# Cointegration

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 cointegrated 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 cointegrated assets are tethered due to the stationarity of their spread.
• Cointegration is a measure of similarity of assets in terms of risk exposure profiles.
• Cointegration describes the long-term relationship between asset prices meanwhile correlation describes the short-term relationship between them.
• Cointegratio 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.

## Thoughts

Created: 2022-03-13 Sun 21:45

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