Estimating the error distribution in multivariate heteroscedastic time-series models

A semiparametric method is studied for estimating the dependence parameter and the joint distribution of the error term in a class of multivariate time-series models when the marginal distributions of the errors are unknown. This method is a natural extension of Genest, C., Ghoudi, K., Rivest, L.-P. [1995. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82, 543–552.] for independent and identically distributed observations. The proposed method first obtains n-consistent estimates of the parameters of each univariate marginal time series, and computes the corresponding residuals. These are then used to estimate the joint distribution of the multivariate error terms, which is specified using a copula. Our developments and proofs make use of, and build upon, recent results of Koul and Ling [2006. Fitting an error distribution in some heteroscedastic time series model. Ann. Statist. 34, 994–1012.] and Koul [2002.Weighted empirical processes in dynamic nonlinear models, Lecture Notes in Statistics, vol. 166. Springer, New York.] for these models. The rigorous proofs provided here also lay the foundation and collect together the technical arguments that would be useful for other potential extensions of this semiparametric approach. It is shown that the proposed estimator of the dependence parameter of the multivariate error term is asymptotically normal, and a consistent estimator of its large sample variance is also given so that confidence intervals may be constructed. A large-scale simulation study was carried out to compare the estimators particularly when the error distributions are unknown, which is almost always the case in practice. In this simulation study, our proposed semiparametric method performed better than the well-known parametric methods. An example on exchange rates is used to illustrate the method.