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I am looking for a rigorous and general treatment of cointegration. Unfortunately, many of the econometrics textbooks and papers I have found in this area either place a lot of restrictions on the timeseries involved or tend to be sloppy (assuming the sum of stationary timeseries is stationary, for example).

Here is one question I am interested in, for example. Consider a set of timeseries $x^i_t$ such that, for each $i$, $\Delta x^i_t$ is strictly stationary. I do not assume, however, that the differenced timeseries are jointly stationary. Assemble these timeseries into a vector $\mathbf{x}_t$ and define a cointegration vector to be any vector $\mathbf{v}$ such that $$ \mathbf{v}^{\top} \mathbf{x}_t $$ is strictly stationary.

The main question is, is the set of cointegration vectors closed under addition? If not, what is the minimal set of conditions one needs to ensure that this is true? Also, what is the situation when one considers weakly stationary timeseries rather than strictly stationary ones?

N.B. This question is based on the following question at mathoverflow, which has, as yet, only received my very non-expert interest.

N.B. 2 I have also crossposted this question here on quantitative finance.

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