In recent years, both oil commodity and US dollar currency have been experiencing an unprecedented high fluctuation while exhibit the significantly opposite trends. This negative relationship will enable the oil commodity and the US dollar currency to be useful tools for strategic asset allocation and risk management. For these reasons, the forecast of the volatility and co-movement structures of oil and exchange rate returns have attracted much attention among academics and institutional investors.
However, it has been demonstrated that oil and exchange rate returns are skew and leptokurtic and perhaps follow extremely dissimilar marginal distributions as well as different degrees of freedom parameters. The relationship structure between the oil and exchange rate returns may also exhibit asymmetric or tail dependence structure. Therefore, in order to improve the drawbacks of conventional multivariate GARCH model, this paper proposes three classes of copula-based GARCH models to elastically describe the volatility and dependence structure of oil and US dollar exchange rate returns. We find that the GARCH model with Student-t copula possesses better explanatory ability of crude oil and USDX futures returns, suggesting that there is symmetric tail dependence structure between crude oil and USDX futures returns. In addition, the leverage effects are demonstrated to be insignificant for both crude oil and USDX futures. Based on the marginal distribution with the component GARCH model, we can find that the persistence of short-run volatility is
allocation problem. In terms of out-of-sample results, we find that the dynamic strategies based on the copula-based GARCH models outperform the static strategy and other dynamic strategies based on the CCC GARCH and DCC GARCH models, which demonstrates that skewness and leptokurtosis of crude oil and USDX futures returns are economically significant. Furthermore, the GARCH model with Frank copula yields highest performance fees and break-even transaction costs to attract investors to switch their trading strategy and performs the most prominent among all selected models. More risk-averse investors are also willing to pay higher fees to switch their strategies from the static strategy to the dynamic strategies based on copula-based GARCH models.
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Table 1. Summary Statistics for Crude oil and USDX rate Excess Returns
Crude oil Futures USDX Futures
Mean(%) 0.0108 -0.0186
SD(%) 2.5404 0.5655
Skewness -0.8964 -0.028
Kurtosis 19.6644 4.6938
Max(%) 16.4097 2.8167
Min(%) -40.072 -2.7401
JB 58617.8208*** 599.3234***
Note: This table reports the descriptive statistics for daily crude oil and USDX futures excess returns for the sample period from January 2, 1990 to December 31, 2009. JB is the Jarque-Bera statistic, which is used to test for normality.
The symbols *, **, and *** represent statistical significance at the 10%, 5%, and 1% levels, respectively.
Table 2. Estimation Results of Copula-Based GARCH Models
GARCH GJR-GARCH Component GARCH
Crude oil USDX Crude oil USDX Crude oil USDX
Panel A: Estimation of Marginals
i 0.03033 -0.00753 0.02846 -0.00793 0.01660 -0.02430***
(0.02761) (0.00722) (0.02718) (0.00725) (0.02696) (0.00718)
ci 0.04376*** 0.00106** 0.04405*** 0.00100**
(0.01186) (0.00047) (0.01189) (0.00047)
ai 0.05639*** 0.02967*** 0.05383*** 0.02763*** 0.01257 0.00019
(0.00650) (0.00399) (0.00819) (0.00506) (0.01486) (0.00065) bi 0.93700*** 0.96772*** 0.93679*** 0.96778*** 0.70329*** 0.99295***
(0.00673) (0.00435) (0.00671) (0.00432) (0.17045) (0.01921)
i 6.74588*** 7.36002*** 6.75076*** 7.35605*** 6.80645*** 7.39712***
(0.11656) (0.25135) (0.12199) (0.24559) (0.18823) (0.53335)
i -0.04734** -0.02554 -0.04806** -0.02584 -0.04662** -0.01543 (0.01931) (0.01893) (0.01946) (0.01902) (0.01942) (0.01872)
di 0.00533 0.00419 Panel B: Estimation of Gaussian Dependence Structure
c 0.00004 0.00005 0.00005
AIC 29622.112 29626.776 29629.609
BIC 29719.894 29737.595 29753.466
Panel C: Estimation of Student-t Dependence Structure
c 0.00004 0.00004 0.00004
AIC 29598.641 29603.089 29605.945
BIC 29702.941 29720.427 29736.321
Table 2. (Continued)
Panel D: Estimation of Clayton Dependence Structure
c -0.00463** -0.00443** -0.00446**
AIC 29715.952 29719.963 29723.991
BIC 29813.734 29830.782 29847.848
Panel E: Estimation of Survival Clayton Dependence Structure
c -0.00051** -0.00052** -0.00050**
AIC 29715.007 29721.328 29722.405
BIC 29812.789 29832.147 29846.262
Panel F: Estimation of Frank Dependence Structure
c 0.000004 0.000006 0.000005
AIC 29627.313 29630.515 29633.982
BIC 29725.094 29741.334 29757.839
Note: The table presents the maximum likelihood estimates of three classes of copula-based GARCH models, which are based on the daily crude oil and USDX futures excess returns for the sample period from January 2, 1990 to December 31, 2009. Three types of marginal distributions (GARCH,
GJR-GARCH and component GARCH models) are used, and they are expressed as follows:
(A) GARCH model:
In addition, five types of copula functions are utilized to describe the dependence structure, and their densities are expressed as follows:
The proper logistic transformation of dependence parameters, t and t, obey the following process
The Akaike information criteria (AIC) and Bayesian information criteria (BIC) are used to evaluate the goodness of fit of the selected models. The numbers in parentheses are standard deviations. The superscripts *, **, and *** indicate statistical significance at the 10%, 5%, and 1% levels, respectively.
Table 3. Out-of-sample Economic Value for Dynamic Strategy Based on selected Models versus Static Strategy with the Minimum Variance Strategy
GARCH GARCHStudent t GARCHClayton GARCHSClayton GARCHFrank
*
GJR GARCH GJR GARCH Student t GJR GARCH Clayton GJR GARCH SClayton GJR GARCH Frank
*
CGARCH CGARCHStudent t CGARCHClayton CGARCHSClayton CGARCHFrank
*
p 1 tc1be 5 tc5be 1 tc1be 5 tc5be 1 tc1be 5 tc5be 1 tc1be 5 tc5be 1 tc1be 5 tc5be
5% 50 25 45 24 46 24 43 22 49 26 53 28 27 14 22 12 54 28 56 29
10% 78 18 59 14 74 17 60 14 100 23 114 27 27 6 3 1 106 24 112 26
15% 93 13 50 7 93 13 61 9 162 23 193 28 7 1 -49 -- 163 23 178 26
Note: The table presents the out-of-sample performance fee () and break-even transaction costs (tcbe) for a dynamic strategy based on selected models versus the static strategy for three target returns (5%, 10% and 15%) with a minimum variance strategy. Each minimum variance strategy builds an efficient portfolio by investing in the daily returns of the crude oil futures, USDX futures, and a risk-free asset. The fees are denoted as the amount which an investor is willing to pay for switching from the static strategy to another dynamic strategy with the relative risk aversion level =1 and 5. The performance fee () is expressed in annualized basis points. The break-even transaction cost (tcbe) is defined as the minimum proportional cost per trade for which the dynamic strategies would have the same utility as the static strategy. In addition, (tcbe) values are reported only when
is positive. The out-of-sample period runs from January 2, 2005 to December 31, 2009.
Panel A: Close Prices
Panel B: Excess Returns
Panel C: Trading Volumes
Figure 1. Daily close prices, daily excess returns and annual trading volumes of crude oil and USDX futures for the sample period from January 2, 1990 to December 31, 2009.
Panel A: Gaussian copula with Normal and Skewed-t marginal distributions
Panel B: Student-t copula with Normal and Skewed-t marginal distributions
Panel C: Clayton copula with Normal and Skewed-t marginal distributions
Panel D: Survival Clayton copula with Normal and Skewed-t marginal distributions
Panel E: Frank copula with Normal and Skewed-t marginal distributions
Figure 2. Contour plot based on two types of marginal distributions (Normal(0,1) and Skewed-t (5,-0.1)) and five types of copula functions (Gaussian, Student-t, Clayton, survival Clayton and Frank) under the specific dependence parameter, 0.2.
Panel A: Crude Oil
Panel B: USDX
Figure 3. Volatility estimates of crude oil and USDX futures excess returns based on the GARCH, GJR-GARCH, and Component GARCH models for the sample period from January 2, 1990 to December 31, 2009.
Panel A: GARCH Model
Panel B: GJR-GARCH Model
Panel C: Component GARCH Model
Figure 4. Correlation estimates between crude oil and USDX futures excess returns based on marginal distributions of the GARCH, GJR-GARCH, and component GARCH models and dependence structures of the Gaussian, Student-t, Clayton, survival Clayton and Frank copulas.
The sample period is from January 2, 1990 to December 31, 2009.