Nonlinear ARIMA Model
Nonlinear Autoregressive Integrated Moving Average Model · Also known as: nonlinear ARIMA, NARIMA, nonlinear time series model, nonlinear Box-Jenkins model
The Nonlinear ARIMA model extends the classical Box-Jenkins ARIMA framework by allowing the conditional mean of a time series to depend on past values and past errors through a nonlinear function. It encompasses families such as Threshold AR (TAR/SETAR), Smooth Transition AR (STAR/LSTAR/ESTAR), and Markov-switching models, capturing asymmetric dynamics, regime changes, and business-cycle asymmetries that linear ARIMA cannot represent.
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When to use it
Use a nonlinear ARIMA model when preliminary tests (Tsay, Terasvirta LM) reject linearity, when the series shows asymmetric responses to positive vs. negative shocks, when there is evidence of regime-dependent dynamics (e.g., recession vs. expansion behavior), or when a linear ARIMA leaves substantial structure in the residuals. Do not use it as a default or when the sample is short (fewer than ~80 observations), when the nonlinearity test fails to reject linearity, or when interpretability is paramount and a simpler linear model fits adequately.
Strengths & limitations
- Captures asymmetric and state-dependent dynamics that linear ARIMA systematically misses.
- Encompasses a rich family of well-studied special cases (SETAR, LSTAR, ESTAR, Markov-switching) with established estimation theory.
- Can improve out-of-sample forecasting for series with regime changes, particularly over business-cycle horizons.
- Formal linearity tests (Tsay, Terasvirta LM) guide whether the added complexity is justified by the data.
- Threshold and transition parameters have economically interpretable meaning, linking statistical structure to economic regimes.
- Requires substantially more data than linear ARIMA; sparse samples yield unreliable threshold estimates.
- Estimation is computationally intensive — grid search over threshold or transition location is often necessary, and local optima are a real risk.
- Model selection across nonlinear families (TAR vs. STAR vs. Markov-switching) is non-trivial and can lead to over-fitting if not guided by formal tests.
- Forecast uncertainty propagates nonlinearly, making analytic confidence intervals difficult to derive.
- Structural change (genuine parameter instability) can masquerade as nonlinearity, leading to mis-specified models.
Frequently asked
How is a Nonlinear ARIMA different from a standard ARIMA?
A standard ARIMA uses a single linear combination of past values and errors to model the conditional mean. A nonlinear ARIMA replaces this with a nonlinear function — for example, switching between two different AR processes depending on whether a threshold variable is above or below a cutoff, or smoothly blending two regimes. The integration and moving-average components still play the same role; only the AR dynamics become nonlinear.
Which nonlinear model should I choose — TAR, LSTAR, or ESTAR?
Terasvirta's LM sequence test helps discriminate: it tests for LSTAR vs. ESTAR given a rejection of linearity. LSTAR is appropriate for asymmetric (boom-bust) dynamics; ESTAR suits symmetric but nonlinear mean reversion (e.g., real exchange rates). TAR/SETAR is preferred when regime shifts appear abrupt rather than smooth.
What sample size do I need?
As a practical rule, at least 80-100 time-series observations are recommended before fitting nonlinear ARIMA models. Threshold location parameters are estimated with high variance in small samples, and nonlinearity tests lose power, making false rejections and spurious threshold estimates more likely.
Can I combine a nonlinear ARIMA with GARCH errors?
Yes, and it is often advisable. Many financial and macroeconomic series exhibit both nonlinear conditional means and time-varying conditional variances. Specifying, say, an LSTAR-GARCH model captures both features simultaneously. Estimate the conditional mean structure first, then fit a GARCH model to the residuals.
Do nonlinear ARIMA models always forecast better than linear ARIMA?
Not automatically. Nonlinear models tend to outperform in forecast comparisons when genuine regime changes occur within the evaluation window. For long horizons or when the series is well-approximated by a linear process, the added estimation variance of the nonlinear model can hurt forecast accuracy. Always compare out-of-sample RMSE against a linear benchmark.
Sources
- Tong, H. (1990). Non-Linear Time Series: A Dynamical System Approach. Oxford University Press. ISBN: 9780198522249
- Terasvirta, T. (1994). Specification, estimation, and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association, 89(425), 208-218. link ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Autoregressive Integrated Moving Average Model. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-arima-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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