Error Correction Models

Two cointegrated stock prices can each wander like random walks on their own, while the gap between them keeps snapping back toward a stable long-run level. Knowing they're cointegrated tells you the gap is mean-reverting; it doesn't tell you how fast, or what to do with a day-to-day change in either price. An error correction model (ECM) answers both.

The two pieces of a price move

For a cointegrated pair with spread $z_{t-1} = y_{t-1} - \beta x_{t-1}$, the ECM writes the change in $y$ as

$$\Delta y_t = \alpha \, z_{t-1} + \gamma \, \Delta x_t + \varepsilon_t$$

In words: today's move in $y$ is partly explained by how far the spread was from equilibrium yesterday ($z_{t-1}$, corrected by a speed-of-adjustment coefficient $\alpha$), partly by today's move in $x$ moving in tandem ($\gamma \, \Delta x_t$), and partly by pure noise ($\varepsilon_t$). The coefficient $\alpha$ is the whole point: it must be negative for the relationship to be mean-reverting, and its size tells you what fraction of yesterday's dislocation gets closed out today.

Worked example

A pair's spread closes at $z_{t-1} = 2.00$ (two points above equilibrium) at the end of Monday. The fitted ECM has $\alpha = -0.25$ and $\gamma \, \Delta x_t \approx 0$ for Tuesday. The predicted correction is $\Delta y_t = -0.25 \times 2.00 = -0.50$: $y$ is expected to fall by 0.50 on Tuesday, closing a quarter of the two-point gap, leaving roughly 1.50 still open. If instead $\alpha = -0.60$, the same two-point dislocation predicts a 1.20 correction — a much faster-reverting pair, and a much shorter expected holding period for a spread trade entered at that level.

What this means in practice

The size of $\alpha$ is directly usable: it implies a half-life of mean reversion ($\ln(0.5)/\ln(1+\alpha)$ periods, roughly), which tells a stat-arb desk how long to expect a position to take to pay off and therefore how much capital it ties up. A very small $|\alpha|$ (close to zero) means the spread is technically cointegrated but corrects so slowly that transaction costs and financing can eat the edge before it mean-reverts.