Data and Empirical Results

3.1 Data and descriptive statistics

This study uses West Texas Intermediate (WTI) crude oil futures and US dollar index (USDX) futures data to stand for oil and exchange rate markets. WTI crude oil, also known as light sweet oil, is the futures contract traded on New York Mercantile Exchange (NYMEX).

The USDX represents the trade-weighted value of the US dollar in terms of a basket of six major foreign currencies. There exist a futures contract and an option contract traded on the New York Board of Trade (NYBOT). Both WTI crude oil and USDX futures prices data⁴ with the nearest to maturity for the period from January 1, 1990 to December 31, 2009 are obtained from DATASTREAM and 5,008 daily return observations are generated for each asset. In addition, we use the three-month Treasury bill as the risk-free rate, which are obtained from the Federal Reserve Board. The daily close prices, daily excess returns, and annual trading volumes of WTI crude oil and USDX futures over the sample period are graphed in Figure 1.

The descriptive statistics of crude oil and exchange rate excess returns are reported in Table 1. Both crude oil and exchange rate returns exhibit the left-skew and leptokurtic phenomenon. So that, we use the Jarque-Bera statistic to test the normality of distribution then we get the conclusion of both oil and exchange rate all reject the null hypothesis of normality.

Consequently, we adapt the Hansen’s skewed-t distribution to dovetail with the feature in our study.

3.2 Estimation results 3.2.1 Marginal distribution

Table 2 presents the estimate results of three classes of copula-based GARCH models.

Panel A reports the parameter estimates of marginal distributions with the GARCH, GJR-GARCH and component GARCH models. We use the Akaike information criteria (AIC) and Bayesian information criteria (BIC) to examine which model has better goodness of fit.

The two methods resolve this problem by adding a corrected term to avoid the overfitting problems caused by different number of parameters in each models. According to these two information criteria, we can find the GARCH model has smaller value than the GJR-GARCH and component GARCH models based on each copula function, which imply that the GARCH model exhibits the best goodness of fit.

As can be seen, the asymmetry parameters, _i, are significant and negative for crude oil returns while insignificant for USDX returns, exhibiting that crude oil returns are skewed to left. In addition, in the GARCH model, the parameters a_i and b_i are significant to explain the crude oil and exchange rate returns have volatility clustering. And the sum of a_i and b_i are very close to 1 implies that there is high volatility persistence in both crude oil and exchange rate markets. Further, the asymmetric parameters d_i in the GJR-GARCH model are insignificant and exhibit no asymmetric effect in the volatility structures of crude oil and exchange rate markets, which is consistent to Manera et al. (2004) and Wang and Yang (2006).

The result may be evidence that the asymmetric reaction to equities markets do not bring into the crude oil and USDX futures market. Next, we adapt component GARCH model to distinguish the return volatilities into permanent and transitory components. The result demonstrates the parameters _i of both crude oil and exchange rate markets are very close to 1 and shows there is high persistence in the permanent component. The result also reveals that it significantly diminishes the value of estimate a_i b_i of crude oil from the GARCH model to the component GARCH model. Such that a_i b_i much less than _i, which signifies the transitory component persistence will decline faster than the permanent component. The

parameters _i and a_i in the component GARCH model are regarded as the reaction of shocks to the permanent and transitory components, respectively. From Table 2, we can also see the impact on the permanent component is significantly greater than that on the transitory component. And the parameter a_i turns to insignificant which explain sudden information will not cause volatility impulsion.

3.2.2 Copula function

The parameter estimates for different copula functions are reported in Panels B-F of Table 2. In terms of AIC and BIC, the Student-t dependence structure exhibits better explanatory ability than other dependence structures no matter what marginal models are employed, while Clayton and survival Clayton copulas have worse explanatory ability. The results imply the tail dependence between oil and exchange rate returns may be significant while not asymmetric. In addition, the GARCH model with Student-t copula perform superior to any other selected model. Moreover, we can see the autoregressive parameter _c is closely to 1 in each copula function, indicating the dependence structure between oil and exchange rate returns is high persistent. And the latent parameter _c is also significant in every copula function which displays that latest return information is a meaningful measure.

Specially, _c in Clayton copula is much larger than others and mean it had more short-run response than others copula functions.

3.2.3 Volatility estimates plot

Figure 3 plots the volatility estimates of crude oil and USDX returns based on GARCH, GJR-GARCH and CGARH models. The crude oil had undergone two periods of larger unrest in our sample period. First period began in August 1990 which commonly known as “The 3rd energy crisis” due to the Persian Gulf War. Because the oil demand of most countries must

rely on imports, the wars of oil-producing countries cause supply diminish so that price soaring. Another period began in September 2008, which stemmed from the American subprime mortgage crisis and then the OPEC claim of oil output reduction. By comparison, the volatility of USDX is much stable. The gravest period is from 2008 to 2009. That reason might come from the purposely control by US government in order to rescue the American economic decline after the financial tsunami. In addition, the volatility estimates based on three different marginal models are similar consistent with the results of goodness of fit. We also can find the circumstance that the crude oil and USDX volatilities usually rise at the same time, implying there exist some connections between crude oil and USDX.

3.2.4 Correlation estimates plot

The correlation parameter estimates between oil and exchange rate returns over the sample period generated from different copula models are plotted in Figure 4⁵. During the period 1990 to 2003, the dependence structure between crude oil and USDX returns keeps a lower level or zero correlation. But since 2003, the correlation started descending continuously to this day due to the crude oil prices have steadily increased caused the international oil price reaches a historical break-through. On the other hand, because the US government wanted to pull the export effectively and reducing the international trade deficit, causing US dollar tremendously decreased in value relative to most other countries’

currencies. Moreover, the depreciation of US dollar against other currencies has helped to drive up the oil price over the past few years. The most major reason is that the US dollar is the main invoicing currency of crude oil futures trading. Thus, the falling of US dollar motivated speculators to buy an abundance of crude oil futures contracts to get greater profits, and then promote raise oil price uncommonly.

5 We transform all dependence structures into the correlation by numerical integral, in order to more clear compare the estimate results.

In addition, in Figure 4, the two paths from Gaussian and Student-t copulas are very close consistent with the results in Panel C of Table 2, which present the degree of freedom of Student-t copula is considerable. The Clayton and survival Clayton copulas exhibit similar dependence trend for each other while display low level dependence relative to the symmetric copulas. Moreover, the main difference in correlation estimates between Clayton and survival Clayton copulas is that the Clayton copula exhibits larger ripples. Finally, the correlation trend based on the Frank copula falls in between and close to Gaussian and Student-t copulas.