TAR / SETAR: Threshold Autoregression for Regime-Switching Time Series
Also known as: Threshold Autoregression, Self-Exciting Threshold Autoregression, SETAR Model, Eşik Otoregresyon
TAR and SETAR are nonlinear autoregressive models introduced by Howell Tong (1990) that allow a time series to follow different linear dynamics in distinct regimes, separated by one or more threshold values. SETAR is the self-exciting variant, in which the threshold variable is a lagged value of the series itself, making it particularly suited to cycles, asymmetric adjustment, and limit-cycle behavior observed in economic and financial data.
Key highlights
- Captures asymmetric dynamics and nonlinear mean reversion that linear AR models miss
- Piecewise linearity makes each regime interpretable and estimable by OLS
- SETAR is endogenously driven — no exogenous regime variable is needed
- Flexible enough to approximate limit cycles and complex nonlinear behavior
Intuition
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How it works
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When to use it
Use TAR/SETAR when theory or prior evidence suggests that a time series behaves differently in distinct regimes — for example, during expansions versus recessions, bull versus bear markets, or above versus below purchasing-power parity. The model requires stationarity within each regime and a reasonably long sample (typically T > 100) to estimate multiple regime-specific parameters reliably. It is most appropriate when regime switches are abrupt; if transitions are gradual, the smooth-transition variant (STAR) is preferable. A linearity pre-test should always precede estimation.
Strengths & limitations
- Captures asymmetric dynamics and nonlinear mean reversion that linear AR models miss
- Piecewise linearity makes each regime interpretable and estimable by OLS
- SETAR is endogenously driven — no exogenous regime variable is needed
- Flexible enough to approximate limit cycles and complex nonlinear behavior
- Grid search for the threshold is computationally intensive for multiple thresholds or high-dimensional models
- Requires a sufficiently large sample in every regime; thin regimes yield imprecise estimates
- Abrupt regime assumption is restrictive — real transitions may be gradual
- Linearity test has nonstandard distribution under the null, complicating inference
Common pitfalls
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Applications
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Frequently asked
What is the difference between TAR and SETAR?
In TAR the threshold variable q_t can be any observable variable, including an exogenous indicator. In SETAR — Self-Exciting TAR — q_t is a lagged value of the dependent series y_{t-d}. SETAR is therefore a purely endogenous nonlinear model, whereas TAR admits externally driven switching. In practice, SETAR is the more common specification in economics and finance.
How do I choose the number of regimes?
Start with two regimes, which is the theoretically motivated minimum and the most commonly encountered case. Additional thresholds can be tested sequentially using a sup-Wald or likelihood-ratio procedure, adding one regime at a time. Information criteria (AIC, BIC) computed over the grid-estimated models can also guide selection, though they do not fully account for the nonstandard distribution of threshold estimates.
Is stationarity required for TAR/SETAR?
Global stationarity of a SETAR process is not guaranteed by within-regime stationarity alone. A sufficient condition for geometric ergodicity is that the AR roots lie inside the unit circle in at least one regime and the process cannot permanently reside in an explosive regime. Tong (1990) provides formal conditions; applied researchers should verify that the estimated model satisfies these before conducting inference.
Sources
- 1.Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach. Oxford University Press.ISBN 978-0-19-852300-6
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Cite this page
ScholarGate. (2026, June 2). TAR / SETAR. ScholarGate. https://scholargate.app/econometrics/tar-setar