# Will an OLS estimator for a regressor differ between a single linear regression and a multiple linear regression? [duplicate]

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?

## 1 Answer

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