12.4 Cointegration
Definition 12.4.1. Two \(I(1)\) series \(Y_t\) and \(X_t\) are cointegrated if there is a \(\theta \) such that \(Y_t - \theta X_t\) is \(I(0)\). The vector \((1, -\theta )\) is the cointegrating vector, and \(Y_t = \theta X_t\) is interpreted as a long-run equilibrium from which the deviations \(Y_t - \theta X_t\) are temporary.
Consumption and income, prices in two markets linked by arbitrage, spot and futures prices, and money and nominal income are standard examples. Each series wanders, but they wander together, because the same stochastic trend drives both. If \(Y_t\) and \(X_t\) are cointegrated, the regression of \(Y_t\) on \(X_t\) is not spurious, and least squares estimates the cointegrating vector remarkably well.
Theorem 12.4.2 (Superconsistency; Stock 1987). Let \(X_t\) be \(I(1)\) and \(Y_t = \alpha + \theta X_t + v_t\) with \(v_t\) stationary, possibly correlated with \(\Delta X_t\). The least squares estimate \(\hat \theta \) satisfies \(n(\hat \theta - \theta ) = O_p(1)\): it converges to \(\theta \) at rate \(n\).
Sketch of proof. \(\hat \theta - \theta = \sum (X_t - \bar X)v_t/\sum (X_t - \bar X)^2\). As in Lemma 12.2.3, the denominator is of order \(n^2\). The numerator is a sum of products of an \(I(1)\) and an \(I(0)\) series, which is of order \(n\), even when the two are correlated, because the correlation contributes a term of order \(n\) and the rest is of the same order as in Lemma 12.2.3(i). The ratio is of order \(1/n\). □
The theorem is the reason why “endogeneity” in the cointegrating regression does not cause inconsistency: the signal in \(X_t\) grows so fast that correlation between \(X_t\) and \(v_t\) is swamped. But in small samples the bias can be substantial, and the limiting distribution of \(\hat \theta \) is non-normal when \(v_t\) and \(\Delta X_t\) are correlated, so the usual \(t\) ratios on \(\hat \theta \) are not valid. Modified estimators (dynamic OLS, which adds leads and lags of \(\Delta X_t\), and fully modified OLS) restore standard inference.
The Engle–Granger test
Engle and Granger (1987) proposed a two-step procedure. First, estimate the cointegrating regression \(Y_t = \alpha + \theta X_t + v_t\) by least squares. Second, apply the ADF test, in the “none” form, to its residuals \(\hat v_t\). The null hypothesis is no cointegration, that \(\hat v_t\) has a unit root.
The Dickey–Fuller critical values cannot be used. Least squares chooses \(\hat \theta \) to minimise \(\sum \hat v_t^2\), which makes the residuals look as stationary as possible even when no cointegrating vector exists, and the test statistic is pushed to the left. The asymptotic critical values depend on the number \(N\) of variables in the cointegrating regression; with a constant they are (MacKinnon 1991):
| \(N\) | \(1\%\) | \(5\%\) | \(10\%\) |
| \(2\) | \(-3.90\) | \(-3.34\) | \(-3.04\) |
| \(3\) | \(-4.29\) | \(-3.74\) | \(-3.45\) |
| \(4\) | \(-4.64\) | \(-4.10\) | \(-3.81\) |
A simulation of \(20\,000\) regressions between independent random walks of length \(500\) gives \(5\%\) points of \(-3.34\), \(-3.77\) and \(-4.13\), in close agreement. With more than two variables there may be more than one cointegrating vector, and the single-equation approach cannot find them all; the system approach of Johansen, based on vector autoregressions, is described in the section on simultaneous equations and VARs.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.