988:. Chan and Tong (1986) rigorously proved that the family of STAR models includes the SETAR model as a limiting case by showing the uniform boundedness and equicontinuity with respect to the switching parameter. Without this proof, to say that STAR models nest the SETAR model lacks justification. Unfortunately, whether one should use a SETAR model or a STAR model for one's data has been a matter of subjective judgement, taste and inclination in much of the literature. Fortunately, the test procedure, based on David Cox's test of separate family of hypotheses and developed by Gao, Ling and Tong (2018, Statistica Sinica, volume 28, 2857-2883) is now available to address this issue. Such a test is important before adopting a STAR model because, among other issues, the parameter controlling its rate of switching is notoriously data-hungry.
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STAR models were introduced and comprehensively developed by Kung-sik Chan and Howell Tong in 1986 (esp. p. 187), in which the same acronym was used. It originally stands for Smooth
Threshold AutoRegressive. For some background history, see Tong (2011, 2012). The models can be thought of in
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of regimes. In both cases the presence of the transition function is the defining feature of the model as it allows for changes in values of the parameters.
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part. Most popular transition function include exponential function and first and second-order logistic functions. They give rise to
Logistic STAR (
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terms of extension of autoregressive models discussed above, allowing for changes in the model parameters according to the value of a
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348:{\displaystyle y_{t}=\gamma _{0}+\gamma _{1}y_{t-1}+\gamma _{2}y_{t-2}+...+\gamma _{p}y_{t-p}+\epsilon _{t}.\,}
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1105:{\displaystyle y_{t}=\mathbf {X_{t}} +G(z_{t},\zeta ,c)\mathbf {X_{t}} +\sigma ^{(j)}\epsilon _{t}\,}
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They can be understood as two-regime SETAR model with smooth transition between regimes, or as
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1792:{\displaystyle G(z_{t},\zeta ,c)=(1+exp(-\zeta (z_{t}-c_{1})(z_{t}-c_{2})))^{-1},\zeta >0}
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144:(AR) parts linked by the transition function. The model is usually referred to as the
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152:) models proceeded by the letter describing the transition function (see below) and
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Chan, K. S.; Tong, H. (1986). "On
Estimating Thresholds in Autoregressive Models".
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1939:"Smooth Transition Autoregressive Models—A Survey of Recent Developments"
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Three basic transition functions and the name of resulting models are:
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Defined in this way, STAR model can be presented as follows:
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first order logistic function - results in
Logistic STAR (
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Exponential transition function for the ESTAR model with
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Exponential transition function for the ESTAR model with
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Logistic transition function for the ESTAR model with
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Van Dijk, D.; Teräsvirta, T.; Franses, P. H. (2002).
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is the transition function bounded between 0 and 1.
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2004:"Threshold Autoregression in Economics"
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2011:Statistics and Its Interface
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894:varying from -10 to +10,
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690:{\displaystyle \gamma \,}
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94:are typically applied to
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181:Consider a simple AR(
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1949:(1): 1–47.
1884:introducing
853:white-noise
851:stands for
485:white-noise
483:stands for
187:time series
124:past values
96:time series
18:Star schema
2064:Categories
1840:References
172:Definition
168:) models.
80:statistics
1964:1765/1656
1892:June 2012
1781:ζ
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