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Two or more non-stationary, integrated variables are cointegrated if there exists a linear combination of those variables which is integrated of a lower order, e.g. stationary.

Two or more non-stationary, integrated variables are cointegrated if there exists a linear combination of those variables which is integrated of a lower order, e.g. stationary. This implies that there is some equilibrium relationship between these variables.

Formally, for a $$k\times 1$$ vector of $$I(d)$$ variables $$x_t$$ with $$d=1,2,\dots$$, cointegration requires that there exists a vector $$\beta$$ such that $$\beta^\top x_t$$ is $$I(d')$$ with $$d'. For $$\beta^i$$, $$i = 1,...,r$$, $$x_t$$ is cointegrated with cointegrating rank $$r$$ and $$\beta^i$$s are called cointegrating vectors. A relevant special case is when $$x_t$$s are all $$I(1)$$ (e.g. random walks) and $$\beta^\top x_t$$ is $$I(0)$$.

Note that usually $$\beta$$ is normalized by restricting one of its elements by setting it equal to one. Also variables must be integrated of the same order, e.g. an $$I(1)$$ and an $$I(2)$$ variable cannot be cointegrated. However, between a set of three (or more) variables which are not integrated of the same order there may be a linear combination of the first two variables that cointegrates with the third.