  A new forecasting method based on the concept of the profile predictive the
likelihood function is proposed for discrete-valued processes. In particular,
generalized autoregressive and moving average (GARMA) models for Poisson
distributed data are explored in details. Highest density regions are used to
construct forecasting regions. The proposed forecast estimates and regions are
coherent. Large sample results are derived for the forecasting distribution.
Numerical studies using simulations and a real data set are used to establish
the performance of the proposed forecasting method. Robustness of the proposed
method to possible misspecification in the model is also studied.
