Periodically stationary multivariate autoregressive models

Abstract

A class of multivariate periodic autoregressive models is proposed where coupling between time series is achieved through linear mean functions. Various response distributions with quadratic mean-variance relationships fit into the framework, including the negative binomial, gamma and Gaussian distributions. We develop an iterative algorithm to obtain unconditional means, variances and auto-/cross-covariances for models with higher order lags. Analytical solutions are given for the univariate model with lag one and multivariate models with linear mean-variance relationship. A special case of the model class is an established framework for modelling multivariate time series of counts from routine surveillance of infectious diseases. We extend this model class to allow for distributed lags and apply it to a dataset on norovirus gastroenteritis in two German states. The availability of unconditional moments and auto/cross-correlations enhances model assessment and interpretation.