Time-Series Modeling and Decomposition

Another type of nonlinear time series models are the Self-Exciting Threshold AutoRegressive (SETAR) models introduced in a seminal paper by Tong and Lim. They are typically applied as an extension of autoregressive models, in order to allow for higher degree of flexibility in model parameters through a regime switching behavior. 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 behavior 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 (hence the Self-Exciting portion of the name). The model consists of $k$ autoregressive (AR) parts, each for a different regime. The model is usually referred to as the SETAR($k$, $p$) model where $k$ is the number of regimes and $p$ is the order of the autoregressive part (since those can differ between regimes, the $p$ portion is sometimes dropped and models are denoted simply as SETAR(k).They allow for changes in the model parameters according to the value of weakly exogenous threshold variable $z_{t}$, assumed to be past values of $y$, e.g. $y_{t-d}$, where $d$ is the delay parameter, triggering the changes. Defined in this way, SETAR model can be presented as follows:

$y_{t}=X_{t} \gamma^{(j)}+\sigma^{(j)} \varepsilon_{t} \text { if } r_{j-1} \text { < } z_{t} \text { < } r_{j}$

where $X_{t}=\left(1, y_{t-1}, y_{t-2}, \ldots, y_{t-p}\right)$ is a column vector of variables; $-\infty=r_{0} \text { < } r_{1} \text { < } \ldots \text { < } r_{k}=+\infty$ are $k-1$ non-trivial thresholds dividing the domain of $z_{t}$ into $k$ different regimes. In each of the $k$ regimes, the AR(p) process is governed by a different set of $p$ variables: $\gamma^{(j)}$. In such setting, a change of the regime (because the past values of the series $y_{t-d}$ surpassed the threshold) causes a different set of coefficients: $\gamma^{(j)}$. to govern the process $y$. 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 Markovchain driven threshold autoregressive model which is now also known as the Markov switching model.