A Poisson INAR(1) process with a seasonal structure

Abstract

This paper introduces a non-negative integer-valued autoregressive (INAR) process with seasonal structure of first order, which is an extension of the standard INAR(1) model proposed by Al-Osh and Alzaid [First-order integer-valued autoregressive (INAR(1)) process. J Time Ser Anal. 1987;8:261–275]. The main properties of the model are derived such as its stationarity and autocorrelation function (ACF), among others. The conditional least squares and conditional maximum likelihood estimators of the model parameters are studied and their asymptotic properties are established. Some detailed discussion is dedicated to the case where the marginal distribution of the process is Poisson. A Monte Carlo experiment is conducted to evaluate and compare the performances of these estimators for finite sample sizes. The standard Yule–Walker approach is also considered for comparison purposes. The empirical results indicate that, in general, the conditional maximum likelihood estimator presents much better performance in terms of bias and mean square error. The model is illustrated using a real data set.