Multivariate GARCH models with spherical parameterizations: an oil price application

In popular Baba-Engle-Kraft-Kroner (BEKK) and dynamic conditional correlation (DCC) multivariate generalized autoregressive conditional heteroskedasticity models, the large number of parameters and the requirement of positive definiteness of the covariance and correlation matrices pose some difficulties during the estimation process. To avoid these issues, we propose two modifications to the BEKK and DCC models that employ two spherical parameterizations applied to the Cholesky decompositions of the covariance and correlation matrices. In their full specifications, the introduced Cholesky-BEKK and Cholesky-DCC models allow for a reduction in the number of parameters compared with their traditional counterparts. Moreover, the application of spherical transformation does not require the imposition of inequality constraints on the parameters during the estimation. An application to two crude oils, WTI and Brent, and the main exchange rate prices demonstrates that the Cholesky-BEKK and Cholesky-DCC models can capture the dynamics of covariances and correlations. In addition, the Kupiec test on different portfolio compositions confirms the satisfactory performance of the proposed models.