Copulas: a Review and Recent Developments (2007)

de¯ned on fy : y>0and(1+» y° ) > 0g. The special case of » = 0, interpreted as¾the limit case » ! 0, gives the constrained modelF (y) =1¡ expµ¡ y°¾; if » =0;for y>0. When °1 induces concavity. The Weibull distribution correspondsto » = 0. Two sub-classes of the Weibull are: the class of sub-exponential distributions,generated by letting °1.The choice of thresholds is a critical issue, not addressed here. After some experimentationobserving the standard errors of marginal parameters estimates, thethresholds were ¯xed as the empirical p i -quantile, (p 1 ;p 2 )=(0:18; 0:17). We note thatthe limit distributional result, the Pickands-Balkema-de Haan theorem, given for theright tail, holds for su±ciently extreme thresholds, but high thresholds may result inlarge variability of estimators (see Embrechts et al., 1997). To estimate the marginaland copula parameters we use the fully parametric approach in two steps, for detailssee Genest et al. (1995) and Joe (1997), based on maximum likelihood estimation.We found the daily marginal (lower tail) excesses follow di®erent MGPD(°;»;¾) distributions,respectively, an MGPD(1:62; 0:56; 6:38) and an MGPD(1:32; 0:33; 1:43).These estimates and respective standard errors are given in Table 2. The ¯rst columngives the threshold values, the second gives the number of joint observationsused (172), and the third, fourth and ¯fth columns show the marginal results, forKorea and Singapore. The marginal excesses are not identically distributed and thusthe excesses are not exchangeable.Table 2: Margins and copula parameters estimates and (standard errors) and ¸U,for the negative co-exceedances of Korea and Singapore.Excess daily returns (raw data)Thresholds # ° » ¾ Copula(parameter) ¸U-1.38 -1.05 172 1.53 1.32 0.56 0.33 6.38 1.43 Gumbel(1.35) 0.33(0.08) (0.05) (0.11) (0.10) (1.32) (0.17) (0.05) (0.14)Excess residuals (¯ltered data)Thresholds # ° » ¾ Copula(parameters) ¸U-0.85 -0.87 147 1.02 1.04 0.13 0.13 0.50 0.56 ALM(1.19, 0.96, 0.82) 0.19(0.06) (0.06) (0.09) (0.08) (0.06) (0.07) (0.07, 0.02, 0.02) (0.09)For the dependence structure we considered four copula families possessing uppertail dependence. They are the Gumbel, see Joe (1997), Joe-Clayton (family BB7 inJoe (1997)), the copula associated to the Kimeldorf and Sampson or Clayton copula,thecopulapertainingtothebivariatePareto distribution, see Frees and Valdez (1998),33