Ordinary Least Squares Regression with binary dependent variable

I know that OLS regression is linear and output expected is continuous and values will fall higher than 1 or less than 0 so is no meaning of values what are not between 0 and 1 (here pointing to values 20, 30 etc not strictly around 1), can’t be interpreted.

My questions is: Mathematically speaking: Why is not appropriate to use OLS regression when we have binary dependent variable?

• Is there a difference between a statistical and mathematical explanation? I believe there are many on this site and others that explain why it is not appropriate statistically. – Jon Aug 15 '17 at 1:21
• By proof is what I’m interested, statistically is clear but can’t find solid math proof ... – n1tk Aug 15 '17 at 1:38
• If the regression predicts one value to be 1.001 and another value to be 0.999, then I personally would not reject the prediction of 1.001 in the case you describe. As I understand your description, one way to interpret the results would be, "Is the regression value above or below 0.5?" – James Phillips Aug 15 '17 at 13:12
• I doubt that you will find a mathematical proof. Sometimes OLS is used for this very thing (see en.wikipedia.org/wiki/Linear_probability_model). Probit and logit are usually used instead because, as you say, the linear model often can't be interpreted at all. – Great38 Aug 15 '17 at 14:10
• Beacuse the assumptions underlying OLS are not fulfilled when you have a binary dependent variable (e.g. the homoscedasticity assumption). See e.g. Basic Econometrics by Gujaratti. – user83346 Aug 15 '17 at 14:57