The switch from one regime to another depends on the past values of the x series (hence the Self-Exciting portion of the name).
The model consists of k autoregressive (AR) parts, each for a different regime.
Consider a simple AR(p) model for a time series yt where: written in a following vector form: where: SETAR models were introduced by Howell Tong in 1977 and more fully developed in the seminal paper (Tong and Lim, 1980).
They can be thought of in terms of extension of autoregressive models, allowing for changes in the model parameters according to the value of weakly exogenous threshold variable zt, assumed to be past values of y, e.g. yt-d, where d is the delay parameter, triggering the changes.
For a comprehensive review of developments over the 30 years since the birth of the model, see Tong (2011).