Autoregressive Integrated Moving Average (ARIMA)

ARIMA is the workhorse of practical forecasting, generalising the autoregressive model in two directions at once. The autoregressive part uses past values of the series, and the moving-average part uses past errors, so the model learns from both the level of the series and the mistakes it just made.

The integrated part handles trends, because many business series, such as cumulative sales or price levels, wander without a stable mean, which breaks the assumptions of a plain autoregressive model. Differencing the series, working with period-to-period changes instead of levels, restores the stable behaviour the model needs.

Every ARIMA model carries three numbers, written (p, d, q), where p counts autoregressive lags, d counts how many times the series is differenced, and q counts the lags of past errors. An ARIMA(1,1,1) is a common starting point for trending monthly data.

The Box-Jenkins methodology gives the model its disciplined workflow: first identify candidate orders from autocorrelation plots, then estimate the coefficients, then diagnose the residuals to confirm they look like noise. Real projects usually iterate the loop several times before trusting the model, and automated selection tools speed this up, but checking residuals remains the quality gate.

The NIST/SEMATECH e-Handbook of Statistical Methods treats Box-Jenkins models as a mature, well-documented technique, which is part of their appeal for auditors and regulators because the assumptions are explicit and the diagnostics are standard. Software made this workflow routine, but judgment still matters.

Two models can fit history equally well while telling different stories about the future, so forecasters compare candidates on out-of-sample accuracy rather than in-sample fit alone, and parsimony usually wins. Practitioners often keep a small stable of approved models, comparing an ARIMA against an exponential smoothing baseline on rolling forecasts to keep everyone honest about whether the extra complexity earns its keep.

Seasonal data extends the framework with seasonal lags, producing SARIMA. A retailer with a strong December peak might difference the series at lag twelve as well as at lag one, letting the model learn that this December will resemble last December.

For managers, ARIMA's strength is honest humility about drivers. The model does not need to know why demand moves, only the demand history, which makes it valuable for inventory, staffing and cash planning when the underlying causes are many, shifting or unknown.

Its limits deserve equal billing, since ARIMA assumes the past's statistical structure persists, so sudden regime changes break it without warning and it offers no explanation anyone can act on, yet used for short horizons with frequent re-estimation it is often the toughest benchmark a forecasting team must beat.