Calculate relative contribution to percent change Let me use a simple example to illustrate my problem. First, assume we are calculating rate $r$ at time $t$ such that $r_t=\frac{x_t}{y_t}$. Furthermore, each measure has two component parts: $x = x_A +x_B$ and $y = y_A + y_B$. We can thus calculate percent change $c$ for the rate between $t_2>t_1$ as $c=\frac{r_2-r_1}{r_1}$.
Next, I want to allocate $c$ to measure the relative contribution of each component $A$ and $B$. When the changes are in the same direction between $t_1$ and $t_2$ this is easy (e.g. $x_{A_1} > x_{A_2}$ and $x_{B_1} > x_{B_2}$ and $y_{A_1} > y_{A_2}$ and $y_{B_1} > y_{B_2}$). You calculate the change for each component, divide that by the absolute change and apply that "share" to the total percent change. That allows me to make a statement, e.g. when the rate changed from $10\%$ to $15\%$, $75\%$ of the $50\%$ change was due to component $A$ and $25\%$ to component $B$.
Here's my question: how can I calculate the relative contribution of these components when the differences are in opposite directions? For example, component $A$ decreased for $x$ and $y$ (and more for $y$, relatively) and component $B$ increased for $x$ and $y$ (and more for $y$, relatively).
I'm sure this is simple but no amount of searching has made me the wiser. If you could point me in the right direction -- or ask questions to better illuminate my subject matter -- I would greatly appreciate your help. Thanks!
PS: I found a few resources, linked below but I'm still not sure of the exact math required....
http://www.bea.gov/papers/pdf/Moulton0603.pdf
http://www.esri.cao.go.jp/en/sna/sokuhou/kako/2007/qe074/kiyoe.pdf

this might be a bit old but I was looking for something similar and I found these two articles which may help someone with a similar question (or so I hope):
Contribution to Growth: growth here could be taken as a change rate
calculating contribution percent change: author proposes a couple of ways to determine how a variable contributes to a change rate when a kpi (like change rate) is additive or multiplicative
