Bounding Pairwise Correlation with Positive Semidefiniteness
Summary
The document presents a mathematical way to screen stock pairs before computing every pairwise correlation. For three variables, their correlation matrix must be positive semidefinite, meaning its eigenvalues cannot be negative. That constraint links the three pairwise correlations and yields a lower bound on one correlation when the other two are known and positive.
This provides a potential search filter: pairs whose correlations would violate the bound can be ruled out without calculating that correlation. The answer gives a formula derived from the matrix condition, but the wording around which pairwise correlations are known and which correlation is bounded is inconsistent, so the variable labels should be checked before applying it. The document does not discuss sample-estimation error, statistical uncertainty, or alternative computational screening methods.
Key ideas
- A valid three-variable correlation matrix must be positive semidefinite.
- The matrix constraint places bounds on one pairwise correlation given the other two.
- Such a bound can help eliminate impossible low-correlation candidates before calculation.
- The formula’s variable labels are ambiguous and should be verified before use.
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# How to quickly estimate a lower bound on correlation for a large number of stocks?
# How to quickly estimate a lower bound on correlation for a large number of stocks?
I would like to find stock pairs that exhibit low correlation. If the correlation between A and B is 0.9 and the correlation between A and C is 0.9 is there a minimum possible correlation for B and C? I'd like to save on search time so if I know that it is mathematically impossible for B and C to have a correlation below some arbitrary level based on A to B and A to C's correlations I obviously wouldn't have to waste time calculating the correlation of B and C.
Is there such a "law"? If not, what are other methods of decreasing the search time?
## Answer by Brian B (score 26, accepted)
https://quant.stackexchange.com/a/531
Yes, there is such a rule and it is not too hard to grasp. Consider the 3-element correlation matrix
$$\left(\begin{matrix} 1 & r & \rho \\ r & 1 & c \\ \rho & c & 1 \end{matrix}\right)$$
which must be positive semidefinite. In simpler terms, that means all its eigenvalues must be nonnegative.
Assuming that $\rho$ and $r$ are known positive values, we find that the eigenvalues of this matrix go negative when
\begin{equation} c<\rho r-\sqrt{1-\rho ^2+\rho ^2 r^2-r^2}. \end{equation}
Therefore the right hand side of this expression is the lower bound for the AC correlation $c$ that you seek, with $\rho$ being the AB correlation and $r$ being the BC correlation.
## Answer by David B (score -2)
https://quant.stackexchange.com/a/3579
The upper bound on BC correlation would be 1 for the example given. B=C would correlate to 1. If AB and Ac are different, I don't know off the top of my head.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.