# Splitting up the variance of Z for Z = X*Y

$Z$ is a function of two dependent random variables, e.g. $X \cdot Y$. Here it is shown that

$$var(Z) = var(XY)=(cov(X^2,Y^2)+E[X^2]E[Y^2])-(cov(X,Y)+E[X]E[Y])^2$$

I am interested in a metric that tells me what percentage of the the variance of $Z$ is caused by $X$ and how much is caused by $Y$, e.g. 68% by $X$ and 32% by $Y$. An intuitive approach like

$$var(X)/var(Z)$$

does not make sense because it can be the case that $var(X)>var(Z)$. Is there a standard solution for this problem? I assume that for $Z = X+Y$ it is similar to the question of a portfolio of stocks and someone wants to know in which way the stocks contribute to the variance of the portfolio. I could not find anything and would be interested in a general formulation for arbitrary functions ($X \cdot Y$,$X-Y$,$X/Y$ etc).

• (+1) Interesting question but... How do you defined "caused by"? The random variables $X$ and $Y$ both vary and possibly impact one another. Commented Apr 16, 2016 at 11:54
• You can also take a look at the decomposition$$\text{var}(XY)=\mathbb{E}[Y^2\text{var}(X|Y)]+\text{var}(Y \mathbb{E} [X|Y])$$that separates the conditional variance of $X$ given $Y$ and the part due to $Y$ Commented Apr 16, 2016 at 11:59
• Thanks a lot for your comment. Based on it, I did some research on variance decomposition. By "caused by" I mean that the measure should include the effect of X on Y. As an example, when Z = X+Y the proportion of X is typically measured as (VAR(X)+COV(X,Y))/var(Z) . I do not see how this effect can be measured for a product. If the first term of your formula only represents the effect of X on Z then I could it divide it by var(Z) analogously. Commented Apr 16, 2016 at 13:41
• If the variables are independent then the variance of the product is $var(X) \cdot var(Y) + var(X) \cdot \mu_Y^2 + var(Y) \cdot \mu_2^X$. The procedure mentioned in my comment above would yield $var(Y) \cdot \mu_2^X / var(Z)$ as the percentage of $Y$. This seems to be wrong. Commented Apr 16, 2016 at 14:11