0.1 Cointegration
Individually non-stationary series can share a common stochastic trend, so that some linear combination of them is stationary even though none of the series is. This is the phenomenon of cointegration, central to modelling long-run equilibrium relationships (e.g. spot vs. futures prices, exchange rates and price levels).
Series \(X_t, Y_t\), each \(I(1)\) (i.e. \(\nabla X_t, \nabla Y_t\) stationary — see Unit Root Tests), are cointegrated if there exists \(\beta \neq 0\) such that \[ u_t = Y_t - \beta X_t \] is stationary, \(I(0)\). More generally, an \(n\)-vector \(I(1)\) process is cointegrated with cointegrating rank \(r\) if there are \(r\) linearly independent vectors \(\beta_1,\dots,\beta_r\) such that \(\beta_i'X_t\) is stationary.
Engle-Granger two-step method. 1. Regress \(Y_t\) on \(X_t\) by OLS to estimate \(\hat\beta\); form the residual \(\hat u_t = Y_t - \hat\beta X_t\). 2. Test \(\hat u_t\) for a unit root using the (Augmented) Dickey-Fuller test with cointegration-adjusted critical values; rejection implies cointegration.
If \(X_t, Y_t\) are cointegrated, their dynamics admit a Vector Error Correction Model (VECM) representation \[ \nabla \mathbf{Z}_t = \alpha\,\beta'\mathbf{Z}_{t-1} + \sum_{j=1}^{k-1}\Gamma_j \nabla\mathbf{Z}_{t-j} + \varepsilon_t, \qquad \mathbf{Z}_t = (X_t,Y_t)', \] where \(\beta'\mathbf{Z}_{t-1}\) is the (lagged) equilibrium error and \(\alpha\) the speed-of-adjustment coefficients pulling the system back toward equilibrium — this links cointegration directly to VAR modelling.
Johansen procedure. For \(n>2\) series, the Engle-Granger method is asymmetric and limited to rank 1; the Johansen test uses a reduced-rank regression (canonical correlation) on a VAR in levels to jointly estimate the cointegrating rank \(r\) and the space spanned by \(\beta\).