Modeling Nonzero Equilibrium Spreads in Cointegrated Systems
Summary
The document addresses how to represent a persistent, nonzero equilibrium spread between prices that are cointegrated, such as the same instrument traded in different markets. In a bivariate vector error-correction model, the proposed approach includes a constant in the cointegrating relationship so that the equilibrium error is centered around zero. The constant can be estimated from the data rather than removing it by manually shifting one series.
The same construction extends to multiple instruments by including a constant for each pairwise cointegrating relationship. The answer notes that this is a standard modeling choice and points to Johansen-style procedures that allow cointegrating relations with or without constants and trends. The discussion is brief and does not cover how to select among deterministic-term specifications, test the estimated relations, or assess whether the sample mean is stable out of sample.
Key ideas
- A nonzero long-run spread can be represented by a constant in the cointegrating relationship.
- In a bivariate model, the constant centers the equilibrium error around zero.
- For multiple instruments, pairwise cointegrating relationships can each include a constant.
- VECM estimation procedures may support alternatives with constants or trends.
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Full text
# Cointegrated time-series with a persistent spread
# 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.
## Answer by Richard Hardy (score 1, accepted)
https://quant.stackexchange.com/a/69029
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.