A bivariate INAR(1) process with application

The study of time series models for count data has become a topic of special interest during the last years. However, while research on univariate time series for counts now flourishes, the literature on multivariate time series models for count data is notably more limited. In the present paper, a bivariate integer-valued autoregressive process of order 1 (BINAR(1)) is introduced. Emphasis is placed on models with bivariate Poisson and bivariate negative binomial innovations. We discuss properties of the BINAR(1) model and propose the method of conditional maximum likelihood for the estimation of its unknown parameters. Issues of diagnostics and forecasting are considered and predictions are produced by means of the conditional forecast distribution. Estimation uncertainty is accommodated by taking advantage of the asymptotic normality of maximum likelihood estimators and constructing appropriate confidence intervals for the h-step-ahead conditional probability mass function. The proposed model is applied to a bivariate data series concerning daytime and nighttime road accidents in the Netherlands.