Modeling multivariate ordinal time series

In this paper, several regression-type models for multivariate ordinal time series are developed. The regression equations are inspired by existing GARCH-type models for univariate discrete-valued time series and include feedback terms in addition to the usual lagged observations to model the memory behavior. The corresponding terms from other individuals (components) are represented by weighted averages which are calculated based on a proximity matrix. The marginal conditional distributions are either binomial (employing the simplifying rank-count formulation) or multinomial. The approach can be generalized to obtain VARMA-type models to allow for more specific dependence between individuals. Additionally, different copulas are considered to model possible cross-dependence explicitly. The main data example concerns the daily air quality (ordinal) in three cities in North China. Here, a spatial dimension is present, which can be exploited in the definition of the proximity matrix and the copulas.