Asymptotic behavior of CLS estimators for unstable INAR(2) models

Abstract

In this paper the asymptotic behavior of the conditional least squares estimators of the autoregressive parameters $(\alpha,\beta)$, of the stability parameter $\varrho := \alpha + \beta$, and of the mean $\mu$ of the innovation $\vare_k$, $k \in \NN$, for an unstable integer-valued autoregressive process $X_k = \alpha \circ X_{k-1} + \beta \circ X_{k-2} + \vare_k$, $k \in \NN$, is described. The limit distributions and the scaling factors are different according to the following three cases: (i) decomposable, (ii) indecomposable but not positively regular, and (iii) positively regular models.