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.

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