As we know from the questions: 1.) is-there-a-difference-between-controlling-for-and-ignoring-other-variables and 2.) regression-with-multiple-variables-pairwise, there is a difference between ignoring multiple variables in linear regression, and controlling for them all simultaneously.
As a follow-up, are there algorithms which makes it possible to control for multiple variables simultaneously by first controlling for each variable individually, and then doing some post processing on the pairwise controlled regression coefficients? For example, if we have $x$ and $z_1, z_2$, then I would envision the algorithm taking in inputs $r_{x,z_1}, r_{x,z_2}$ and $r_{z1,z2}$ and then does something to condition $x$ on both $z_1$ and $z_2$ simultaneously with the pairwise correlation coefficients $r_{x,z_1}, r_{x,z_2}$ and $r_{z1,z2}$? For clarity, $r_{a,b}$ is the correlation coefficient between $a$ and $b$.
If such an algorithm exists, can someone post references and/or explain it? Does this algorithm work in the case where $|\mathbf{z}| > 2$?
