The question is probbaly easy, but I don't want to fool myself.

If there are three (univariate real) time series $A$, $B$ and $C$ such that $A$ and $B$ both highly correlated with $C$ at some arbitrary lags, does it implies that $A$ and $B$ also highly (near as high) correlated at some lag?

I presume it does, because I can't imagin a counter example.


It seems to me I've found a proof.

By talking "correlation" I refer to Pearson product-moment correlation coefficient. It's possible to express the correlation coefficient as follows (see Geometric interpretation):

$$\rho_{A, B}=\frac{A \cdot B}{\|A\| \|B\|}$$

Which is the cosine of the angle $\theta_{A, B}$ between the (high-dimensional) vectors $A$ and $B$. Note that $\rho_{A, B}=1$ for $\theta_{A, B}=0$ which means collinearity.

Now I refer to the transitivity of collinearity: if $A\|C$ and $B\|C$ then $A\|B$. That's enough for $\rho_{A, C}=\rho_{B, C}=1$.

It's even possible to estimate $\rho_{A, B}$ from $\rho_{A, C}$ and $\rho_{B, C}$:

$$\rho_{A, B} \geq cos(\theta_{A, C}+\theta_{B, C})$$


$$\rho_{A, B} \geq cos(arccos(\rho_{A, C})+arccos(\rho_{B, C}))$$

Where from I can say that if $A$ and $B$ both highly correlated with $C$ then $A$ and $B$ are as well near as high correlated.

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