# Co-integration

Let an \(I(d)\) series be an integrated series of order \(d\). For most financial applications, an \(I(0)\) series is weak-sense stationary, i.e. finite and time-invariant mean and variance, with covariance depending only on the time lag.

Two series \(x(t), y(t)\) are co-integrated if they are both \(I(1)\) series such that \(\exists \beta : z(t)=x(t)-\beta y(t)\) is an \(I(0)\) series.

These are useful for mean-reversion trading strategies that depend on the time invariance of the mean of \(I(0)\) series, for example the minimum profit maximization strategy.

## Properties

- Prices of co-integrated assets are tethered due to the stationarity of their spread.
- Co-integration is a measure of similarity of assets in terms of risk exposure profiles.
- Co-integration describes the long-term relationship between asset prices meanwhile correlation describes the short-term relationship between them.
- Co-integration specifies the ratio of long leg to short leg \(\beta\), also called the hedge ratio, which can be computed using the Engle-Granger test or Johansen test.