Assume that I’m going to estimate a linear regression where I assume u∼N(0,σ2). What is the benefit of OLS against ML estimation? I know that we need to know a distribution of u when we use ML Methods, but since I assume u∼N(0,σ2) whether I use ML or OLS this point seems to be irrelevant. Thus the only advantage of OLS should be in the asymptotic features of the β estimators. Or do we have other advantages of the OLS method?

**Answer**

Using the usual notations, the log-likelihood of the ML method is

l(β0,β1;y1,…,yn)=∑ni=1{−12log(2πσ2)−(yi−(β0+β1xi))22σ2}.

It has to be maximised with respect to β0 and β1.

But, it is easy to see that this is equivalent to minimising

∑ni=1(yi−(β0+β1xi))2.

Hence, both ML and OLS lead to the same solution.

More details are provided in these nice lecture notes.

**Attribution***Source : Link , Question Author : MarkDollar , Answer Author : ocram*