SETAR (model)

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In statistics, Self-Exciting Autoregressive (SETAR) 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 (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).

1 Definition
2 See also
3 References

[edit] Definition

[edit] Autoregressive Models

Consider a simple AR(p}) model for a time series y_t

$y_{t}=\gamma_{0}+\gamma_{1}y_{t-1}+\gamma_{2}y_{t-2}+...+\gamma_{p}y_{t-p}+\epsilon_{t}.\,$

where:

$\gamma_{i}\,$ for i=1,2,...,p are autoregressive coefficients, assumed to be constant over time;

$\epsilon_{t}\sim^{iid}WN(0;\sigma^{2})\,$ stands for white-noise error term with constant variance.

written in a following vector form:

$y_{t}=\mathbf{X_{t}\gamma}+\sigma\epsilon_{t}.\,$

where:

$\mathbf{X_{t}}=(1,y_{t-1},y_{t-2},\ldots,y_{t-p})\,$ is a column vector of variables;

$\gamma \,$ is the vector of parameters : $\gamma_{0}, \gamma_{1},\gamma_{2},..., \gamma_{p}\,$ ;

$\epsilon_{t}\sim^{iid}WN(0;1)\,$ stands for white-noise error term with constant variance.

[edit] SETAR as an Extension of the Autoregressive Model

SETAR models 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 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:

$y_{t}=\mathbf{X_{t}}\gamma^{(j)}+\sigma^{(j)}\epsilon_{t}\,$ if $r_{j-1}<z_{t}<r_{j}.\,$

where:

$X_{t}=(1,y_{t-1},y_{t-2},...,y_{t-p})\,$ is a column vector of variables;

$-\infty=r_{0}<r_{1}<\ldots<r_{k}=+\infty\,$ are k-1 non-trivial thresholds dividing the domain of z_t into k different regimes.

[edit] Basic Structure

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.

[edit] See also

Logistic Smooth-Transmision Model

[edit] References

Hansen, B.E. (1997). Inference in TAR Models, Studies in Nonlinear Dynamics and Econometrics, 2, 1-14.
Tong, H. (1990). Non-Linear Time Series: A Dynamical System Approach. Oxford University Press.
Tsay, R.S. (1989). Testing and Modeling Threshold Autoregressive Processes, Journal of the American Statistical Association, 84 (405), 231-240.

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Categories: Non-linear systems | Time series analysis