Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification

We introduce a new class of semiparametric copula-based multivariate dynamic (SCOMDY) models, which specify the conditional mean and the conditional variance of a multivariate time series parametrically, but specify the multivariate distribution of the standardized innovation semiparametrically as a parametric copula evaluated at nonparametric marginal distributions. We first study large sample properties of the estimators of SCOMDY model parameters under a misspecified parametric copula, then propose pseudo likelihood ratio (PLR) tests for model selection between two SCOMDY models with possibly misspecified copulas, and finally develop PLR tests for model selection between more than two SCOMDY models. The limiting null distributions of the PLR tests do not depend on the estimation of conditional mean and conditional variance parameters, hence are very easy to simulate. Empirical applications to three and higher dimensional daily exchange rate series indicate that a SCOMDY model with a tail-dependent copula is generally preferred.