OLS and MLE
When does the Ordinary Least Squares (OLS) estimator coincide with the Maximum Likelihood Estimator (MLE)?
Answer
OLS coincides with MLE when we have normality of errors (the 6th condition).
Let the model be
\[
y = X\beta + \varepsilon,
\qquad
\varepsilon \sim \mathcal{N}(0,\sigma^2 I).
\]
Under normality of the errors,
\[
y \mid X \sim \mathcal{N}(X\beta,\ \sigma^2 I).
\]
In this case, the estimator that maximizes the likelihood is exactly the OLS estimator:
\[
\hat{\beta}_{\text{MLE}} = \hat{\beta}_{\text{OLS}} = (X^{T}X)^{-1}X^{T}y.
\]
Hence, OLS equals MLE when the errors are normally distributed.