Nonlinear ARMA Model (NARMA)
Nonlinear Autoregressive Moving Average Model · Also known as: NARMA, nonlinear ARMA, NLARMA, nonlinear autoregressive moving average
The Nonlinear ARMA (NARMA) model extends the classical linear ARMA framework by allowing the conditional mean to depend on past observations and past errors through an arbitrary nonlinear function. It captures complex dynamics — such as regime changes, asymmetric cycles, and threshold effects — that linear models miss, making it valuable for economic and financial time series.
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When to use it
Use a NARMA model when diagnostic tests — such as the BDS test, Tsay's nonlinearity test, or an ARCH-LM test — provide evidence that the conditional mean is nonlinear, or when economic theory predicts asymmetric responses (e.g., recessions vs. expansions). It suits moderate to large samples (at least 100–200 observations) and univariate or low-dimensional multivariate series. Do not use it when a well-specified linear ARMA passes all diagnostic tests, when the sample is small (fewer than ~80 observations), or when interpretability is paramount and a complex function cannot be justified to stakeholders.
Strengths & limitations
- Captures asymmetric, threshold, and regime-switching dynamics that linear ARMA models systematically miss.
- Encompasses a broad class of specific models (TAR, STAR, bilinear, NN-AR) as special cases, so it provides a unifying theoretical framework.
- Can substantially improve out-of-sample forecast accuracy for economic and financial series with known nonlinear patterns.
- Accommodates both autoregressive and moving-average nonlinearity, giving more flexibility than pure nonlinear AR models.
- Supported by well-developed specification tests (BDS, Tsay, RESET) that guide model selection before estimation.
- Estimation is computationally intensive and may converge to local minima; global search strategies are needed.
- Model selection (choice of functional form, orders p and q) requires expertise and careful testing, increasing the risk of over-fitting.
- Requires reasonably large samples — at least 100–200 observations — for reliable parameter estimation and inference.
- Multi-step forecasts are harder to compute and interpret than in the linear case, often requiring simulation.
- Interpretability of the nonlinear function f is limited compared to linear coefficients, reducing transparency for policy use.
Frequently asked
How is a NARMA model different from a standard ARMA model?
A linear ARMA model expresses y_t as a fixed linear combination of past values and past errors. A NARMA model replaces that linear combination with an arbitrary nonlinear function f, allowing the impact of past observations and shocks to vary with the current state of the series.
How do I know whether I need a NARMA model?
Run nonlinearity tests before committing: the BDS test for general nonlinear dependence, Tsay's test for threshold-type nonlinearity, and ARCH-LM for variance nonlinearity. If the conditional mean tests reject linearity, a NARMA specification is warranted. If only variance tests reject, GARCH may be sufficient.
What specific models fall under the NARMA umbrella?
Common special cases include the Threshold AR (TAR), Smooth Transition AR (STAR), bilinear models (where f includes products of lags and errors), and neural-network AR models. The choice depends on the form of nonlinearity suggested by tests and economic theory.
Is NARMA suitable for short time series?
Generally no. Nonlinear estimation is parameter-hungry; fewer than about 80–100 observations typically yields unreliable estimates with wide confidence intervals. For short series, consider a simpler regime-switching specification or a linear model with structural breaks.
How are multi-step forecasts computed for a NARMA model?
Because the expectation of a nonlinear function differs from the function of the expectation, iterative point substitution underestimates forecast uncertainty. Simulation-based methods — Monte Carlo integration or bootstrap residual resampling — are preferred for constructing reliable multi-step forecast intervals.
Sources
- Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach. Oxford University Press. ISBN: 978-0198522300
- Granger, C. W. J., & Terasvirta, T. (1993). Modelling Nonlinear Economic Relationships. Oxford University Press. link ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Autoregressive Moving Average Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-arma-model
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