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Suppose our true model is:

$Y=\alpha +\beta_{1} X_{1}+\beta_{2} X_{2}+\beta_{3} X_{3}+u$

but instead, we omit $X_{3}$ and estimate the following by OLS:

$Y=\alpha +\beta_{1} X_{1}+\beta_{2} X_{2}+v$

Can anybody help me obtain an expression for the bias in the estimate of $b_{1}$ in the following format?

$b_{1}=\beta_{1}+bias$

The reason I am asking is to get a sense of the bias for the case when $X_{3}$ is correlated with $X_{2}$ but not with $X_{1}$. Presumably, the coefficient estimate of $X_{1}$ will still be biased to the extent that $X_{1}$ is correlated with $X_{2}$.

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The formula must be $$ \mathrm{E}\tilde{\beta_1}=\beta_1+\beta_3\frac{\sum\hat{r}_{i1}x_{i3}}{\sum\hat{r}_{i1}^2} $$ Where $\tilde{\beta_1}$ is the biased estimator of $\beta_1$, $\hat{r}_{i1}$ are the OLS residuals from the regression of $x_1$ on $x_2$ and $x_{i3}$ are the sample values of $x_3$.

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