What is an ARIMA model?
An ARIMA (AutoRegressive Integrated Moving Average) model explains the current value of a time series using its own past values and past forecast errors. Series with trends are first differenced to make them stable. George Box and Gwilym Jenkins set out the full model-building procedure in *Time Series Analysis: Forecasting and Control* (1970), so it is also called the Box–Jenkins method.
- AR(p)
- regression on p past values
- I(d)
- differenced d times to reach stationarity
- MA(q)
- regression on q past errors
- Notation
- ARIMA(p, d, q)
Stationarity and differencing
ARIMA assumes a stationary series whose mean and variance do not drift. Trending series such as GDP are differenced once (period-to-period change) or occasionally twice. Unit-root tests, above all the augmented Dickey–Fuller (ADF) test building on Dickey and Fuller (1979), help decide.
The Box–Jenkins cycle
- Identify candidate p, d, q from ACF and PACF plots.
- Estimate coefficients, comparing models with AIC or BIC.
- Diagnose residuals with the Ljung–Box test (1978) to check for white noise.
- Forecast with the model that passes.
| Model | ACF | PACF |
|---|---|---|
| AR(p) | Decays | Cuts off after p |
| MA(q) | Cuts off after q | Decays |
| ARMA | Decays | Decays |
How do utilities forecast tomorrow's power demand? The simplest assumption is that past patterns continue. ARIMA turns that intuition into a model; when patterns repeat every summer, seasonal ARIMA (SARIMA) adds seasonal differencing.
Limitations
ARIMA is linear and struggles with structural breaks. When volatility clusters, as in financial returns, analysts add a GARCH model for the variance. The claim that prices are hard to forecast from past data links to the efficient market hypothesis.
Frequently asked questions
- What does ARIMA(1,1,1) mean?
- The once-differenced series is explained by one lagged value and one lagged error.
- How many times should I difference?
- Usually once, rarely twice, guided by unit-root tests and plots.
- ARIMA vs ARMA?
- ARMA is for stationary series; ARIMA adds differencing for non-stationary ones.
References
- Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: Forecasting and control. Holden-Day.
- Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366), 427–431.
- Ljung, G. M., & Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika, 65(2), 297–303.