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Finance & economics · Topic
Cointegration
Cointegration describes a stable long-run relationship between two or more non-stationary time series. Even though each series wanders randomly (I(1)), a linear combination of them is stationary. This underpins pairs trading, purchasing-power-parity tests, and ECMs.
Definition
Series $y_t$ and $x_t$ are cointegrated (CI(1,1)) if:
- Both are integrated of order 1: $y_t \sim I(1)$, $x_t \sim I(1)$.
- There exists $\beta$ such that $z_t = y_t - \beta x_t \sim I(0)$.
The vector $(1, -\beta)^\top$ is called the cointegrating vector.
Error Correction Model (ECM)
Granger’s representation theorem says that every cointegrated system has an ECM:
\[\Delta y_t = \alpha(y_{t-1} - \beta x_{t-1}) + \text{lags} + \varepsilon_t\]The coefficient $\alpha < 0$ is the speed of adjustment — how fast the system corrects deviations from equilibrium. This gives cointegration its economic content: shocks are transitory, not permanent.
Testing for cointegration
| Test | H₀ | Notes |
|---|---|---|
| Engle–Granger | No cointegration | Two-step; residual-based; limited to one cointegrating vector |
| Johansen | $r$ cointegrating vectors | Full-system; handles multiple vectors; preferred in practice |
The Johansen trace statistic tests:
\[\Lambda_{trace}(r) = -T\sum_{i=r+1}^n \log(1 - \hat{\lambda}_i)\]where $\hat{\lambda}_i$ are the estimated eigenvalues of the companion matrix.