# Cointegrated time-series with a persistent spread

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.

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.
• 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 $c_{ij}$. Dec 10, 2021 at 16:12