ARIMA Forecasting: Stationarity Tests, Order Selection and Prediction
What is ARIMA?
ARIMA (autoregressive integrated moving average) is the classic Box-Jenkins time-series forecasting model, combining an autoregressive part p, a differencing part d and a moving-average part q, written ARIMA(p,d,q). For seasonal data it extends to SARIMA(p,d,q)(P,D,Q,s), where s is the seasonal period. It is widely used for sales forecasting, quality trend monitoring, inventory demand and economic indicator forecasting.
When to Use It
ARIMA requires a stationary input series, so use it after checking stationarity: the ADF test has a unit-root null hypothesis (rejecting it suggests stationarity) while the KPSS test has a stationarity null hypothesis (rejecting it suggests non-stationarity). If the tests contradict each other the series is usually trend-stationary. Apply first- or second-order differencing (d = 1 or 2) as needed; the tool recommends the differencing automatically.
How to Use It (Step by Step)
Paste the time series and run the ADF/KPSS tests, apply the suggested differencing, then identify the order from the ACF (a cutoff suggests q) and PACF (a cutoff suggests p), or let the tool search the best (p,d,q) by AIC or BIC. Fit the model, check the residual diagnostics with the Ljung-Box white-noise test, then set the forecast horizon and read the point forecasts with their 95% confidence intervals.
Key Formulas / Example
ARIMA(p,d,q): (1 − ΣφᵢBⁱ)(1 − B)ᵈx_t = (1 + ΣθⱼBʲ)ε_t, where B is the backshift operator. AIC = 2k − 2ln(L̂) and BIC = k·ln(n) − 2ln(L̂), with smaller values indicating a better balance of fit and parsimony. Short-horizon forecasts (1–5 steps) are the most reliable; validate longer forecasts out of sample with MAE or RMSE before using them for decisions.