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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?

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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 $\hat\beta$.

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    $\begingroup$ Perhaps should mention why modelling $\pi$ as a linear regression is a bad idea since it will violate $\pi \in (0,1)$. $\endgroup$ – Xiaomi Oct 10 '18 at 0:56
  • $\begingroup$ It depends on the situation. If $0.3<\pi_i<0.7$, $\hat\pi_i$ goes beyond (0,1) is rear. One advantage is easier to explain the meaning of the $\beta$ than that from logistic. Of curse, WLS is required. $\endgroup$ – user158565 Oct 10 '18 at 1:41

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