Hows does coefficient $b_1$ change when estimating $b_1 x_1+b_2 x_2+b_3 x_3$ instead of $b_1 x_1+b_2 x_2$ My question is related to [1], [2] and [3].
Assume we estimate a multiple regression,
$$ y = a + b_1x_1 + b_2x_2 + u $$
and are mainly interested in the value of $\hat{b}_1$ (lets denote this specific estimate $\hat{b}_{1; \text{model 1}}$).
If we run a different model by including an additional independent variable $x_3$
$$ y = a + b_1x_1 + b_2x_2 + b_3x_3 + u $$
we will observe a different estimate $\hat{b}_1$ (denoted as $\hat{b}_{1; \text{model 2}}$), because the answer in [1] states that

A parameter estimate in a regression model will change if a variable is added to the model that is:

*

*correlated with that parameter's corresponding variable (which was already in the model), and

*correlated with the response variable


Question:
Does there exist a closed formula for the change in the estimated coefficient $\hat{b}_1$ when including additional independent variables?

Edit:
Assume we just include one additional indep. variable $x_3$ where all observations are known. Of course, one could run both regressions in that case, but does there exist a way to directly calculate the change in the estimated $\hat{b}_1$?

We can find both $\hat{b}_{1, \text{model } 1}$ and $\hat{b}_{1, \text{model } 2}$ are closed form (take the appropriate element of $(X^T X)^{-1}X y$) thus the difference between them is also available in closed form.
I do not think we can, in general, offer a simpler or more elegant solution than computing their differences because matrix inversion is a complex operation. Nonetheless, it is a closed form solution.
