that the conditional variance for each return, , , follows a univariate GARCH process, that is
(6)
where represents the ARCH effect, or short-run persistence of shocks to return i, and represents the GARCH effects, or the contribution of shocks to return i to long-run persistence, namely
The conditional correlation matrix of CCC is , where for . From (5), , , and , where is the conditional covariance matrix. The conditional correlation matrix is defined as , and each conditional correlation coefficient is estimated from the standardized residuals in (5) and (6). Therefore, there is no multivariate estimation involved for CCC, except in the calculation of the conditional correlations.
Although the CCC specification in (6) is a computationally straightforward multivariate GARCH model, it assumes independence of the conditional variances across returns and does not accommodate asymmetric behavior. To contain interdependencies, Ling and McAleer (2003) proposed a vector autoregressive moving average (VARMA) specification of the conditional mean in (5) and the following specification for the conditional variance:
(7)
where , , and W, for and for are matrices. As in the univariate GARCH model, VARMA-GARCH assumes that negative and positive shocks have identical impacts on the conditional variance. To separate the asymmetric impacts of the positive and negative shocks, McAleer, Hoti and Chan (2008) proposed the VARMA-AGARCH specification for the conditional variance, namely
(8)
where are matrices for , and , where
.
If , (7) collapse to the asymmetric GARCH, or GJR model. Moreover, VARMA-AGARCH reduces to VARMA-GARCH when for all i. If and and being diagonal matrices for all i and j then VARMA-AGARCH reduces to CCC-MGARCH. The parameters of model (5)-(8) are obtained by maximum likelihood estimation (MLE) using a joint normal density. When does not follow a joint multivariate normal distribution, the appropriate estimator is defined as the Quasi-MLE (QMLE).
Unless is a sequence of iid random vectors, or alternatively a martingale difference process, the assumption that the conditional correlations are constant may seen unrealistic. In order to make the conditional correlation matrix time-dependent, Engle (2002) proposed a dynamic conditional correlation (DCC) model. The DCC model is defined as:
, (9)
(10)
where is a diagonal matrix of conditional variance, and is the information set available to time t. The conditional variance, , can be defined as a univariate GARCH model as follow:
(11)
If is a vector of i.i.d. random variables, with zero mean and unit variance, in (10) is the conditional covariance matrix (after standardization, ). The are used to estimate the dynamic conditional correlation, as follows:
(12)
where the symmetric positive definite matrix is given by
(13)
in which and are scalar parameters to capture the effect of previous shocks and previous dynamic conditional correlations on current dynamic conditional correlation, and and are non-negative scalar parameters satisfying . As in is conditional on the vector of standardized residuals, (13) is conditional covariance matrix. is the unconditional variance matrix of .
4.Data
The data utilized in this study is the dailysynchronous closing price of spot, forward and futures crude oil prices from 3 major crude oil markets namely, Brent, WTI and Dubai. All 4,659 price observations are starting from 2 January 1991 to 10 November 2008 and are obtained from DataStream database service. The sample ranges and number of observation of each market are presented in Table 1. The returns of crude oil prices i of market j at time t in a continuous compound basis are calculated as , where and are the closing prices of crude oil price i of market j for day and , respectively. The univariate and multivariate conditional volatility models are estimated by EViews 6 and RATS software package.
The descriptive statistics for the return series of crude oil returns are summarized in Table 1. The sample mean is quite small, but the corresponding variance of return is much higher. Both negative skewness and high kurtosis statistics in the Table 1 present that the returns is not distributed normally. Similarly, under the null hypothesis of normality, the distributions of the sample return series of the Jarque-Bera test (J-B) statistics are also rejected. The logarithm of crude oil prices are plotted in Fig. 1. It is certainly clear that there is substantial clustering in the volatilities, meaning a turbulent trading day tends to be followed by another turbulent day, while a tranquil period tends to be followed by anther tranquil period.
The empirical results of the unit root tests for all sample returns in each market are summarized in Table 2. The Augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) test are used to explore the existence of unit roots in the individual series. Both tests have same the null hypothesis being used to check nonstationarity in each time series data. The large negative values for all cases show that all of the individual return series reject the null hypothesis at the 1% significant level, which means all returns series are stationary.
[Insert Table 1-2 here]
[Insert Figure 1 here]
5.Estimation
The univariate estimates of the conditional volatilities, GARCH (1,1) and GJR(1,1) model with different mean equations model based on the spot, forward and futures return in each market are given in Tables 3 to 5. Their respective estimates and Bollerslev-Woodridge (1992) robust t - ratios of each parameter are presented. All models also present log-moment and second moment condition which are consistent to statistical properties. Estimated second moment value of GARCH(1,1) and GJR(1,1) model namely and are less than 1, and estimated log-moment value of GARCH(1,1) and GJR(1,1) model given by and are less than 0.
The univariate GARCH estimates of Brent market in Table 3, coefficients in mean equations of Panel 3a. are not all statistically significant. The mean equation of AR(1)-GARCH(1,1) is significant only forward return, while ARMA(1,1)-GARCH(1,1) in every return series are significant. In addition, the coefficient in conditional variance equations from both AR(1)-GARCH(1,1) and ARMA(1,1)-GARCH(1,1) are all significant.Consequently, the ARMA(1,1)-GARCH(1,1) model is preferred than AR(1)-GARCH(1,1). In case of asymmetric GARCH(1,1) of Panel 3b, only coefficients of mean equation of ARMA(1,1) are significant. The estimates of the asymmetric effect at the univariate model are not statistically significant except spot return.
The results from the univariate estimation of WTI market are reported in Table 4. The robust t ratios show that the ARMA(1,1)-GARCH(1,1) specification of all returns is statistically adequate for both the conditional mean and conditional variance. While the coefficients of conditional mean equation of AR(1)-GARCH(1,1) are insignificant. The univariate GJR models are present in Panel 4b. in Table 4. Only forward of ARMA-GARCH model is significant. However, the variance equations do not present the asymmetric effect of negative and positive shock on conditional variance.
In case of Dubai market from Table 5, coefficients in mean equation of spot and forward return in Panels5a. and 5b. are significant only in AR(1)-GARCH(1,1) and AR(1)-GJR(1,1) model. Panel 5a presents the coefficients in conditional variance equation from AR(1)-GARCH(1,1) are all statistically significant. Whereas, in Panel 5b conditional variance coefficients are all significant only spot return, this show the asymmetric effect of negative and positive shock on conditional variance.
[Insert Table 3-5 here]
Table 6 presents the constant conditional correlation across spot, forward and futures returns in each market using the CCC model based on estimating univariate GARCH(1,1). For three returns in Brent and WTI market in Panel 6a and 6b provide 6 conditional correlation, while two returns of Dubai market in Panel 6c give one conditional correlation. The highest estimated conditional correlation in Brent market is 9.40 between standardized shocks to the volatility of spot and forward. In case of WTI market, the highest estimated conditional correlation in Brent market is 8.83 between standardized shocks to the volatility of spot and futures, and futures and forward. In addition, the conditional correlation between spot and forward return of Tapis market is 0.936.
[Insert Table 6 here]
The estimates of the dynamic conditional correlation and descriptive statistics for DCC across prices in each market are presented in Table 7 Panel 7a and 7b respectively. Based on robust t-ratio, the estimates of the DCC parameters, and , of each market are always statistically significant. There indicates that the assumption of constant conditional correlation is not maintain empirically. In addition, the average of dynamic conditional correlation of each pairs is identically to the constant conditional correlation reported in Table 6. The short-run persistence of shocks on the dynamic conditional correlation is greatest across the returns in WTI being 0.264, whereas the largest long-run persistence of shocks to the conditional correlations is across returns in Brent market being 0.995 = 0.027 + 0.968.