# Cointegration of (multiple) time series

Some time ago I stumbled upon the notion of cointegration of time series (see the wiki for some basic fact).

Unfortunately, my knowledge of time series is a bit sketchy, and moreover I was able to find only some refs (mostly from economists) which are not general enough (anyone who can point me to some mathematically-savvy presentation would do me a great favor).

So, assume that I have a finite collection $$\{T_i(t)\}$$ of time series, each of them of order of integration 1. The first question is:

If $$\alpha T_i(t) + \beta T_j(t)$$ is stationary, are $$\alpha$$ and $$\beta$$ unique ? ( I suspect they are not, in which case, which additional constraints I have to add to ensure uniqueness? )

Now, suppose I look at three of them, each pairwise cointegrated. As stationarity is not in general kept under linear combinations (see here for an example), I do not think that the triple is multicointegrated, ie there is a linear combination with non-zero coefficients of the triplet which is a stationary series. Hence the next question:

which conditions should be added to the triplet so that pairwise cointegration implies cointegration of the entire triplet? More generally, what about cointegration of $$N +1$$ times series knowing that any subset of size $$N$$ is cointegrated?

Lastly, the converse of the last question:

When multi-cointegration of a N+1 time series implies multi-cointegration of any subset of $$N$$ elements?

Any help is greatly appreciated

• What do you mean by collective cointegration? Certainly, there are stationary time series that are linear combinations of all three (with one coefficient equal to zero). – Aaron Bergman yesterday
• Aaron, I have been sloppy. I mean that there exist a non trivial (no zero coefficients) linear combination of the three such that it is stationary. Edited as needed – Mirco A. Mannucci 23 hours ago
• For what it's worth, in the references I can find on cointegration, among other additional assumptions, the initial set of timeseries is considered to be $I(1)$ as a vector which means that the vector of first differences is stationary, so each component is jointly stationary with the rest. This assumption, I think, makes your questions straightforward. I do not know what happens when it is relaxed. – Aaron Bergman 12 hours ago
• Hayashi's textbook was the most careful presentation I could find -- I now share your frustration with the lack of good definitions in this literature (much less the sloppiness with the difference between stationary and weakly stationary in many textbooks). – Aaron Bergman 12 hours ago
• I think you need to provide some more context before you make the claim that "all your questions are straightforward". Yes, all times series here are assumed to be L(1), that is true, which means that each becomes stationary after a single derivation. But that simply means that each element in the vector is L(0), ie stationary, not that they are jointly stationary. Anyway, I hear you: I think that for a mathematician such confusions between weak stationarity and strong one are difficult to swallow... – Mirco A. Mannucci 11 hours ago