Modeling the joint dynamics of multiple positive valued time series is of great interest across diverse application domains, including but not limited to, epidemiology, econometrics, finance, insurance, and signal processing. One prominent example is in jointly modeling different intraday volatility measures such as realized variance, daily ranges, and absolute returns of multiple firms or assets over time.

The core challenge in modeling multivariate positive-valued time series is in estimating the dependence between the components as well as temporal dependence, for which a multivariate distributional formulation (such as multivariate gamma) is required and in such cases the likelihood computation can be cumbersome and time consuming. Existing methods for dynamic modeling of multivariate positive-valued time series uses Markov Chain Monte Carlo (MCMC)
methods for parameter estimation which can be computationally intensive.

To address these challenges, the authors develop a computationally feasible dynamic modeling framework for multivariate positive-valued time series. They discuss a flexible level correlated model (LCM) framework for building hierarchical models which allows combining marginal gamma distributions for the positive-valued component responses, while accounting for association among the components at a latent level. The authors introduce vector autoregression
(VAR) evolution of the latent states, building custom functions to enable its estimation using integrated nested Laplace approximation (INLA) for fast Bayesian inference using R-INLA package. This LCM approach circumvents expensive calculation of multivariate gamma likelihoods, and R-INLA facilitates fast approximate Bayesian inference.

As a case study, they model the interdependencies between intraday volatility measures from several stock indexes. The authors have provided an illustration in https://github.com/chiranjit2324/Modeling-Multivariate-Positive-Valued-Time-Series-Using-R-INLA