Will an OLS estimator for a regressor differ between a single linear regression and a multiple linear regression?

Question

Thought question that I am having a difficult time formulating mathematically.

Knowing that $y = \beta_0 + \beta_1X_1 +\beta_2X_2 + \varepsilon$, where $X_1$ and $X_2$ are non-random and $\beta_2$ is non-zero.

If I construct a regression using both $X_1$ and $X_2$ in a multivariable regression, but then re-fit my model using only $X_1$ (going from a multivariable regression to a single linear regression), will my OLS estimator for $\beta_1$ remain unchanged (e.g. will the estimator for $\beta_1$ be equal in both models)?

Intuitively, I would assume that I would obtain different results across models as more information is being used to predict $y$ in the multivariable regression.

Should I be looking at the FWL theorem? Or does this get to the root of my question?

Answer 1

The $X_1$ coefficient will only remain unchanged in the two models if $X_2$ has 0 covariance with either $Y$ or with $X_1$.