A parametric time series model with covariates for integers in Z

While models for integer valued time series are now abundant, there is a shortage of similar models when the time series refer to data defined on Z, i.e., in both the positive and negative integers. Such data occur in certain disciplines and the need for such models also appear when taking differences of a positive integer count time series. In addition one would often like to include covariates to explain variations in the variable of interest. In this article we construct a model doing all these assuming a specific innovation distribution and provide fully parametric inference, including prediction. Real data applications on accidents and financial returns are given. Finally we also discuss alternative models and extensions.