  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.
