We give a general formulation of a non-Gaussian conditional linear AR(1) model subsuming most of the non-Gaussian AR(1) models that have appeared in the literature. We derive some general results giving properties for the stationary process mean, variance and correlation structure, and conditions for stationarity. These results highlight similarities and differences with the Gaussian AR(1) model, and unify many separate results appearing in the literature. Examples illustrate the wide range of properties that can appear under the conditional linear autoregressive assumption. These results are used in analysing three real data sets, illustrating general methods of estimation, model diagnostics and model selection. In particular, we show that the theoretical results can be used to develop diagnostics for deciding if a time series can be modelled by some linear autoregressive model, and for selecting among several candidate models.
Keywords: autoregression, data analysis, non-Gaussian time series, Poisson time series, Gamma time series, Exponential time series.