Thursday, May 31, 2018

The Uniqueness of the Cointegrating Vector

Suppose that we have (only) two non-stationary time-series, X1t and X2t (t = 1, 2, 3, .....). More specifically, suppose that both of these series are integrated of order one (i.e., I(1)). Then there are two possibilities - either X1 and X2 are cointegrated, or they aren't.

You'll recall that if they are cointegrated, then there is a linear combination of X1 and X2 that is stationary. Let's write this linear combination as Zt = (X1t + αX2t). (We can normalize the first "weight" to the value "one" without any loss of generality.) The vector whose elements are 1 and α is the so-called "cointegrating vector".

You may be aware that if such a vector exists, then it is unique.

Recently, I was asked for a simple proof of this uniqueness. Here goes.........