Publication Date: August 2015
Revision Date: October 2019
This paper considers estimation of semi-nonparametric GARCH ﬁltered copula models in which the individual time series are modelled by semi-nonparametric GARCH and the joint distributions of the multivariate standardized innovations are characterized by parametric copulas with nonparametric marginal distributions. The models extend those of Chen and Fan (2006) to allow for semi-nonparametric conditional means and volatilities, which are estimated via the method of sieves such as splines. The ﬁtted residuals are then used to estimate the copula parameters and the marginal densities of the standardized innovations jointly via the sieve maximum likelihood (SML). We show that, even using nonparametrically ﬁltered data, both our SML and the two-step copula estimator of Chen and Fan (2006) are still root-n consistent and asymptotically normal, and the asymptotic variances of both estimators do not depend on the nonparametric ﬁltering errors. Even more surprisingly, our SML copula estimator using the ﬁltered data achieves the full semiparametric eﬀiciency bound as if the standardized innovations were directly observed. These nice properties lead to simple and more accurate estimation of Value-at-Risk (VaR) for multivariate ﬁnancial data with flexible dynamics, contemporaneous tail dependence and asymmetric distributions of innovations. Monte Carlo studies demonstrate that our SML estimators of the copula parameters and the marginal distributions of the standardized innovations have smaller variances and smaller mean squared errors compared to those of the two-step estimators in ﬁnite samples. A real data application is presented.
Keywords: Semi-nonparametric dynamic models, Residual copulas, Semiparametric multistep, Residual sieve maximum likelihood, Semiparametric efficiency
JEL Classification Codes: C14, C22, G32.