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

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