ARIMA

A classical forecasting model combining autoregression, differencing and moving averages.

Also known as: ARIMA, autoregressive integrated moving average, Box-Jenkins

ARIMA forecasts a series from its own history through three ideas, one per letter: autoregression — predict from past values, i.e. lags; integrated — difference the series until it is stationary; moving average — correct using past forecast errors. The (p, d, q) order says how many of each: ARIMA(2,1,1) uses two lags, one differencing, one error term.

ARIMA(p,d,q):  difference d times → predict from p lags + q past errors

ARIMA shines on short, stable, univariate series with clear short-term structure — the classic operations-forecasting workhorse. Its discipline is its value: checking stationarity, reading autocorrelation plots to choose orders, and validating on a holdout forces you to understand the series rather than throwing lags at it.

The classic mistakes:

Fitting levels that trend. ARIMA needs stationarity; fitting raw trending data gives nonsense orders and drifting forecasts. Difference (or detrend) first, and verify.
Order-picking by in-sample fit. More parameters always fit history better. Choose (p,d,q) by holdout error or an information criterion, not by how pretty the fitted line looks.
Ignoring the residuals. If leftover errors still show structure (autocorrelation, season), the model missed something the data is still saying. Diagnose residuals; they are the model telling on itself.
Univariate tunnel vision. ARIMA sees one series. Promotions, prices, holidays and campaigns need exogenous inputs (ARIMAX/regression hybrids) or scenarios — a pure ARIMA will “learn” the promo effect as noise.
Seasonality assumed away. Plain ARIMA has no seasonal machinery; seasonal data needs seasonal orders or explicit deseasonalising first.

When not to use it: long, rich, multi-driver problems (use regression with features), intermittent demand (use specialised methods), or any series where a naive forecast already wins on the holdout. ARIMA earns its keep on short-horizon, stable, single-series forecasts — measure it against naive before trusting it.