Bounds of the correlation between 2 random variables given their correlations with a third variable?

Question

Suppose we have corr(X,Y) = p1 and corr(Y,Z) = p2, how do we determine the range of possible values for corr(X,Z)?

Answer 1

Letting $\rho=\operatorname{corr}(X,Z), p_1=\operatorname{corr}(X,Y), p_2=\operatorname{corr}(Y,Z) $, the matrix $$M=\pmatrix{1&p_1&\rho\\p_1&1&p_2\\\rho&p_2&1}$$ must be positive semidefinite, and conversely, if $M$ is psd, there exist random variables $X,Y,Z$ for which $M$ is the correlation matrix. So your answer is: for given $p_1,p_2$, the set of possible correlation coefficiets $\rho$ is the set of numbers that make $M$ psd.

Answer 2

I don't think there's any restrictions on what $\operatorname{corr}(X, Z)$ could be, without further information. For any $c \in \mathbb{R}$, let $A, B$ be two random variables which are uncorrelated with $Y$ and for which $\operatorname{corr}(A, B) = c$.

Then taking $X = A + p_1 Y/\operatorname{Var}(Y)$, $Z = B + p_2 Y/\operatorname{Var}(Y)$ gives:

• $\operatorname{corr}(X, Y) = p_1$
• $\operatorname{corr}(Y, Z) = p_2$
• $\operatorname{corr}(X, Z) = c + \frac{p_1 p_2}{\operatorname{Var}(Y)},$

and since $c$ was arbitrary, so is $\operatorname{corr}(X, Z)$.

Answer 3

The covariance is $$ \operatorname E((X-\operatorname EX)(Y-\operatorname EY)). $$ This satisfies the definition of an inner product on the vector space of random variables with finite variance. And the inner product of two vectors is the product of the norms times the cosine of the angle between them. Thus the cosine is $$ \frac{\operatorname E((X-\operatorname EX)(Y-\operatorname EY))}{\operatorname{sd}(X)\operatorname{sd}(Y)} $$ i.e. it is the correlation.

So suppose $\theta_1= \arccos(\operatorname{corr}(X,Y))$ and $\theta_2 = \arccos(\operatorname{corr}(Y,Z)).$ If the angle between $X$ and $Y$ is $\theta_1$ and that between $Y$ and $Z$ is $\theta_2$ then the angle between $X$ and $Z$ cannot exceed $\theta_1+\theta_2,$ so $$ \operatorname{corr}(X,Z) \le \cos(\theta_1+\theta_2). $$