AbstractIn this article, we describe fast Bayesian statistical analysis of vector positive‐valued time series, with application to interesting financial data streams. We discuss a flexible level correlated model (LCM) framework for building hierarchical models for vector positive‐valued time series. The LCM allows us to combine marginal gamma distributions for the positive‐valued component responses, while accounting for association among the components at a latent level. We introduce vector autoregression evolution of the latent states, deriving its precision matrix and enabling its estimation using integrated nested Laplace approximation (INLA) for fast approximate Bayesian modeling via the R‐INLA package, building custom functions to handle this setup. We use the proposed method to model interdependencies between intraday volatility measures from several stock indexes.