Bayesian Causality Test for Integer-Valued Time Series Models with Applications to Climate and Crime Data

We investigate the causal relationship between climate and criminal behaviour. Considering the characteristics of integer-valued time series of criminal incidents, we propose a modified Granger causality test based on the generalized auto-regressive conditional heteroscedasticity type of integer-valued time series models to analyse the relationship between the number of crimes and the temperature as an environmental factor. More precisely, we employ the Poisson, negative binomial and log-linear Poisson integer-valued generalized auto-regressive conditional heteroscedasticity models and particularly adopt a Bayesian method for our analysis. The Bayes factors and posterior probability of the null hypothesis help to determine the causality between the variables considered. Moreover, employing an adaptive Markov chain Monte Carlo sampling scheme, we estimate model parameters and initial values. As an illustration, we evaluate our test through a simulation study and, to examine whether or not temperature affects crime activities, we apply our method to data sets categorized as sexual offences, drug offences, theft of motor vehicles, and domestic-violence-related assault in Ballina, New South Wales, Australia. The result reveals that more sexual offences, drug offences and domestic-violence-related assaults occur during the summer than in other seasons of the year. This evidence strongly advocates a causal relationship between crime and temperature.