Autoregressive Models Consider a simple AR(
p) model for a
time series yt :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
row 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.
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 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: : y_{t}=\mathbf{X_{t}}\gamma^{(j)}+\sigma^{(j)}\epsilon_{t}\quad \textrm{if} \quad r_{j-1} where: : X_{t}=(1,y_{t-1},y_{t-2},...,y_{t-p})\, is a column vector of variables; : -\infty=r_{0} are
k+1 non-trivial thresholds dividing the domain of
zt 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 :\gamma^{(j)}\,. In such setting, a change of the regime (because the past values of the series
yt-d surpassed the threshold) causes a different set of coefficients :\gamma^{(j)}\, to govern the process
y. ==See also==