SETAR (model)

A Self-Exciting Threshold Autoregressive (SETAR) model is a non-linear time series model. It is an extension of the autoregressive (AR) model, allowing for different AR regimes based on the past values of the time series itself. The model is "self-exciting" because the transition between regimes is determined by the time series data.

In its simplest form, a SETAR model consists of two or more regimes, each defined by a different AR model. The regime in which the model operates at any given time is determined by a threshold variable (usually a lagged value of the time series) and one or more threshold values. When the threshold variable crosses a threshold value, the model switches to a different regime.

The mathematical representation of a two-regime SETAR(p,d) model is as follows:

y_t = φ_0,1 + φ_1,1y_t-1 + ... + φ_p,1y_t-p + ε_t, if y_t-d ≤ r

y_t = φ_0,2 + φ_1,2y_t-1 + ... + φ_p,2y_t-p + ε_t, if y_t-d > r

Where:

• y_t is the time series value at time t.
• p is the order of the autoregressive model.
• d is the delay parameter, indicating which lagged value of y_t is used as the threshold variable.
• r is the threshold value.
• φ_i,j are the autoregressive coefficients for regime j (j=1,2) and lag i (i=0,1,...,p). φ_0,j represents the intercept term for regime j.
• ε_t is a white noise error term.

SETAR models are particularly useful for modeling time series that exhibit non-linear behavior, such as asymmetry, limit cycles, and sudden jumps or breaks. They are commonly used in economics, finance, and hydrology to model phenomena like business cycles, interest rates, and river flows.

Key aspects of specifying a SETAR model include determining the number of regimes, the order of the autoregressive model within each regime (p), the delay parameter (d), and the threshold values (r). These parameters are typically estimated using statistical techniques such as least squares or maximum likelihood estimation. The choice of these parameters significantly impacts the model's ability to accurately represent the underlying time series dynamics.