Multivariate zero-inflated INGARCH models: Bayesian inference and composite likelihood approach

In this paper, we propose a framework for modeling multivariate count time series data that accommodates zero-inflated components. Our approach is based on a novel class of bivariate distributions (ZMP \(_2\) $_2^{}$ ) which is formulated via the mixed Poisson representation and a four-point mixture. In the ZMP \(_2\) $_2^{}$ , shared latent effects from some exponential family distribution act on the Poisson rates in order to introduce dependency. Bivariate zero-inflation is then posed via the mixture approach by Chin-Shang et al. (Technometrics 41:29–38, 1999), where excess of zeros can arise in different ways. This can be with respect to either one of the marginals, or simultaneous for both components. The ZMP \(_2\) $_2^{}$ class admits a flexible zero-inflation behaviour and a tractable hierarchical structure achieved with augmentation of the data to the mixture components. It is placed in the time series context by adopting a bivariate integer-valued GARCH (INGARCH) structure, becoming the first model in this class capable to handle zero inflation. The ZMP \(_2\) $_2^{}$ is then extended to dimensions higher than two via a composite likelihood approach, which is a proxy for an underlying p-variate ( \(p>2\) $p>2$ ) distribution. The latter is constructed by the product of ZMP \(_2\) $_2^{}$ INGARCH terms, and provides a trade-off between complexity and information. Inferential procedures are carefully outlined from a Bayesian perspective via Markov Chain Monte Carlo. We demonstrate that an efficient Metropolis-within-Gibbs sampler can be constructed availing of the model’s hierarchical structure and conjugate priors. Our proposal is explored in a real data application concerning the weekly cases of measles reported in three states of Germany. Comparison to other bivariate INGARCH models is pursed and demonstrates that the ZMP INGARCH substantially improves the analysis of measles infections.