Bayesian modelling of integer-valued transfer function models

External events are commonly known as interventions that often affect times series of counts. This research introduces a class of transfer function models that include four different types of interventions on integer-valued time series: abrupt start and abrupt decay (additive outlier), abrupt start and gradual decay (transient shift), abrupt start and permanent effect (level shift), and gradual start and permanent effect. We propose integer-valued transfer function models incorporating a generalized Poisson, log-linear generalized Poisson, or negative binomial to estimate and detect these four types of interventions in a time series of counts. Utilizing Bayesian methods, which are adaptive Markov chain Monte Carlo (MCMC) algorithms to obtain the estimation, we further employ deviance information criterion (DIC), posterior odd ratios, and mean squared standardized residual for model comparisons. As an illustration, this study evaluates the effectiveness of our methods through a simulation study and application to crime data in Albury City, New South Wales (NSW) Australia. Simulation results show that the MCMC procedure is reasonably effective. The empirical outcome also reveals that the proposed models can successfully detect the locations and types of interventions.