Semiparametric dynamic max‐copula model for multivariate time series

The paper presents a novel non‐linear framework for the construction of flexible multivariate dependence structure (i.e. copulas) from existing copulas based on a straightforward ‘pairwise max‐’rule. The newly constructed max‐copula has a closed form and has strong interpretability. Compared with the classical ‘linear symmetric’ mixture copula, the max‐copula can be viewed as a ‘non‐linear asymmetric’ framework. It is capable of modelling asymmetric dependence and joint tail behaviour while also offering good performance in non‐extremal behaviour modelling. Max‐copulas that are based on single‐factor and block factor models are developed to offer parsimonious modelling for structured dependence, especially in high dimensional applications. Combined with semiparametric time series models, the max‐copula can be used to develop flexible and accurate models for multivariate time series. A new semiparametric composite maximum likelihood method is proposed for parameter estimation, where the consistency and asymptotic normality of estimators are established. The flexibility of the max‐copula and the accuracy of the proposed estimation procedure are illustrated through extensive numerical experiments. Real data applications in value‐at‐risk estimation and portfolio optimization for financial risk management demonstrate the max‐copula's promising ability to capture accurately joint movements of high dimensional multivariate stock returns under both normal and crisis regimes of the financial market.