A Mixed Dependence Between the Exchange Rate and International Crude Oil Returns: An Application of Dynamic Mixture Copula

ABSTRACT

In this study, the dynamic dependence between the international crude oil return and the exchange rate return for Taiwan is examined. Two mixture copulas (symmetric Joe–Clayton, SJC, and mixture of Gumbel and survival Gumbel, GSG) and two dynamic dependences (a Markov-switching type and an AR-like type) are considered in order to study whether the dynamic dependence is mixed and asymmetric. The empirical results show that the Markov-switching GSG copula performs the best when compared to other specifications investigated in this article. The relationship is positive and symmetric during periods of volatile crude oil prices, while it is independent during periods of stable crude oil prices.

KEY WORDS:

• Crude oil
• dependence
• exchange rate
• Markov switching
• mixture copula

Acknowledgments

The author gratefully acknowledges three anonymous referees for constructive comments and suggestions.

Notes

1. For example, Reboredo (2012) and Aloui et al. (2013) consider six copula functions (Normal, Student-t, Plackett, Frank, Clayton, and Gumbel). Wu, Chung, and Chang (2012) employ five copula functions (Normal, Student-t, Clayton, survival Clayton, and mixture Clayton). The relative efficacy of Normal, Student-t, and mixture Clayton copulas has been examined by Chen, Choudhry, and Wu (2013a).

2. See, for instance, Krugman (1983), Golub (1983), Chen and Chen (2007), and Reboredo and Rivera-Castro (2013). A detailed introduction can be found in Buetzer, Habib, and Stracca (2012) and Beckmann and Czudaj (2013a, 2013b).

3. Some studies, such as Lee (2009), Garcia and Tsafack (2011), Silva Filho, Ziegelmann, and Dueker (2012), and Wang, Wu, and Lai (2013), employ Markov-switching copulas to investigate the state-dependent tail structure.

4. The author thanks an anonymous referee for suggesting that I provide a comparison between the AR(1)-Jump-GARCH(1,1) model and the AR(1)-GARCH(1,1) model.

5. The author thanks an anonymous referee for suggesting that I provide a reason for choosing the SJC and GSG copulas.

6. See Chollete, De La Pena, and Lu (2011) for briefly introducing the characteristics of popular copulas, including normal copula, Student-t copula, Gumbel copula, Clayton copula, survival Gumbel copula, and survival Clayton copula.

7. Patton (2006) points out that the maximum likelihood estimators have an asymptotic normal distribution.

8. See Proposition 3.2 of Garcia and Tsafack (2011) for the step-by-step procedure.

9. The time-series plots are available upon request from the author.

10. The author thanks an anonymous referee for suggesting the Jarque–Bera test and the ARCH-LM test.

11. The daily return was obtained by multiplying the first-order difference of the log price index by 100%.

12. The author thanks an anonymous referee for this suggestion.

13. Results are not reported here due to limited space, but are available upon request.

14. Following Chan and Maheu (2002), the maximum number of jumps is set to 20. After calculating the conditional jump probability $\mathrm{P}(n_{it}=\mathrm{w}|{\mathrm{\Omega }}_t)$, this article observes that the probability of more than seven jumps occurring is almost zero.

15. Although Chen and Fan (2006) and Genest, Remillard, and Beaudoin (2009) derive test statistics in order to test whether or not a copula with fixed parameters is mis-specified, their statistics are not valid when the copula parameters have Markov-switching dynamics or AR-like dynamics.

16. Following Patton (2006), when the parameter estimate approaches the left boundary of the parameter space, the value of the parameter is restricted to 0.

17. They are available upon request.

18. This article attempts to estimate a three-regime specification. However, a third regime is hardly to be identified in all cases. Hence, only a two-regime specification is analyzed in this article.

19. The filtered probabilities are defined in Equation (26).

20. The test results for symmetric dependence are available upon request.

21. In the copula specification, Zhang (2008) introduces the formulas for LDM and UDM.

22. The author thanks an anonymous referee for suggesting the implementation of a robustness check.

23. Estimation results are available upon request.