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If the first order conditions of MLE and OLS are identical is MLE as efficient as OLS?

It seems that they should be equal in terms of efficiency however if OLS is the best linear unbiased estimator does that mean that there is in fact a difference and OLS is slightly more efficient than MLE.

If there is a difference why?

If there is no difference how do you formally show this. Do you need to show the asymptotic distribution etc.

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Ordinary Least Squares (OLS) is the maximum likelihood estimator (MLE) when the conditional distribution of the $Y$ is normal. However, the proof of the Gauss Markov Theorem (which shows that OLS is BLUE) does not require the conditional $Y$ to be normally distributed, so the BLUE-ness of OLS is a nonparametric result. In fact, if the conditional $Y$ were any other distribution, maximum likelihood would be more efficient, it's just that the estimator would not be a linear one. It should match our intuition: using the knowledge of the actual distribution should help us. The MLE is always the asymptotically efficient estimator.

Reminder: a linear estimator is any estimator of the form $\hat{Y} = bY$ i.e. a projection. There are other forms of linear estimators like the average slope. This is an elusive fact, one must delve into Seber and Lee "Linear Regression Analysis" for the proper definition.

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