You cannot mean-revert a random walk. Log prices of individual stocks are non-stationary: their variance grows without bound over time, their historical averages drift, and any threshold you set for "overpriced" or "underpriced" has no statistical content — prices have no level to return to. Statistical arbitrage makes spreads tradeable by constructing linear combinations of non-stationary price series whose stochastic trends cancel, leaving a residual whose variance is bounded and whose mean is stable. The formal name for this cancellation is cointegration; the formal name for the property of the residual is stationarity; and whether a candidate spread has that property is decided by a unit-root test.
This note builds the econometric toolkit that underpins everything from pair selection to S-score calculation: the definition of stationarity, the random walk as the canonical non-stationary process, orders of integration, and the Augmented Dickey–Fuller test. It closes with the key insight that two I(1) series can form an I(0) spread — the bridge into cointegration and the common-trends model.
Learning objectives. By the end of this note you should be able to:
- state the definitions of strict and weak (covariance) stationarity and identify which properties each requires;
- demonstrate that a random walk is I(1) by showing its variance grows linearly in time;
- explain why log stock prices are modelled as random walks and why log returns are approximately white noise;
- execute the Augmented Dickey–Fuller test and interpret its output (test statistic, critical values, order of augmentation);
- determine whether a spread is I(0) or I(1) from ADF output and explain what the result implies for tradeability;
- state the key insight of cointegration: that two I(1) series can have an I(0) linear combination.
In this lesson
-
- Stationarity: the condition for statistical predictability
-
- White noise and moving averages as building blocks
-
- The random walk as the canonical non-stationary series
-
- Log prices as random walks; log returns as white noise
-
- Orders of integration
-
- The Augmented Dickey–Fuller test
-
- Why a stationary spread is the prerequisite for tradeability
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
Read the full lessonNew to SelfMadeQuant.com? See membership options or browse the Statistical Arbitrage syllabus.