Dynamic generalized linear models for non-Gaussian time series forecasting

The purpose of this paper is to provide a discussion, with illustrating examples, on Bayesian forecasting for dynamic generalized linear models (DGLMs). Adopting approximate Bayesian analysis, based on conjugate forms and on Bayes linear estimation, we describe the theoretical framework and then we provide detailed examples of response distributions, including binomial, Poisson, negative binomial, geometric, normal, log-normal, gamma, exponential, Weibull, Pareto, beta, and inverse Gaussian. We give numerical illustrations for all distributions (except for the normal). Putting together all the above distributions, we give a unified Bayesian approach to non-Gaussian time series analysis, with applications from finance and medicine to biology and the behavioural sciences. Throughout the models we discuss Bayesian forecasting and, for each model, we derive the multi-step forecast mean. Finally, we describe model assessment using the likelihood function, and Bayesian model monitoring. Some key words: Bayesian forecasting, non-Gaussian time series, dynamic generalized linear model, state space, Kalman filter.