We propose an additive time-varying (or partially time-varying) multivariate model of volatility, where a time-dependent component is added to the extended vector GARCH process for modelling the dynamics of volatility interactions. Volatility co-dependence is allowed to change smoothly between two extreme states, and second-moment interdependence is identified through these structural changes. The estimation of the new time-varying vector GARCH process is simplified using an equation-by-equation estimator for the volatility equations in the first step and estimating the correlation matrix in the second step. A new Lagrange multiplier test is derived for testing the null hypothesis of constant volatility co-dependence against a smoothly time-varying interdependence between financial markets. Monte Carlo experiments show that the test statistic has satisfactory finite-sample properties. An empirical application to sovereign bond yields illustrates the modelling strategy and the usefulness of the new specification. • We propose an additive time-varying (or partially time-varying) multivariate model of volatility, where a time-dependent component is added to the extended vector GARCH process for modelling the dynamics of volatility interactions. • The estimation of the new time-varying vector GARCH process is simplified using an equation-by-equation estimator. • A Lagrange multiplier test is derived for testing the null hypothesis of constancy co-dependence volatility against a smoothly time-varying interdependence between financial markets. • An application to sovereign bond yields illustrates the modelling strategy and the usefulness of the new specification.