Cointegration Under Strictly Stationary Differences

Summary

The document raises a theoretical question about cointegration when each component time series has strictly stationary first differences, but the differenced series are not assumed to be jointly stationary. It defines a cointegration vector as a vector whose linear combination of the original series is strictly stationary, then asks whether the set of such vectors is closed under addition. It also asks what additional assumptions would guarantee closure and how the answer changes for weak stationarity.

No answer, proof, example, or empirical application is included. The contribution is the formulation of a boundary case for standard cointegration reasoning, especially the role of joint stationarity assumptions in properties of linear combinations. It is relevant to quantitative researchers working with time-series models, but it does not establish a result or give a procedure for testing cointegration. Any use in trading research would require resolving the stated mathematical questions and checking that the chosen stationarity definition fits the data and model.

Key ideas

• The question assumes each series has strictly stationary first differences but not joint stationarity across differences.
• Cointegration vectors are defined as coefficients producing a strictly stationary linear combination.
• The central issue is whether the set of cointegration vectors is closed under addition.
• The document asks how the answer differs under weak stationarity but supplies no solution.

Tags

# Cointegration where first differences are not jointly stationary


# Cointegration where first differences are not jointly stationary












Note: This is a crosspost from this post on cross-validated, where it did not receive an answer. I thought I might have better luck here.

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.