Nonlinear Moving Average (NMA) Model
Nonlinear Moving Average Model · Also known as: NMA model, nonlinear moving average, NLMA model, nonlinear MA
The Nonlinear Moving Average (NMA) model extends the classical linear MA model by allowing the current observation to depend on past innovations through a nonlinear function rather than a simple weighted sum. It is used in time series analysis when error shocks transmit to outcomes in an asymmetric or state-dependent fashion.
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When to use it
Use a Nonlinear MA model when diagnostic tests (BDS, Hinich bispectrum, Tsay test) reject linearity in the MA dynamics of a series, or when theoretical reasoning suggests that shocks propagate asymmetrically. It is well-suited to financial returns with leverage effects, commodity prices exhibiting threshold behavior, and macro series with regime-dependent shock transmission. Do not apply it routinely as a default: the linear MA model is preferable when nonlinearity tests are not significant, sample sizes are modest (fewer than 200 observations), or when interpretability and parsimony are primary concerns.
Strengths & limitations
- Captures asymmetric and state-dependent shock propagation that linear MA models miss.
- Flexible enough to nest linear MA as a special case, allowing formal model comparison.
- Can substantially improve multi-step forecast accuracy for series with nonlinear dynamics.
- Grounded in a coherent theoretical framework with links to bilinear, GARCH, and threshold models.
- Diagnostic tests (BDS, Tsay) provide principled guidance on whether nonlinearity is warranted.
- Innovations are latent; estimation requires iterative numerical methods and is sensitive to starting values.
- Overfitting risk is high with flexible nonlinear specifications on moderate samples.
- Model selection (choice of q, functional form of f) lacks the standardized guidance available for linear ARMA models.
- Theoretical properties such as stationarity and invertibility are harder to establish than in the linear case.
- Interpretation of estimated parameters is less transparent than linear MA coefficients.
Frequently asked
How is a Nonlinear MA model different from a standard (linear) MA model?
A linear MA(q) model writes y_t as a weighted sum of q past innovations. A Nonlinear MA model replaces that weighted sum with a general nonlinear function of those same innovations, allowing the impact of a shock to depend on its sign, magnitude, or the sign of earlier shocks.
How do I test whether a nonlinear MA specification is needed?
Apply the BDS test to residuals from a fitted linear MA model. Rejection indicates unexplained nonlinear dependence. The Tsay test and Hinich bispectrum test are additional diagnostics. Only if these tests reject linearity is a nonlinear extension clearly warranted.
What software can estimate Nonlinear MA models?
R packages such as tsDyn and TSA, as well as MATLAB's Econometrics Toolbox and EViews, support various nonlinear time series specifications. Custom NLS or ML routines are often required for less standard functional forms of f.
Can I combine a Nonlinear MA with autoregressive terms?
Yes. Mixed specifications — sometimes called Nonlinear ARMA or bilinear ARMA — include both past observations and past innovations in a nonlinear function. These are more general but require larger samples and careful regularisation to avoid overfitting.
Is a Nonlinear MA model stationary?
Stationarity conditions for NMA models are more complex than for linear MA and depend on the specific functional form of f and the distribution of innovations. Sufficient conditions exist for some special cases (e.g., bilinear models) but must be verified analytically or by simulation for each specification.
Sources
- Granger, C. W. J., & Andersen, A. P. (1978). An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht, Gottingen. link ↗
- Tong, H. (1990). Non-Linear Time Series: A Dynamical System Approach. Oxford University Press. ISBN: 978-0198522300
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Moving Average Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-ma-model
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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