Autoregressive Models

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

${\displaystyle y_{t}=\gamma _{0}+\gamma _{1}y_{t-1}+\gamma _{2}y_{t-2}+...+\gamma _{p}y_{t-p}+\epsilon _{t}.\,}$

where:

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

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

written in a following vector form:

${\displaystyle y_{t}=\mathbf {X_{t}\gamma } +\sigma \epsilon _{t}.\,}$

where:

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

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

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

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 (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 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:

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

where:

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

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

The SETAR model is a special case of Tong's general threshold autoregressive models (Tong and Lim, 1980, p. 248). The latter allows the threshold variable to be very flexible, such as an exogenous time series in the open-loop threshold autoregressive system (Tong and Lim, 1980, p. 249), a Markov chain in the Markov-chain driven threshold autoregressive model (Tong and Lim, 1980, p. 285), 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 (2011).

Basic Structure

In each of the k regimes, the AR(p) process is governed by a different set of p variables :${\displaystyle \gamma ^{(j)}\,}$. 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 :${\displaystyle \gamma ^{(j)}\,}$ to govern the process y.