We review and synthesize the wide range of non-Gaussian first order linear autoregressive models that have appeared in the literature. Models are organized into broad classes to clarify similarities and differences and facilitate application in particular situations. General properties for process mean, variance and correlation are derived, unifying many separate results appearing in the literature. Examples illustrate the wide range of properties that can appear even under the autoregressive assumption. These results are used in analysing a variety of real data sets, illustrating general methods of estimation, model diagnostics and model selection.
Keywords: autoregression, data analysis, non-Gaussian time series, Poisson time series, Gamma time series, Exponential time series.