Time series models with univariate margins in the convolution-closed infinitely divisible class

Abstract

A unified way of obtaining stationary time series models with the univariate margins in the convolution-closed infinitely divisible class is presented. Special cases include gamma, inverse Gaussian, Poisson, negative binomial, and generalized Poisson margins. ARMA time series models obtain in the special case of normal margins, sometimes in a different stochastic representation. For the gamma and Poisson margins, some previously defined time series models are included, but for the negative binomial margin, the time series models are different and, in several ways, better than previously defined time series models. The models are related to multivariate distributions that extend a univariate distribution in the convolution-closed infinitely divisible class. Extensions to the non-stationary case and possible applications to modelling longitudinal data are mentioned.