This is probably an elementary question. Let's see.
Say variable $y_t$ depends on two other processes, as follows:
$$y_t=x_tz_t$$
I observe these variables in period 0 and period 1. I want to evaluate how much did $x$ and $z$ contribute to the change in $y$. For any variable $w$, we can define:
$$ \frac{w_1}{w_0} = 1+g_w$$
where $g_w$ is the growth rate of $w$ over the period. Using the above, it is trivial to show that:
$$ g_y = g_x + g_z + g_xg_z $$
This however is not entirely helpful, because the contribution of $x$ and $z$ to $y$ cannot be disentangled. I fear there is simply no way around this. Is there?
I have seen decompositions of trend, cycle and seasonality, but in my case $x$ and $z$ do not represent such sharp differentiations. Any ideas? Can we ever say something like C% of the change in $y$ is due to $x$? If not, what is the best alternative?
Notice that in the economic literature there is something called the "growth accounting" method, whereby the components of economic growth are decomposed into factors like labour, capital and technological change, based on a multiplicative production function. This uses the same decomposition as above, but under the assumption that multiplicative growth rate terms are very small (and thus it works only when components grow at a small rate). Thus, this method uses:
$$g_y \approx g_x + g_z $$
This is surely sufficient in settings of low change, but on other ones it is not.