I have a single output variable $y$ and a number of inputs $x_1$, $x_2$, etc. These are time series. Each $x_i$ explains the changes in $y$ in specific circumstances, and the goal is to have a linear model that looks like $y=b_1x_1+b_2x_2+...$ The point is that each $x_i$ must be sampled on different clock $c_i$. Imagine that $y$ is continuous, but each $x_i$ is somewhat discrete, most of the time is 0, an occasionally it has an impulse, i.e. 'ticks'. Then, it decays quickly back to steady state of 0. We want to sample $x_i$ only then, when it "jumps", and decide the influence of these impulses of separate $x_i$s individually.
My approach is to take one sampling method $s_1$, sample $y$ and $x1$ on it, specific to $x_1$. Do the regression $y$ on $x_1$. Then, take the error $y-b_1x_1$, sample the error on another sampler $s2$, and regress this on $x_2$.
I can decide to sample $x$s and $y$ whenever I want, I can examine their value at any point in time, and decide whether to sample based on that. Most of time the $x$s will be 0, so I will choose to sample $x_i$ when it's actually trying to tell me something about $y$.
What's a good approach to solve this problem? Is there a regression form specific to this sort of situations?
Due to practical constraints, I want my model to include all $x$s at the same time, rather than switch between different models on the fly.