Copulas: a Review and Recent Developments (2007)

Table 3: ARMA + FIGARCH ¯ts to daily returnsfrom Korea and Singapore indexes.Korea : ARMA(1; 0)+FIGARCH(1;d;1) Singapore: ARMA(1; 0)+FIGARCH(2;d;1)Value Std.Error t value Pr(> jtj) Value Std.Error t value Pr(> jtj)AR(1) 0.11763 0.02084 5.646 9.110e-009 0.1151 0.019923 5.775 4.298e-009A 0.06975 0.02772 2.517 5.954e-003 0.0295 0.008441 3.495 2.410e-004GARCH(1) 0.53477 0.06354 8.416 0.000e+000 0.7562 0.046910 16.121 0.000e+000ARCH(1) 0.22228 0.04286 5.186 1.154e-007 0.3750 0.057193 6.557 3.303e-011ARCH(2) 0.1561 0.029486 5.295 6.453e-008fraction 0.39866 0.05110 7.802 4.441e-015 0.4855 0.043910 11.057 0.000e+000Nelsen (2005a) studies non-exchangeability and proposes a non-exchangeabilitymeasure for the case of identically distributed margins. The degree of exchangeabilitymay be measured by computing the maximum of the absolute value of the di®erencesH(x; y) ¡ H(y; x). For identically distributed margins with joint distribution H andcopula C, the set of values of jH(x; y) ¡ H(y; x)j are the same of jC(u; v) ¡ C(v; u)j.Thus he proposes to compute 3maxjC(u; v) ¡ C(v; u)j, for all u; v 2 [0; 1] 2 .Herewesuggest to use an empirical version of this statistics, using the observed pairs andthe ¯tted ALM copula. Note that, by the Glivenko-Cantelli theorem, as the samplesize goes to in¯nity, the sample version should approach the true value. The valueobtained using the empirical version of Nelsen's measure of non-exchangeability was0:0058.Summarizing, we provided an example where non-exchangeability was found foridentically distributed random variables, and long memory in volatility was responsiblefor changes in dependence structure, increasing extremal dependence. Workscombining copulas and short memory processes (GARCH type) modelling includeGoorbergh et al. (2005) and Breiman et al. (2003) among others.5 ConclusionsThe independence assumptions, which are typical in many statistical models are oftendue more to convenience rather than to the problem in hand. Furthermore, there aresituations where neglecting dependence e®ects may occur into a dramatic underestimationfor quantity of interest (some appropriate risk measure, for example). Takingcare of dependencies becomes therefore important in order to extend standard modelstowards more e±cient ones. However, relaxing the independence assumption yieldsmuch less tractable models. The pitfall of the copula approach is that it is usuallydi±cult to choose or ¯nd the appropriate copula for the problem in hand. An alternativeis suggested in Section 3.3. Often, the only possibility is to start with someguess such a parametric family of copulas and then to try to ¯t the parameters. Asa consequence, the model obtained may su®er a certain degree of arbitrariness.35