Statistical inference for multivariate residual copula of garch models

Recently a flexible class of semiparametric copula-based multivariate GARCH models has been proposed to quantify multivariate risks, in which univariate GARCH models are used to capture the dynamics of individual financial series, and parametric copulas are used to model the contemporaneous dependence among GARCH residuals with nonparametric marginals. In this paper we address two questions regarding statistical inference for this class of models. (1) Under what mild sufficient conditions is the asymptotic distribution of the pseudo maximum likelihood estimator (MLE) of the residual copula parameter of Chen and Fan (2006a) justified? (2) How do we test the correct specification of a parametric copula for the GARCH residuals? In order to answer both questions rigorously, we establish a new weighted approximation for the empirical distributions of the GARCH residuals, which is of interest in its own right. Simulation studies and data examples are provided to examine the finite sample performance of the pseudo MLE of the residual copula parameter and the proposed goodness-of-fit test.