Modelling Z-valued time series with Skellam thinning-based INAR(1) process

Non-stationary count time series, which are small in value and show a trend having relatively large fluctuation, are commonly encountered in real-world applications. Differencing is a commonly employed method to transform a non-stationary time series into a stationary one. However, the differenced count time series is defined on Z and cannot be fitted by the mainstream count time series models, which are defined on N-0. Although some integer-valued autoregressive (INAR) models based on the signed thinning operator have been proposed for fitting Z-valued time series, these models still encounter difficulties in terms of data generation mechanism, model selection, and parameter estimation. These difficulties may pose challenges for practitioners in practical applications. Therefore, this article utilizes the reparameterized version of the Skellam (Poisson difference) distribution to construct a new thinning operator, thereby introducing a class of Z-valued time series models. Strict stationarity, ergodicity, statistical properties, and parameter estimation methods of the processes are detailedly discussed. An application to the monthly counts of newly listed companies illustrates that the new model is superior in data analysis of finance and economics.