On the Ergodicity of First-Order Threshold Autoregressive Moving-Average Processes

We introduce a certain Markovian representation for the threshold autoregressive moving-average (TARMA) process with which we solve the long-standing problem regarding the irreducibility condition of a first-order TARMA model. Under some mild regularity conditions, we obtain a complete classification of the parameter space of an invertible first-order TARMA model into parametric regions over which the model is either transient or recurrent, and the recurrence region is further subdivided into regions of null recurrence or positive recurrence, or even geometric recurrence. We derive a set of necessary and sufficient conditions for the ergodicity of invertible first-order TARMA processes.