What are the problems if you estimate a linear regression model using OLS, when the dependent is a dummy ? What Models are used to overcome it What are the problems if you estimate a linear regression model using OLS, when the dependent is a dummy ? What Models are used to overcome it

This is called a linear probability model. Despite people liking it, there are issues.

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*It can predict illegal probabilities outside of $[0,1]$.


*When we have a binary $y$, we are in the lucky situation of knowing the conditional distribution, and we can model that conditional distribution. Unfortunately, OLS is equivalent to maximum likelihood estimation with Gaussian conditional distributions, not binomial, and the OLS solution is not the minimum-variance solution among consistent estimators.
A solution that remedies both of these is to use maximum likelihood estimation of a binomial conditional distribution while applying a link function that compresses predicted values to $(0,1)$. The most common example of this is logistic regression.
