12.5 Error correction models
A static cointegrating regression describes the long run. The short-run dynamics, how \(Y\) moves when it is away from equilibrium, are described by an error correction model.
Theorem 12.5.1. The autoregressive distributed lag model \[ Y_t = \alpha + \phi Y_{t-1} + \beta _0X_t + \beta _1X_{t-1} + \varepsilon _t, \qquad |\phi | < 1, \] is identical to the error correction model \[ \Delta Y_t = \beta _0\Delta X_t - (1 - \phi )\bigl (Y_{t-1} - \mu - \theta X_{t-1}\bigr ) + \varepsilon _t, \qquad \mu = \frac {\alpha }{1 - \phi }, \quad \theta = \frac {\beta _0 + \beta _1}{1 - \phi } . \] In a static equilibrium with \(Y_t = Y^*\) and \(X_t = X^*\) for all \(t\) and \(\varepsilon _t = 0\), \(Y^* = \mu + \theta X^*\). The common factor restriction of Theorem 11.4.2, \(\beta _1 = -\phi \beta _0\), holds if and only if \(\theta = \beta _0\).
Proof. Subtract \(Y_{t-1}\) from both sides and add and subtract \(\beta _0X_{t-1}\): \[ \Delta Y_t = \alpha - (1 - \phi )Y_{t-1} + \beta _0\Delta X_t + (\beta _0 + \beta _1)X_{t-1} + \varepsilon _t , \] and factor \(-(1 - \phi )\) out of the terms in \(Y_{t-1}\), \(X_{t-1}\) and the constant. Setting \(Y_t = Y_{t-1} = Y^*\) and \(X_t = X_{t-1} = X^*\) in the original equation gives \((1 - \phi )Y^* = \alpha + (\beta _0 + \beta _1)X^*\). Finally \(\theta = \beta _0\) means \(\beta _0 + \beta _1 = \beta _0(1 - \phi )\), that is \(\beta _1 = -\phi \beta _0\). □
The model has a clear interpretation. \(\beta _0\) is the impact effect of a change in \(X\); \(\theta \) is the long-run effect; and \(1 - \phi \) is the speed of adjustment, the fraction of last period’s disequilibrium \(Y_{t-1} - \mu - \theta X_{t-1}\) removed in this period. The half-life of a disequilibrium is \(\log 0.5/\log \phi \). The last part of the theorem shows what the AR(1) error model of the previous section assumes: that the long-run effect equals the impact effect, so that there is no adjustment to speak of. That is rarely plausible, and it is a further reason to prefer the general dynamic model.
When \(Y\) and \(X\) are \(I(1)\) and cointegrated, every term of the error correction model is \(I(0)\): \(\Delta Y_t\), \(\Delta X_t\), and the lagged disequilibrium. The regression is balanced, and its coefficients have standard distributions. The converse is the Granger representation theorem (Engle and Granger 1987): if two \(I(1)\) series are cointegrated, they have an error correction representation in which at least one of them responds to the lagged disequilibrium. Cointegration and error correction are two descriptions of the same thing.
Two-step estimation
Engle and Granger’s second step replaces the unknown disequilibrium by the lagged residual from the cointegrating regression: \[ \Delta Y_t = c + \beta _0\Delta X_t - \lambda \hat v_{t-1} + \text {lags of }\Delta Y_t, \Delta X_t + \varepsilon _t . \] Because \(\hat \theta \) is superconsistent, the estimation error in \(\hat v_{t-1}\) is negligible, and the coefficients of this regression have the usual asymptotic normal distributions. A significantly negative \(\lambda \) is itself evidence of cointegration. If \(\Delta Y_t\) and \(\Delta X_t\) have means, the intercept \(c\) is not zero: averaging the equation, \(E(\Delta Y) = c + \beta _0E(\Delta X)\), while in the long run \(E(\Delta Y) = \theta E(\Delta X)\), so \(c = (\theta - \beta _0)E(\Delta X)\).
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