Asymptotic distribution of the Yule–Walker estimator for INAR [formula omitted] processes

Abstract

The INteger-valued AutoRegressive (INAR) processes were introduced in the literature by Al-Osh and Alzaid [1987. First-order integer-valued autoregressive (INAR(1)) process. J. Time Ser. Anal. 8, 261–275] and McKenzie [1988. Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Probab. 20, 822–835] for modelling correlated series of counts. These processes have been considered as the discrete counter part of AR processes, but their highly nonlinear characteristics lead to some statistically challenging problems, namely in parameter estimation. Several estimation procedures have been proposed in the literature, mainly for processes of first order. For some of these estimators the asymptotic properties as well as finite sample properties have been obtained and studied. This paper considers Yule–Walker parameter estimation for the pth-order integer-valued autoregressive, INAR ( p ) , process. In particular, the asymptotic distribution of the Yule–Walker estimator is obtained and it is shown that this estimator is asymptotically normally distributed, unbiased and consistent.