Using a DAG to understand omitted variable bias in OLS vs Binary Dependent Variable Regression Suppose I have three variables. $A$ and $U$ are continuous variables but $U$ is unobserved. $Y$ is the binary outcome. $A$ and $U$ are independent.
Let the true model be from the typical probit or logit setup,
$$Y = \mathbf{1}\{\beta_0 + \beta_1A + \beta_2U + \epsilon = Y^* > 0 \},$$
where $\epsilon$ is normal or logistic noise.
I want to say the DAG simply features two edges, $A \rightarrow Y$ and $U \rightarrow Y$. Then, there is no backdoor path featuring $A$ so it would seem that $\beta_1$ is identified without controlling for $U$.
However, I know this is true of OLS (supposing we observed the latent $Y^*$), and not true of logistic or probit models. (previously covered here)
Does this mean the DAG should be written differently for binary $Y$? We could say $A\rightarrow Y^*$, $U \rightarrow Y^*$, and $Y^* \rightarrow Y$? But I don't see how that shows there is an identification problem, and is it even necessary?
Is there a way to write the DAG that highlights why there is OVB in the binary $Y$ case but not for continuous $Y^*$?

I would argue that there is no Omitted variable bias here. What you have is a bias on the coefficients produced by a different scale on the errors.
Because probit model imposes a constrain on the variance (=1), the "bias" you see is more related to the new unobserved scale in the errors. However, what you are interested is not the coefficient on the latent variable, but the effect on p(y=1|X).
If you concentrate on that parameter, there shouldn't be an OVB
One way to see this is the following
Consider 2 variables x1 and x2, and the error u. All are IID N(0,1)
your latent $y^*=x_1-x_2 + u$ and your dummy simply $y=(y^*>0)$
Now run the following regressions:
$ p(y|x1,x2) = F(a_0+a_1 x1 + a_2 x_2)$
$ p(y|x1) = F(b_0+b_1 x1 )$
$ p(y|x2) = F(c_0+        c_2 x_2)$
What you will see is that $b_1$ will be smaller than $a_1$ because now the latent error is now larger (var=2), but probit model standardizes the variance to be =1. So all coefficients have to be adjusted by the square root of 2.
However, if you estimate the marginal effects. You will find that they are still the same in all models.
F
