# 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). Oct 19 '20 at 3:03
• 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 Oct 19 '20 at 9:18
• Question is here: stats.stackexchange.com/questions/493070/… Oct 21 '20 at 19:48
• Because I haven't given up on this, I have also posted the question here: quant.stackexchange.com/questions/59685/… Dec 1 '20 at 16:39
• But then why dont you guys post on Math Stackexchange?
– Tom
Feb 4 at 17:26