Cointegration

Remarks

The cointegrating vectors, together with $0$, form a linear subspace of $\R^k$; its dimension $r$, with $0 \le r \le k - 1$, is the cointegrating rank. More generally, components that are I($d$) are cointegrated of order ($d$, $b$) when some nonzero combination is I($d - b$) with $b > 0$.

Example: let $w_t$ be a random walk and $e_t$ a white noise independent of it, and set $x_{1,t} = w_t$ and $x_{2,t} = 2 w_t + e_t$. Both components are I($1$), and $\beta = (2, -1)^T$ gives $\beta^T x_t = -e_t$, which is stationary, so the pair is cointegrated.

Non-example: two independent random walks $w_t$ and $v_t$ with step variance $\sigma^2 > 0$ are not cointegrated. For $\beta = (a, b)^T \ne 0$, the combination $a w_t + b v_t$ is again a random walk with step variance $(a^2 + b^2) \sigma^2 > 0$, which is not stationary.

In the Engle–Granger method, $\beta$ is estimated by regressing one component on the others. Then a Dickey–Fuller type test is applied to the residuals, with critical values adjusted for the estimation step. The Granger representation theorem links cointegration with error-correction models.