ARIMA Models: How Nonstationary Models Are Built from Stationary Ones#

Stage 6 · 🏗️ Building & Forecasting Models · Lesson 15 of 18 · advanced

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Important

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The core idea#

ARMA needs a stationary series, but real data trends and drifts. ARIMA bridges the gap with one move: difference the series until it is stationary, fit an ordinary ARMA to the differenced version, and the model inherits ARMA’s whole toolkit. The “I” stands for Integrated — the series must be un-differenced (integrated) to recover the original.

The model#

An ARIMA(p, d, q) applies the \(d\)-th difference \((1-B)^d\) before the ARMA machinery:

\[\phi(B)\,(1 - B)^d\, x_t = \theta(B)\, w_t.\]

Here \((1-B)^d\) is the differencing operator, \(\phi(B)\) the AR polynomial and \(\theta(B)\) the MA polynomial. A series needing \(d\) differences to become stationary is called integrated of order \(d\), or \(I(d)\).

Choosing d#

One difference (\(d = 1\)) removes a linear trend; two (\(d = 2\)) removes a quadratic one; seasonal patterns need a seasonal difference (next lesson). Pick the smallest \(d\) that makes the ADF / KPSS tests read stationary — over-differencing inflates the variance and injects artificial correlation, so more is not better.

Forecasting back#

Fitting happens on the differenced scale, but forecasts are wanted on the original one. The model “integrates” — cumulatively sums — its differenced-scale predictions back up to the level of the raw series, carrying the forecast uncertainty with it. In statsmodels this is all handled by ARIMA(y, order=(p, d, q)).

Hint