Linear regression - iterative approach 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 criteria specific to its qualities, and its relationship to $y$ to be established only at those situations when that particular $x_i$ is the lead influence on $y$.
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 $x_2$.
What's the name of this method, is there a regression of this sort? Does it sound like the right solution to my problem?
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.
 A: I'm posting as an answer since it's too long for a comment.
If I understand correctly, you have data coming from, let's say, $N$ multiple sensors $ x_i^{(t)} \in \mathbb{R}^{T \times N} $ where each sensor is a time series with $T$ data points. You also have a target output variable $y^{(t)} \in \mathbb{R}^T$. 
You would like to perform a prediction using a linear model and you would like to adjust the subsequence sample size (or sliding window size) for each $x_i$ sensor.
One option is to take a larger fixed size sliding window over your time series, $ X^{(t)}  \in \mathbb{R}^{W \times N}$ where $W$ is the window size.

Then, there are methods that perform regularization for all $x_i$ such as group lasso or group elastic net or ridge regression.
The subsequence for a sensor is one  group.
In effect, this adjusts the $ b_i^{(w)} $ for all $x_i^{(w)}$ 
giving a specific weight to each one of the subsequence points in your window, effectively adjusting the window size or sample size separately per $x_i$. 
