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I am trying to compare the following two terms and determine which one is larger than the other.

Term 1: ${\operatorname{cov}(X,Y)} \over {\operatorname{var}(Y)}$

Term 2: ${\operatorname{cov}(\operatorname{cov}(X,Y), Y)} \over {(\operatorname{var}(Y))^2}$

If x and y are uncorrelated, then $\operatorname{cov}(X,Y) = 0$, so both of these terms would be $0$. What would happen if they are positively correlated? I think I'm missing some information on the properties of covariances that is preventing me from manipulating the second term.

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    $\begingroup$ With respect to Term 2: the covariance of $x$ and $y$ is a number, e.g., 4.25, and as such, has no covariance with anything else. $\endgroup$
    – jbowman
    Feb 3, 2020 at 20:08
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    $\begingroup$ @jbowman Would that make it equal to 0? $\endgroup$
    – melbez
    Feb 3, 2020 at 20:12
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    $\begingroup$ Yes, so Term 2 is equal to 0 also. $\endgroup$
    – jbowman
    Feb 3, 2020 at 20:14
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    $\begingroup$ Could you please explain the second numerator? Assuming $X$ and $Y$ refer to random variables, then $\operatorname{Cov}(X,Y)$ is either a fixed number or matrix, whence its covariance with $Y$ cannot be anything other than zero. That strongly suggests you mean something else by your symbols, but what is it? $\endgroup$
    – whuber
    Feb 3, 2020 at 20:49
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    $\begingroup$ @whuber I meant exactly what I wrote. This was the result of a proof I'm working on. I didn't realize it would be 0. $\endgroup$
    – melbez
    Feb 3, 2020 at 21:30

1 Answer 1

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Statistics like $\operatorname{cov}(X,Y)$, $\operatorname{var}(X), E[X], E[Y]$ are all constants. Only when we use random variables on givens side, they become functions (in general) of RVs: e.g. $E[X|Y], \operatorname{var}(X|Y)$.

So, since $\operatorname{cov}(X,Y)=a$ is a constant, $\operatorname{cov}(\operatorname{cov}(X,Y), Y)=\operatorname{cov}(a,Y)=0$, which would make your second term $0$. Based on your positive/negative correlation, Term 1 is either greater or smaller than Term 2.

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