First‐Order Integer‐Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

A. Alzaid and M. Al‐Osh

Statistica Neerlandica, 1988, vol. 42, issue 1, 53-61

Abstract: Some properties of a first‐order integer‐valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self‐decomposability and unimodality of the 1‐dimensional marginals of the process {Xn} generated according to the scheme Xn=α°Xn‐i +en, where α°Xn‐1 denotes a sum of Xn ‐ 1, independent 0 ‐ 1 random variables Y(n‐1), independent of Xn‐1 with Pr‐(y(n ‐ 1)= 1) = 1 ‐ Pr (y(n‐i)= 0) =α. The distribution of the innovation process (en) is obtained when the marginal distribution of the process (Xn) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.

Date: 1988
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