Article ID: | iaor19932371 |
Country: | Israel |
Volume: | 29 |
Issue: | 2 |
Start Page Number: | 363 |
End Page Number: | 373 |
Publication Date: | Jun 1992 |
Journal: | Journal of Applied Probability |
Authors: | Liu Jian, Susko Ed |
Keywords: | time series & forecasting methods |
Two recent papers by Petruccelli and Woolford and Chan et al. showed that the key element governing ergodicity of a threshold AR(1) model is the joint behavior of the two linear AR(1) pieces falling in the two boundary threshold regimes. They used essentially the necessary and sufficient conditions for ergodicity of a general Markov chain of Tweedie in a rather clever manner. However, it is difficult to extend the results to the more general threshold ARMA models. Besides, irreducibility is also required to apply Tweedie’s results. In this paper, instead of pursuing the ideas in Tweedie’s results, the authors shall develop a criterion similar in spirit to the technique used by Benes in the context of continuous-time Markov chains. Consequently, they derive a necessary and sufficient condition for existence of a strictly stationary solution of a general non-linear ARMA model to be introduced in Section 2 of this paper. This condition is then applied to the threshold ARMA(1,