I have a number of different graphs of causality relationships amongst equities (one measure I use is granger based). I was interested in knowing to what extent if A ↔ B ↔ C implies a weak A ↔ C, where am using ” ↔” to indicate granger causes in either or both directions.

In most cases the spearman correlations of A ↔ B are high, so thought as a very rough analog to observe the falloff in correlation as one progresses farther along the relationship path. So if one has a graph of “directs” from A with associated correlations to A, what are the correlations of the directs directs back to A, and so on:

**A**

ρ(A,A) = 1
**A ↔ {B,C,D}**

ρ(A, {B,C,D}) = 0.84, 0.76, 0.90
**B ↔{E,F}, C↔G, D↔{H,I,J}**

ρ(A, {E,F,G,H,I,J}) = 0.62, 0.54, 0.59, …

Well, it turns out that the correlation falls off dramatically. This does not prove anything directly about causality, but seems to imply that transitivity is limited or near non-existant or at least falls off very quickly.

Here is an analysis done for 2 different equities:

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