Generalized Poisson difference autoregressive processes

Abstract

This paper introduces a novel stochastic process with signed integer values. Its autoregressive dynamics effectively captures persistence in conditional moments, rendering it a valuable feature for forecasting applications. The increments follow a Generalized Poisson distribution, capable of accommodating over- and under-dispersion in the conditional distribution, thereby extending standard Poisson difference models. We derive key properties of the process, including stationarity conditions, the stationary distribution, and conditional and unconditional moments, which prove essential for accurate forecasting. We provide a Bayesian inference framework with an efficient posterior approximation based on Markov Chain Monte Carlo. This approach seamlessly incorporates inherent parameter uncertainty into predictive distributions. The effectiveness of the proposed model is demonstrated through applications to benchmark datasets on car accidents and an original dataset on cyber threats, highlighting its superior fitting and forecasting capabilities compared to standard Poisson models.