Cointegration – time series with a common trend

The concept of an integrated multivariate series is complicated by the fact that all the component series of the process may be individually integrated but the process is not jointly integrated in the sense that one or more linear combinations of the series exist that produce a new stationary series. 

In other words, a combination of two co-integrated series has a stable mean to which this linear combination reverts. A multivariate series with this characteristic is said to be co-integrated. This also applies when the individual series are integrated of a higher order and the linear combination reduces the overall order of integration. 

cointegration is different from correlation: two series can be highly correlated but need not be co-integrated. For example, if two growing series are constant multiples of each other, their correlation will be high but any linear combination will also grow rather than revert to the mean.

The VAR analysis can still be applied to integrated processes using the error-correction form of a VAR model that uses the first differences of the individual series plus an error correction term in levels.

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