# Endogeneity - Omitted variable bias in OLS

If a have a true model $$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon$$ but $$x_3$$ is unobservable.

What are the consequences of having a unobservable variable which correlates with only $$x_2$$ relative to $$x_3$$ being correlated with both $$x_2$$ and $$x_1$$, assuming $$x_1$$ and $$x_2$$ also being correlated in both cases?

True equations:

\begin{align} y &{}= \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \epsilon\\ x_3 &{}= \gamma_0 + \gamma_1x_1 + \gamma_2x_2 + \eta \end{align}

Assuming $$\epsilon$$ is unrelated to all $$x$$'s and $$\eta$$ is unrelated to $$x_1$$ and $$x_2$$. If we omit $$x_3$$, we have:

\begin{align} y &{}= \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3(\gamma_0 + \gamma_1x_1 + \gamma_2x_2 + \eta) + \epsilon\\ &{}= (\beta_0 + \beta_3\gamma_0) + (\beta_1+\beta_3\gamma_1)x_1 + (\beta_2+\beta_3\gamma_2)x_2 + (\beta_3\eta + \epsilon)\\ \end{align}

If $$\gamma_1 = 0$$ then:

$$y = (\beta_0 + \beta_3\gamma_0) + \beta_1x_1 + (\beta_2+\beta_3\gamma_2)x_2 + (\beta_3\eta + \epsilon)$$

And the coefficient of $$x_1$$ is estimated as $$\beta_1$$. Otherwise, the estimated coefficient of $$x_1$$ will be biased by $$\beta_3\gamma_1$$.

And the estimated coefficient of $$x_2$$ is always biased by $$\beta_3\gamma_2$$.

That $$x_2$$ and $$x_1$$ are correlated is irrelevant since both of them are present in the regression.