Assume $X_t$ and $Y_t$ represent the prices of the same financial instrument traded in two different markets (in particular they are cointegrated). For some reason the long run equilibrium between $X$ and $Y$ is not zero but some constant dollar ammount, i.e. $X_\infty - Y_\infty = c$. This $c$ I can estimate by looking at the distribution of the differences $X_t - Y_t$ and taking the mean difference. However, I do not know the source of it.

What is the methodologically correct way of modelling the relationship in a VECM framework? Would I simply substract this constant from one of the $X_t$ or $Y_t$, or add deterministic terms to the VECM model?

I am also interested in a more general situation where there are $N$ instruments representing the same asset each having its own equilibrium spread $c_{ij} = X_i - X_j$ at infinity.


1 Answer 1


In the bivariate case, define the cointegrating relationship as $c+Y_t-X_t$ such that the mean of it is zero and then estimate $c$ from the data. Similarly, in the multivariate case, define the cointegrating relationships as $c_{ij}+X_{i,t}-X_{j,t}$ for different pairs $(i,j)$.

This is fairly standard in general, though not necessarily in the context you are looking at. E.g. the ca.jo function in the urca package in R considers several types of cointegrating relationships, including ones with or without a constant and/or a trend.

  • $\begingroup$ The problem with python implementation of VECM from statmodels one could only add one common deterministic factor, it will estimate the sum of all the $c_{ij}$. $\endgroup$
    – vkrouglov
    Dec 10, 2021 at 16:12
  • $\begingroup$ @vkrouglov, thank you for the note. $\endgroup$ Dec 10, 2021 at 16:29

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