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Skew t-Copula and its Estimation: For Application to Risk Aggregation Yoshiba, Toshinao


Correlation structure of risk factors matters to financial portfolio risk management. When the risk factors are specified by some assets’ return, risk managers have to care about lower tail dependence of those factors rather than upper. On the other hand, it is practical to specify overall nontail dependence by sample linear or rank correlation matrix. Based on this background, we focus on the application of skew t-copula. Skew t-copula is defined by a multivariate skew t-distribution and its marginal. As indicated in Kotz and Nadarajah (2004), various types of multivariate skew t-distribution have been proposed. That implies various types of skew t-copula can exist. Three types of skew t-copula are known so far. The first type was mentioned in Demarta and McNeil (2005), which is based on multivariate version of generalized hyperbolic (GH) skew t-distribution proposed in Barndorff-Nielsen (1977) (See also Aas and Haff (2006)). The second type was constructed in Smith et al. (2012) based on Sahu et al. (2003). Kollo and Pettere (2010) tried to construct the third type based on Azzalini and Capitanio (2003). This paper is constructed as follows. First, we improve the skew t-copula approach proposed in Kollo and Pettere (2010) and derive log-likelihood function to estimate the parameters by maximum likelihood estimation (MLE). Second, we indicate that MLE for the skew t-copula requires fast and accurate quantile calculation for univariate skew t-distribution. Third, we show the difference between upper and lower tail dependence of the skew t-copula citing Fung and Seneta (2010) and Bortot (2010). Finally, we introduce a method of moment approach for the parameter estimation of the skew t-copula.

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