OLS weight bias with binary outcome

Question

The typical approach when you have a binary outcome variable is to use logistic regression. If you use OLS regression then it becomes easy to violate various assumptions (normality of residuals, constant variance)

What would happen if you were to run OLS regression anyway? Specifically, does violating those assumptions cause a systematic bias in the estimated weights, and is there a way to prove whether the estimates are over or under estimates?

Answer 1

For a binary outcome random variable, we have $$E(Y) = \pi$$.

If we assume that $$\pi = X\beta$$

Then the OLS $\hat\beta = (X'X)^{-1}X'Y$ will give the unbiased estimate of $\beta$, because

$E(\hat\beta) = (X'X)^{-1}X'X\beta = \beta$

Of course, it is possible that estimate of variance of $\hat\beta$ may be incorrect, because of violations.

*** Assume "the estimated weights" in the question means $\hat\beta$.