Modeling time series of count with excess zeros and ones based on INAR(1) model with zero-and-one inflated Poisson innovations

Abstract

The first-order non-negative integer-valued autoregressive (INAR(1)) process has been applied to model the counts of events for a long time, and the Poisson model provides a standard and popular framework. However, this model may not be suitable for a data set with excess zeros and excess ones. We introduce a new stationary INAR(1) process with zero-and-one inflated Poisson (ZOIP) innovations. The proposed model is assumed to be a mixture of three separate data generation processes: one generates only zeros, one generates only ones, and the last one is a Poisson data-generating process. We present some structural properties such as the mean, variance, marginal and joint distribution functions of the process. We develop a method to test whether zero and one inflated under a Poisson INAR(1) model, which is based on dispersion index, zero index and one index. We also give the asymptotic distribution of the resulting test statistics under the null hypothesis of a Poisson INAR(1) model. Conditional maximum likelihood estimators are given, and the asymptotic properties of the estimators are established. In addition, the forecasting problem is addressed. Finally, a simulation study shows that the estimation method is accurate and reliable as long as the sample size is reasonably large. Two real data examples lead to superior performances of the proposed model compared with other competitive models in the literature.