Interpreting Cointegration Coefficients on Log Prices
Summary
This note asks how to interpret cointegration vectors estimated from log prices, especially when coefficients are not equal and opposite. Exponentiating a relation such as one log price minus a multiple of another yields a stationary relationship involving powers of the underlying prices. Such a relation can look unintuitive when translated into everyday price ratios, and the coefficient vector does not automatically specify a simple share or unit hedge ratio for a pairs trade.
The response suggests that a fitted relationship may be weakly identified when the observed price range is narrow: over a limited range, competing functional relationships can fit similarly, and an anomalous observation can affect which relation appears plausible. It recommends examining more varied data and the underlying economic drivers before assigning a real-world interpretation. This is a conceptual caution rather than a method for converting any estimated vector into trade quantities; the note provides no trading test or evidence that a particular cointegrating relationship is stable.
Key ideas
- Cointegration vectors on log prices translate into multiplicative relationships among prices, including powers.
- A log-price coefficient vector does not by itself give a simple units-based hedge ratio.
- A narrow observed price range can make different functional relationships difficult to distinguish.
- Interpretation of a cointegrating relation should consider data coverage and underlying economic drivers.
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Full text
# How to interpret the physical meaning of cointegration vectors of log prices in real world
# How to interpret the physical meaning of cointegration vectors of log prices in real world
I'm trying to understand the physical meaning of cointegration vectors of log prices in the real world.
For example, if I have two assets $A$ and $B$, and Johansen test gives us a cointegration coefficient of [1 -1] for their log prices:$$log(A)-log(B)=\eta$$ When we convert this relationship back to real prices to understand its physical meaning, we have $$\frac{A}{B}=e^\eta$$ This makes sense because a fixed ratio between two real assets is intuitive.
However, for cointegration coefficient other than [1 -1], say [1 -2]: $$log(A)-2log(B)=\eta$$ when we convert it back to real price, we will have a relationship of $$\frac{A}{B^2} = e^\eta$$ This is nonintuitive to understand. How is this relationship possible between two assets in the real world like this: An asset's price is quadratic to another asset's price?
It is even worse if more assets are involved in the cointegration, giving relationships like $(A^2 B^3)/(C^4 D^{1.5}) = \eta$ in the real price.
In other word, for example, in a pairs trading, if the cointegration vector is found to be [1 -2] of the log prices, what ratio should I trade for the real price? It should not be [1 -2].
I understand how cointegration works with log prices, but I'm struggling to grasp its real-world meaning. Could someone shed some light on this?
## Answer by Arshdeep (score 0)
https://quant.stackexchange.com/a/80253
One possibility is you have not explored a good amount of data, and in a small region $sqrt(A)$ can approximate $A$, and if there is an anomaly in the data, $sqrt(A)$ can very well be a candidate. You will probably observe that the likelihood function is very flat in this case, meaning the data does not imply a massive confidence on the result that you have, and it is worthwhile revisiting data.
For example, say the relationship is $B=2A$. Now if A is in [10000,10050], you can say $B=200*sqrt(A)$ and really the data will not be able to say one is much better than the other. If I add an anomaly here which is (10025,20025), it will be more difficult to determine the relationship.
It is very much likely that the result is due to range of A not explored thoroughly. If you are keen that the true relationship is $B=sqrt(A)$, then it is not possible to explain without looking at the underlying factors that govern $A$ and $B$.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.