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

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 {X n } generated according to the scheme X n =α° X n ‐i +e n , where α° X n‐1 denotes a sum of X n ‐ 1 , independent 0 ‐ 1 random variables Y (n‐1) , independent of X n‐1 with Pr ‐( y (n ‐ 1) = 1) = 1 ‐ Pr ( y (n‐i) = 0) =α. The distribution of the innovation process ( e n ) is obtained when the marginal distribution of the process ( X n ) 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.