ARMA Model (Autoregressive Moving Average)
Autoregressive Moving Average Model · Also known as: ARMA, Box-Jenkins model, autoregressive moving average, AR(p)MA(q)
The ARMA(p,q) model describes a stationary time series as a combination of two components: an autoregressive part that regresses the current value on its own past p values, and a moving average part that accounts for past q error terms. It is the foundational framework of the Box-Jenkins methodology for univariate time series modelling and short-run forecasting.
Read the full method
Sign in with a free account to read this section.
Method map
The neighbourhood of related methods — select a node to explore.
+10 more
When to use it
Use ARMA when the series is already stationary (or has been made so by a small number of differences) and the goal is parsimonious in-sample modelling or short-to-medium-horizon forecasting of a single variable. It excels when ACF/PACF patterns suggest bounded autoregressive or moving-average memory. Do not use ARMA when the series is non-stationary without differencing (use ARIMA or SARIMA), when seasonality is present (use SARIMA), when the series has conditional heteroscedasticity in volatility (add a GARCH layer), when explanatory variables are needed (use ARIMAX or regression with ARMA errors), or when multiple interrelated series are to be modelled jointly (use VAR).
Strengths & limitations
- Combines autoregressive and moving-average dynamics in one parsimonious model with few parameters.
- Provides optimal linear forecasts for stationary series under the Box-Jenkins framework.
- Well-understood theoretical properties: identifiability, invertibility, and asymptotic normality of MLE estimates.
- Widely implemented in R (stats, forecast), Python (statsmodels), EViews, Stata, and SAS.
- ACF/PACF diagnostics give interpretable guidance for model selection.
- Requires strict covariance stationarity; a unit root or trend violates the assumptions and demands differencing or detrending first.
- Linear by construction — it cannot capture asymmetries, threshold effects, or nonlinear dynamics without extension.
- Order selection (p, q) can be ambiguous when ACF/PACF do not cut off cleanly; AIC and BIC sometimes favour different orders.
- Forecasting accuracy degrades rapidly at long horizons for volatile or structurally changing series.
- Cannot model volatility clustering; use ARMA-GARCH when conditional variance is time-varying.
Frequently asked
What is the difference between ARMA and ARIMA?
ARMA(p,q) is for stationary series. ARIMA(p,d,q) adds d rounds of differencing to make a non-stationary series stationary before fitting an ARMA(p,q). If your series has a unit root, use ARIMA; if it is already stationary, ARMA suffices.
How do I choose p and q?
Start with the ACF and PACF plots to get candidate orders, then compare models by AIC or BIC over a grid of small (p, q) values. Prefer the model with the lowest information criterion that also passes the Ljung-Box residual test.
Can ARMA handle seasonal patterns?
Pure ARMA cannot capture seasonality. Use SARIMA, which adds seasonal AR and MA terms at the seasonal lag, or apply seasonal differencing before fitting ARMA.
Does ARMA assume normally distributed errors?
Maximum likelihood estimation assumes Gaussian errors for finite-sample inference and interval forecasts. However, the AR and MA parameter estimates are consistent and asymptotically normal even under non-Gaussian errors, so point forecasts remain valid. Heavy-tailed or skewed errors weaken the reliability of the confidence intervals.
What is the invertibility condition for the MA part?
Invertibility requires that the roots of the MA polynomial lie outside the unit circle. Without invertibility, the model is not uniquely identified and the MA parameters cannot be consistently estimated. Most software enforces invertibility during estimation.
Sources
- Box, G. E. P., & Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control. Holden-Day. link ↗
- Brockwell, P. J., & Davis, R. A. (2002). Introduction to Time Series and Forecasting (2nd ed.). Springer. ISBN: 978-0387953519
How to cite this page
ScholarGate. (2026, June 3). Autoregressive Moving Average Model. ScholarGate. https://scholargate.app/en/econometrics/arma-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.
- ARIMA modelEconometrics↔ compare
- Autoregressive modelEconometrics↔ compare
- Moving Average ModelEconometrics↔ compare
- SARIMA modelEconometrics↔ compare
- Vector AutoregressionEconometrics↔ compare