First‐order Integer‐valued Autoregressive (Inar(1)) Process

Abstract

A simple model for a stationary sequence of integer-valued random variables with lag-one dependence is given and is referred to as the integer-valued autoregressive of order one (INAR(1)) process. The model is suitable for counting processes in which an element of the process at time t can be either the survival of an element of the process at time t -1 or the outcome of an innovation process. The correlation structure and the distributional properties of the INAR(1) model are similar to those of the continuousvalued AR(1) process. Several methods for estimating the parameters of the model are discussed, and the results of a simulation study for these estimation methods are presented.

Time series count data models: An empirical application to traffic accidents

“…The class of integer-valued autoregressive processes denoted by INAR have been studied by many authors (e.g., Al-Osh and Alzaid, 1987;McKenzie, E., 1988, Brännäs, Hellström, 2001, Karlis, 2006. …”

“…The properties of the model in equation (5) can be found in Al-Osh and Alzaid (1987) andMaKenzie (1988). The mean and variance of the process { } t Y are equal to ) 1 /( α λ − .…”

“…The mean and variance of the process { } t Y are equal to ) 1 /( α λ − . Equation (5) is termed as the Poisson INAR(1) which assumes that the underlying time series process is a stationary (Al-Osh and Alzaid, 1987;MaKenzie , 1988;Hall, 2001, Hellstrom, 2002). (1) model, and the INARMA(1,1,) NB model.…”

Estimation and Forecasting in INAR(P) Models Using Sieve Bootstrap

“…The most common approach to build an integer-valued autoregressive (INAR) process is based on a probabilistic operator called binomial thinning, as reported in Al-Osh and Alzaid (1987) and McKenzie (1985) who first introduced INAR processes. While theoretical properties of INAR models with Poisson innovations have been extensively studied in the literature (see, for instance, Freeland and McCabe (2004a), , and the references therein), relatively few contributions discuss the development of methods for INAR models with innovations distributed differently from the Poisson.…”

Note on integer-valued bilinear time series models

“…Until the mid seventies modeling discrete valued time series did not attract much attention since most traditional representations of dependence become either impossible or impractical. The last two decades there were many developments in the literature on integer-valued time series; see McKenzie (2003) for a detailed review; for example, INteger-valued Moving Average processes (Al-Osh and Alzaid (1991)), INteger-valued AutoRegressive processes (Al-Osh and Alzaid (1987), Al-Osh and Alzaid (1990), and Du and Li (1991)), and Generalized INteger-valued AutoRegressive processes (Latour (1998)) are common choices. Doukhan et al (2006) introduced the class of nonnegative INteger-valued BiLinear time series, which contains the three aforementioned classes.…”