Heterogeneous INAR(1) model with application to car insurance

The bonus-malus scheme shows how the history of claim arrivals determines the dynamics of insurance premium. It is important to distinguish to what extent changes in the insurance premium are explained by lagged claim counts introduced among explanatory variables and by unobservable heterogeneity included in the model, which needs to be regularly updated. For this purpose, we introduce the integer valued autoregressive (INAR) model with unobserved heterogeneity. The model is applied to premium updating in car insurance and compared to the standard method based on the negative binomial distribution. We find that the premium depends on the claim history and that the timing of claim arrivals matters. This result is different from the outcome of the standard framework in which the average number of claims per year is the only relevant factor.