Stability of nonlinear AR(1) time series with delay

The stability of generally defined nonlinear time series is of interest as nonparametric and other nonlinear methods are used more and more to fit time series. We provide sufficient conditions for stability or nonstability of general nonlinear AR(1) models having delay d⩾1. Our results include conditions for each of the following modes of the associated Markov chain: geometric ergodicity, ergodicity, null recurrence, transience and geometric transience. The conditions are sharp for threshold-like models and they characterize parametric threshold AR(1) models with delay.