14.6 Cointegrated systems: the vector error correction model
Theorem 14.6.1. The VAR(\(k\)) in levels \(y_t = c + A_1y_{t-1} + \dots + A_ky_{t-k} + \varepsilon _t\) can be written as the vector error correction model \[ \Delta y_t = c + \Pi y_{t-1} + \sum _{j=1}^{k-1}\Gamma _j\Delta y_{t-j} + \varepsilon _t, \qquad \Pi = -\Bigl (I - \sum _{i=1}^kA_i\Bigr ), \qquad \Gamma _j = -\sum _{i=j+1}^kA_i . \] If each element of \(y_t\) is \(I(1)\), then
- (i)
- \(\rank \Pi = 0\) means that there is no cointegration, and the model is a VAR in differences;
- (ii)
- \(0 < \rank \Pi = r < G\) means that there are exactly \(r\) linearly independent cointegrating vectors, and \(\Pi = \alpha \beta '\) with \(\alpha \) and \(\beta \) of order \(G \times r\): the columns of \(\beta \) are the cointegrating vectors and \(\alpha \) holds the speeds of adjustment;
- (iii)
- \(\rank \Pi = G\) is impossible, since it would make \(y_t\) stationary.
Proof. The algebra is that of Theorem 12.3.1, with matrices in place of scalars. For (ii) and (iii): every term other than \(\Pi y_{t-1}\) in the error correction form is \(I(0)\), so \(\Pi y_{t-1}\) must be \(I(0)\) too. If \(\Pi = 0\) this is automatic. If \(\Pi \) has rank \(r\), write \(\Pi = \alpha \beta '\) with \(\alpha \) and \(\beta \) of full column rank \(r\); then \(\beta 'y_{t-1} = (\alpha '\alpha )^{-1}\alpha '\Pi y_{t-1}\) is \(I(0)\), so the \(r\) columns of \(\beta \) are cointegrating vectors. If \(r = G\), \(\Pi \) is invertible and \(y_{t-1} = \Pi ^{-1}(\Pi y_{t-1})\) would be \(I(0)\), contradicting the assumption. □
Johansen’s procedure
Johansen (1988) estimated the VECM by maximum likelihood with the rank of \(\Pi \) restricted. Let \(R_{0t}\) and \(R_{1t}\) be the residuals from regressing \(\Delta y_t\) and \(y_{t-1}\) on a constant and the lagged differences, and \(S_{ij} = \frac 1n\sum _tR_{it}R_{jt}'\). The problem reduces to \(R_{0t} = \alpha \beta 'R_{1t} + \varepsilon _t\), a regression with a coefficient matrix of reduced rank.
Theorem 14.6.2 (Johansen). Let \(\hat \lambda _1 > \dots > \hat \lambda _G\) be the eigenvalues of \(S_{11}^{-1}S_{10}S_{00}^{-1}S_{01}\), and \(\hat v_1, \dots , \hat v_G\) the corresponding eigenvectors. Under normality, the maximum likelihood estimate of \(\beta \) with rank \(r\) is \([\hat v_1 \cdots \hat v_r]\), and the maximised log-likelihood is \[ \ell _r = \text {const} - \frac n2\log |S_{00}| - \frac n2\sum _{i=1}^r\log (1 - \hat \lambda _i) . \] The likelihood ratio statistic for \(H_0\colon \rank \Pi \leq r\) against \(\rank \Pi = G\) is therefore the trace statistic \[ \text {trace}(r) = -n\sum _{i=r+1}^G\log (1 - \hat \lambda _i) . \]
Sketch of proof. For fixed \(\beta \), the regression of \(R_{0t}\) on \(\beta 'R_{1t}\) gives \(\hat \alpha (\beta ) = S_{01}\beta (\beta 'S_{11}\beta )^{-1}\) and residual covariance \(S_{00} - S_{01}\beta (\beta 'S_{11}\beta )^{-1}\beta 'S_{10}\), and the concentrated log-likelihood is \(-\frac n2\) times the log of its determinant. By the formula for the determinant of a partitioned matrix, \[ \bigl |S_{00} - S_{01}\beta (\beta 'S_{11}\beta )^{-1}\beta 'S_{10}\bigr | = |S_{00}|\,\frac {\bigl |\beta '(S_{11} - S_{10}S_{00}^{-1}S_{01})\beta \bigr |}{|\beta 'S_{11}\beta |} . \] Minimising the ratio over \(\beta \) is a generalised eigenvalue problem, solved by the eigenvectors of \(S_{11}^{-1}S_{10}S_{00}^{-1}S_{01}\) belonging to the \(r\) largest eigenvalues, and its minimum is \(\prod _{i\leq r}(1 - \hat \lambda _i)\). Differencing \(\ell _G\) and \(\ell _r\) gives the trace statistic. □
The \(\hat \lambda _i\) are the squared canonical correlations between \(\Delta y_t\) and \(y_{t-1}\) after the short-run dynamics have been removed: a large \(\hat \lambda _i\) means a combination of the levels that strongly predicts the changes, which is what a cointegrating relation does. The tests are applied in sequence, \(r = 0, 1, \dots \), and the first \(r\) not rejected is the estimated rank. The trace statistic has a non-standard distribution, of the Dickey–Fuller type; with an unrestricted constant, the \(5\%\) critical values of Osterwald-Lenum (1992) for \(G - r = 1, 2, 3, 4\) are \(3.76\), \(15.41\), \(29.68\) and \(47.21\). A simulation of \(5000\) pairs of independent random walks with drift, of length \(150\), gives \(3.30\) and \(15.74\) for the first two, close to these asymptotic values. If the \(\alpha \) coefficient of a variable is zero, that variable does not adjust to the disequilibrium: it is weakly exogenous for \(\beta \), and drives the system’s common trend.
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