On non-stationarity of the Poisson gamma state space models

TL;DR

This paper investigates the non-stationary properties of Poisson-gamma state space models, revealing that their predictive variance diverges over time and forecasts tend to zero, with implications for long-term predictions.

Contribution

The study uncovers the non-stationary behavior of PGSS models' predictive distributions and analyzes how hyper-parameters influence long-term forecast convergence.

Findings

Predictive variance diverges with forecast horizon.

Predictive distribution converges to zero over time.

Hyper-parameters affect long-term forecast behavior.

Abstract

The Poisson-gamma state space (PGSS) models have been utilized in the analysis of non-negative integer-valued time series to sequentially obtain closed form filtering and predictive densities. In this study, we show the underlying mechanics and non-stationary properties of multi-step ahead predictive distributions for the PGSS family of models. By exploiting the non-stationary structure of the PGSS model, we establish that the predictive mean remains constant while the predictive variance diverges with the forecast horizon, a property also found in Gaussian random walk models. We show that, in the long run, the predictive distribution converges to a zero-degenerated distribution, such that both point and interval forecasts eventually converge towards zero. In doing so, we comment on the effect of hyper-parameters and the discount factor on the long-run behavior of the forecasts.