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 $\hat{b}_1$?

By the Frisch–Waugh(–Lovell) theorem and the well-know formula $\left(X^{\top}X \right)^{-1}X^{\top}y$ for the OLS estimator we have
$$
\begin{align}
\hat{b}_{1; \text{model 1}}&=\left( \left(M_{2}x_1 \right)^{\top}M_{2}x_1 \right)^{-1} \left(M_{2}x_1 \right)^{\top}M_{2}y
=\left( x_1^{\top}M_{2}x_1 \right)^{-1} x_1^{\top}M_{2}y,\\
\hat{b}_{1; \text{model 2}}&=\left( \left(M_{2,3}x_1 \right)^{\top}M_{2,3}x_1 \right)^{-1} \left(M_{2,3}x_1 \right)^{\top}M_{2,3}y
=\left( x_1^{\top}M_{2,3}x_1 \right)^{-1} x_1^{\top}M_{2,3}y,
\end{align}
$$
with symmetric and idempotent matrices
$$
\begin{align}
M_{2}&=I-\left( \mathbf{1}\,x_2 \right) \left( \left( \mathbf{1}\,x_2 \right)^{\top} \left( \mathbf{1}\,x_2 \right) \right)^{-1} \left( \mathbf{1}\,x_2 \right)^{\top},\\
M_{2,3}&=I-\left( \mathbf{1}\,x_2\; x_3 \right) \left( \left( \mathbf{1}\,x_2\; x_3 \right)^{\top}\left( \mathbf{1}\,x_2\; x_3 \right) \right)^{-1} \left( \mathbf{1}\,x_2\; x_3 \right)^{\top}.
\end{align}
$$
Hence, the change in the OLS estimate of $b_1$ is given by
$$
\hat{b}_{1; \text{model 2}}-\hat{b}_{1; \text{model 1}}=\left( \left( x_1^{\top}M_{2,3}x_1 \right)^{-1} x_1^{\top}M_{2,3}-\left( x_1^{\top}M_{2}x_1 \right)^{-1} x_1^{\top}M_{2} \right)y
$$
or simply by
$$
\left( x_1^{\top}M_{2,3}x_1 \right)^{-1} x_1^{\top}M_{2,3}y - \hat{b}_{1; \text{model 1}} 
$$
if you already know the value of $\hat{b}_{1; \text{model 1}}$.
