Nonlinear Autoregressive (NAR) Model
Nonlinear Autoregressive Model · Also known as: NAR model, nonlinear autoregression, NLAR, threshold autoregressive model
The Nonlinear AR model extends the classical autoregressive framework by allowing the mapping from past values to the current value to follow an arbitrary or regime-switching nonlinear function. Major families include the Self-Exciting Threshold AR (SETAR), Smooth Transition AR (STAR), and neural network AR, each capturing different forms of asymmetry, regime shifts, or smooth nonlinear dynamics in univariate time series.
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When to use it
Use a Nonlinear AR model when a univariate time series shows evidence of asymmetric cycles, regime shifts, or nonlinear mean reversion that a linear AR fails to capture — confirmed by a statistical linearity test. It is well suited to financial returns exhibiting asymmetric volatility clustering, macroeconomic series showing expansion-recession asymmetry, and commodity prices with threshold-driven reversals. Do not apply it when the linearity test is insignificant, when the series is very short (fewer than 80-100 observations give unreliable threshold estimates), or when the nonlinearity is clearly attributable to a structural break rather than a recurring regime mechanism.
Strengths & limitations
- Captures asymmetric and regime-dependent dynamics that linear AR models systematically miss.
- SETAR and STAR forms are interpretable: each regime has clearly identifiable AR coefficients and economic meaning.
- Nested inside the linear AR framework, so a formal statistical test can justify the added complexity.
- Flexible enough to approximate a wide variety of nonlinear time-series processes with relatively few parameters.
- Produces conditional forecasts that reflect regime uncertainty, improving multi-step prediction accuracy for asymmetric series.
- Requires substantially longer series than a linear AR to estimate threshold or transition parameters reliably; short samples lead to poorly identified regimes.
- Model selection (number of regimes, lag order, threshold variable, delay d) is complex and involves multiple nested choices.
- Neural network AR variants are highly flexible but prone to overfitting and lack interpretable coefficients.
- Standard information criteria (AIC, BIC) are less reliable for comparing non-nested nonlinear models.
Frequently asked
How do I choose between SETAR and STAR?
SETAR assumes an abrupt switch between regimes, while STAR uses a smooth logistic or exponential transition. The LST linearity test and inspection of the estimated transition speed parameter c can guide choice; if regime changes appear gradual, STAR is preferred. In practice, both models are estimated and compared by AIC or residual diagnostics.
What lag order and delay parameter should I use?
Lag order p is typically chosen by AIC or BIC on the linear AR first, then confirmed on the nonlinear model. The delay parameter d (which lag acts as the threshold variable) is usually set to d in {1,...,p} and selected by minimising the residual sum of squares over the grid search.
Can I apply a Nonlinear AR model to a non-stationary series?
Threshold nonlinearity and unit-root non-stationarity can coexist (e.g., a threshold unit-root process). For clearly unit-root series, first test for cointegration or difference the series, then fit the NAR model to the stationary representation. Applying SETAR directly to a strongly non-stationary series can produce spurious regime identification.
Does the Nonlinear AR model handle multivariate data?
The standard NAR/SETAR/STAR framework is univariate. For multivariate settings use a Threshold VAR (TVAR) or Nonlinear VAR, which extend the regime-switching idea to a system of equations.
How many observations do I need?
A general rule of thumb is at least 80-100 observations for a two-regime SETAR with a moderate lag order. Each regime should contain at least 15-20% of observations; otherwise threshold estimates are imprecise and inference on regime coefficients is unreliable.
Sources
- Tong, H. (1990). Non-Linear Time Series: A Dynamical System Approach. Oxford University Press. ISBN: 9780198522201
- Terasvirta, T. (1994). Specification, estimation, and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association, 89(425), 208-218. DOI: 10.1080/01621459.1994.10476462 ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Autoregressive Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-ar-model
Which method?
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