A note on the asymptotic behavior of a mildly unstable integer-valued AR(1) model

This note examines the conditional least squares (CLS) estimators for integer-valued autoregressive (INAR) models, with a focus on the INAR(1) model. Our investigation reveals that the joint limiting distribution of the CLS estimators for the autoregressive coefficient and the mean of the model errors is degenerate in the case of mildly unstable INAR(1) models with moderate deviations from unity. This degeneracy may pose challenges for statistical inference. We provide commenting notes based on the asymptotic results to address this issue. Additionally, we conduct simulation studies and analyze a real-world dataset to validate the theoretical findings.