A Gentle Introduction to Stationarity#

Stage 2 · 📐 Stationarity · Lesson 03 of 18 · beginner

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Important

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What it is#

A series is stationary when its statistical behaviour does not depend on time: loosely, the mean, the variance, and the way points co-vary with their lagged selves all stay constant as the observation window slides. A stationary series has no predictable long-run pattern — no trend and no fixed seasonal swing.

The intuition#

Stationarity is what makes a series learnable from a single realisation. If the rules do not change over time, the past is a fair guide to the future, and one long stretch of data is enough to estimate them. Trend and seasonality break this because the distribution of the value shifts with when you look. (The subtlety worth remembering: a series with cycles of no fixed length can still be stationary — you cannot say in advance where the peaks will fall.)

Achieving stationarity#

The classical fix is differencing — model the change rather than the level:

\[\nabla x_t = x_t - x_{t-1},\]

which removes a linear trend; a seasonal difference \(x_t - x_{t-s}\) removes a period-\(s\) pattern. (Detrending by regression is an alternative.) This “integrate” step is the I in ARIMA. Over-differencing injects artefacts, so use the smallest order that works.

How to check#

Plot first: a stationary series looks flat and stable around a constant level, with no drift. Then formalise with tests — the Augmented Dickey–Fuller (ADF) test (null hypothesis: non-stationary, so a small p-value argues for stationarity) and the KPSS test (null: stationary), commonly read together. In statsmodels these are adfuller(x) and kpss(x).

Hint