Existence of a common term if time series are pairwise cointegrated

Question

Let's suppose that we have $n$ time series that are integrated of order one:

$y_t^i\sim I(1)$ for $i=1, 2, \dots, n$

The difference between any two series is stationary:

$y_t^i-y_t^j\sim I(0)$ for $i,j=1, 2, \dots, n$

I can express each time series as the sum of a $I(1)$ common term (which is shared by all of the time series) and an $I(0)$ stationary process (which can be unique to each series):

$y_t^1=y_t^\text{common} + y_t^\text{1, stationary}$

$y_t^2=y_t^\text{common} + y_t^\text{2, stationary}$

$y_t^3=y_t^\text{common} + y_t^\text{3, stationary}$

etc.

Does a common term always exist if time series are pairwise cointegrated? Is there a simple proof for that?

Answer 1

In general you can say that in a system with $r$ cointegrating relationships, there are $n-r$ random walks (or common trends, as you prefer) driving the non-stationary part of the system ($n$ being the dimension of the vector you are assessing). This result comes from Stock and Watson common trends representation.