Suppose the DAG in the population is as follows:
We observe both $X_1$ and $X_2$.
We are interested in the effect of $X_1$ on $Y$. We want to use OLS to estimate the relationship.
Now if I take $X_2$ into account:
$Y=\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$
We have a multicollinearity problem between $X_1$ and $X_2$.
If we omit $X_2$:
$Y=\beta_0 + \beta_1 X_1 + \epsilon$
We have correlation between the independent variable and the error term.
What is the appropriate course of action?