ARIMA (Autoregressive Integrated Moving Average) Model
Autoregressive Integrated Moving Average Model · Also known as: Box-Jenkins model, ARIMA(p,d,q), ARIMA Modeli
ARIMA is a univariate time-series forecasting model that combines autoregressive, integrated (differencing), and moving-average components to predict a single continuous series from its own past. It is the centrepiece of the Box-Jenkins methodology set out in Box, Jenkins, Reinsel & Ljung's Time Series Analysis (5th ed., 2015).
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When to use it
Use ARIMA when you have a single continuous time series of reasonable length (at least about 50 observations) and want to forecast its future from its own past. The series must be made stationary through differencing, and after fitting the residuals should behave like white noise with no remaining autocorrelation (checked with a Ljung-Box test). Identify the orders beforehand with stationarity tests (ADF, KPSS) and ACF/PACF analysis. It is not the right tool for short series, where simpler exponential smoothing is preferred, nor for series with strong seasonality, which call for SARIMA.
Strengths & limitations
- Captures momentum, shocks, and trend in one compact univariate model with only the orders (p, d, q) to set.
- Backed by the well-established Box-Jenkins identification-estimation-diagnostic workflow.
- Produces forecasts with calibrated prediction intervals and a clear diagnostic check on the residuals.
- Needs a stationary series, achieved through differencing; non-stationary data must first be transformed.
- Parameter estimates are unreliable on short series (n < 50), where simpler methods do better.
- Standard ARIMA does not model seasonality; seasonal patterns require SARIMA instead.
Frequently asked
What do the (p, d, q) orders mean?
p is the number of autoregressive lags (how many past values feed the forecast), d is how many times the series is differenced to remove trend and reach stationarity, and q is the number of moving-average lags (how many past forecast errors feed the forecast). ACF and PACF plots help identify p and q, while stationarity tests guide d.
How do I know if my series is stationary?
Run an Augmented Dickey-Fuller (ADF) test and a KPSS test. If the series is non-stationary you apply differencing (raise d); if differencing alone is not enough, a log transformation or detrending can help before fitting.
How do I check that the fitted ARIMA is adequate?
Examine the residuals: a Ljung-Box test should not reject the null of no autocorrelation. A significant result (p < 0.05) means the model is misspecified, so the order should be increased or alternatives considered. An ARCH-LM test can additionally flag variance clustering.
When should I use SARIMA or another model instead?
If the series shows clear seasonal peaks in the ACF, switch to SARIMA, which extends ARIMA with seasonal terms. For very short series, simple exponential smoothing is more reliable, and if residual autocorrelation persists a multivariate VAR model may fit better.
Sources
- Box, G. E. P., Jenkins, G. M., Reinsel, G. C. & Ljung, G. M. (2015). Time Series Analysis: Forecasting and Control (5th ed.). Wiley. ISBN: 978-1118675021
How to cite this page
ScholarGate. (2026, June 1). Autoregressive Integrated Moving Average Model. ScholarGate. https://scholargate.app/en/econometrics/arima
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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