Asymptotic properties of CLS estimators in the Poisson AR(1) model

Many papers have been written on count valued ARMA models, since they were introduced by Al-Osh and Alzaid [1987. J. Time Ser. Anal. 8, 261-275] and McKenzie [1988. Adv. Appl. Probab. 20, 822-835]. However surprisingly little has been written about estimation of these models and even less about the asymptotic properties of the parameter estimates. In fact, some of the asymptotic properties that do appear and are cited in the literature are incorrect. In this paper we derive a corrected explicit expression for the asymptotic variance matrix of the conditional least squares estimators (CLS) of the Poisson AR(1) process. We also show that the distribution of the CLS estimators is asymptotically equivalent to that of estimators based on the Yule-Walker equations and thus neither is more efficient than the other to this order.