ARIMA Models
ARIMA forecasts a series by combining autoregression, differencing for trend, and a moving average of past errors.
Three letters, three ideas
ARIMA stands for AutoRegressive Integrated Moving Average, and its three parameters (p, d, q) name three mechanisms. The AR part regresses the value on its own p past values. The I part differences the series d times to remove trend and make it stationary. The MA part regresses on the last q forecast errors, capturing short shocks that persist briefly.
Reaching stationarity
ARIMA requires a stationary series, one whose statistical properties do not drift over time. Differencing (subtracting the previous value) removes linear trend; differencing twice handles curvature. Tests such as the augmented Dickey-Fuller test help decide how many differences d are needed. Over-differencing injects spurious structure, so the smallest d that achieves stationarity is preferred.
Choosing p and q
The autocorrelation and partial autocorrelation functions guide the choice: a sharp cutoff in the partial autocorrelation suggests the AR order p, while a cutoff in the autocorrelation suggests the MA order q. In practice, candidate models are compared by information criteria such as AIC or BIC, and automated procedures search the (p, d, q) grid to minimize them.
Seasonality and extensions
Seasonal ARIMA (SARIMA) adds a second set of terms at the seasonal lag, for example capturing that each December resembles prior Decembers. ARIMAX includes exogenous regressors. These models are transparent and give principled prediction intervals, and a well-fit ARIMA is a strong, interpretable baseline against which more complex forecasters must justify themselves.
- p, d, q set autoregression, differencing, and error-averaging
- Series must be made stationary before fitting the ARMA part
- Use AIC or BIC and residual checks to select orders
- SARIMA adds seasonal terms; ARIMAX adds external drivers