A trinomial difference autoregressive model and its applications

Abstract

This paper considers the autoregressive modelling problem of ‐valued time series of counts, whose counting sequence consists of −1, 0 and 1. Most existing methods are based on the signed binomial thinning operator and its some extensions, in which the counting sequence consists of 0 and 1, that is, the cases of −1 is ignored. To fill this gap, we first construct the trinomial difference thinning operator and then propose the trinomial difference Z‐valued autoregressive (TDZAR) model and give some stochastic properties. An attractive merit of the TDZAR model is that the incorporated trinomial difference thinning operator makes the conditional maximum likelihood estimate more wieldy and easy. Second, we discuss the two‐step conditional least squares estimate and the conditional maximum likelihood estimate and establish their asymptotic properties of the estimators. Third, the performances of these estimators are compared via simulation study. Finally, we apply the proposed model to a real data set.