Abstract
This paper develops a new time-varying mixture copula, in which the dynamic weights of four distinct copulas are determined by a two-stratum process, to investigate the magnitude of tail dependence in four independent quadrants. In the two-stratum process, the weight of each copula is determined firstly by the relative importance of positive and negative dependence structures, and then by its own past values and adjustment processes. The weighting mechanism is time-varying in each stratum. This new specification is applied to analyze the asymmetric tail dependencies between the stock and exchange rate markets. Empirical results show four interesting findings. First, the quasi-maximum likelihood estimation (QMLE) has a better fitting ability than does the inference function for margins. The relative efficiency of the QMLE is irrespective of marginal specifications. Second, the goodness-of-fit tests of the new time-varying mixture copula are crucially affected by the marginal specifications. Third, estimation methods impact mixture weights. Four distinct tail dependencies are observed, revealing the importance of considering all four tails concurrently, and not just parts of the four tails. Fourth, the asymmetric positive and negative dependencies are significant. Each country shows a similar pattern of asymmetric negative dependence, but a different pattern of asymmetric positive dependence. These empirical findings provide important portfolio allocation implications.
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Notes
See, for instance, Ng (2008), Ning (2010), Michelis and Ning (2010a, b), Aloui et al. (2011), Chollete et al. (2011), Chang (2012), Zhou and Gao (2012), Chen and Tu (2013), Ross and de Magistrias (2013), Wang et al. (2013), Sukcharoen et al. (2014), Wu and Lin (2014), Buscher et al. (2015), Hung et al. (2016), Reboredo et al. (2016), Chang (2017a, b), Chang and Yu (2017), and Liu et al. (2017a, b).
The 10 copulas are Normal, Student-t, R_Clayton, Gumbel, Clayton, R_Gumbel, R2_Clayton, R1_Gumbel, R1_Clayton, and R2_Gumbel. Here, R_C refers to the survival copula of copula C, R1_C refers to the 90° anticlockwise-rotated copula C, and R2_C refers to the 90° clockwise-rotated copula C.
Although Joe (2005) and Patton (2006b) have shown that, under regularity conditions, the IFM has a similar efficiency property to the maximum likelihood method, their simulation experiments are based on the single copula framework. Whether the conclusion obtained from their specifications is still valid needs further investigation.
There are many different copula functions, such as Gumbel, Clayton, survival Gumbel, survival Clatyon, Student-t, BB7, etc. Gumbel and survival Clayton focus on the upper-upper dependence, whereas Clayton and survival Gumbel observe the lower-lower dependence. Student-t and BB7 can detect upper-upper and lower-lower tail dependencies simultaneously; the former is symmetric and the latter is asymmetric.
The exchange rate is defined as the units of US dollar to one local currency.
These countries include Canada, France, Germany, Italy, Japan, and the UK.
The eight emerging countries are Brazil, Chile, Colombia, India, Mexico, Russia, South Africa, and Turkey.
Australia, Canada, Chile, Norway, and New Zealand are mainly commodity-exporting countries. However, the exchange rate system of Chile, Norway and New Zealand was not a free-floating exchange rate regime during the sample period employed in this paper. For example, New Zealand did not adopt a floating exchange rate regime until 1985. Norway adopted a floating exchange rate system in 2001. The central bank of Chile carries out some exchange rate interventions, which are commonly observed in developing countries.
It is worth emphasizing that a mixture copula, which is composed of \({C}_{00}\), \({C}_{90}\), \({C}_{180}\), and \({C}_{270}\), is analyzed here. This paper does not use each of the four copulas to analyze the dependencies, nor does it discuss the differences in estimation results for different single copulas. Only the differences in estimation results for two different estimation methods are analyzed here. Hence, this paper does not encounter a non-nested testing problem. The information criteria are still valid.
The author thanks an anonymous referee for suggesting that the paper implements a robustness check.
The ARCH model with normal distribution is employed for Canada due to the convergence problem.
The variance equation for the GJR-GARCH process is as follows:
$${\sigma }_{it}^{2}={c}_{i}+{d}_{i}{\varepsilon }_{i,t-1}^{2}+{e}_{i}{\sigma }_{i,t-1}^{2}+{f}_{i}{d}_{it}{\varepsilon }_{i,t-1}^{2},$$where \({d}_{it}=1\) if \({\varepsilon }_{i,t-1}<0\), and \({d}_{it}=1\) if \({\varepsilon }_{i,t-1}\ge 0.\)
Due to convergence difficulty, which may be related to small sample observations, a simplified exponential volatility clustering model, EARCH, is employed here. The variance equation for the EARCH process is as follows:
$${\text{ln}}\sigma _{{it}}^{2} = k_{i} + \gamma _{i} {\raise0.7ex\hbox{${\left| {\varepsilon _{{i,t - 1}} } \right|}$} \!\mathord{\left/ {\vphantom {{\left| {\varepsilon _{{i,t - 1}} } \right|} {\sigma _{{i,t - 1}} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\sigma _{{i,t - 1}} }$}}.$$The estimations results for different marginal specifications are available upon request.
The goodness-of-fit tests for marginal specifications are reported in Sect. 4.2.2.
The author thanks an anonymous referee for providing this suggestion.
The testing results for \({u}_{1t}\) and \({u}_{2t}\) have been shown in Table 10.
The author thanks an anonymous referee for providing this suggestion.
The results for Model A and Model C are available upon request.
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Acknowledgements
The author would like to acknowledge two anonymous referees for their constructive suggestions and comments. The financial support from the Ministry of Science and Technology of Taiwan (MOST 107-2410-H-415-003) is also greatly acknowledged.
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Chang, KL. A New Dynamic Mixture Copula Mechanism to Examine the Nonlinear and Asymmetric Tail Dependence Between Stock and Exchange Rate Returns. Comput Econ 58, 965–999 (2021). https://doi.org/10.1007/s10614-020-09981-5
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DOI: https://doi.org/10.1007/s10614-020-09981-5