Hysteretic Poisson INGARCH model for integer-valued time series

This study proposes a new model for integer-valued time series-the hysteretic Poisson integer-valued generalized autoregressive conditionally heteroskedastic (INGARCH) model-which has an integrated hysteresis zone in the switching mechanism of the conditional expectation. Our modelling framework provides a parsimonious representation of the salient features of integer-valued time series, such as discreteness, over-dispersion, asymmetry and structural change. We adopt Bayesian methods with a Markov chain Monte Carlo sampling scheme to estimate model parameters and utilize the Bayesian information criteria for model comparison. We then apply the proposed model to five real time series of criminal incidents recorded by the New South Wales Police Force in Australia. Simulation results and empirical analysis highlight the better performance of hysteresis in modelling the integer-valued time series.