SETAR (model)
In statistics, Self-Exciting Threshold AutoRegressive 'models are typically applied to time series data as an extension of autoregressive models, in order to allow for higher degree of flexibility in model parameters through a regime switching behaviour.
Given a time series of data x''t, the SETAR model is a tool for understanding and, perhaps, predicting future values in this series, assuming that the behaviour of the series changes once the series enters a different regime. The switch from one regime to another depends on the past values of the x series.
The model consists of k autoregressive parts, each for a different regime. The model is usually referred to as the SETAR' model where k'' is the number of threshold, there are k+1 number of regime in the model, and p is the order of the autoregressive part.
Definition
Autoregressive Models
Consider a simple AR model for a time series ytwhere:
written in a following vector form:
where:
SETAR as an Extension of the Autoregressive Model
SETAR models were introduced by Howell Tong in 1977 and more fully developed in the seminal paper. 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.Defined in this way, SETAR model can be presented as follows:
where:
The SETAR model is a special case of Tong's general threshold autoregressive models. The latter allows the threshold variable to be very flexible, such as an exogenous time series in the open-loop threshold autoregressive system, a Markov chain in the Markov-chain driven threshold autoregressive model, which is now also known as the Markov switching model.
For a comprehensive review of developments over the 30 years
since the birth of the model, see Tong.