OLS Loss Function - MLE assumptions? What are the model assumptions necessary for p(y|x,w) in order for the MLE estimate to provide the OLS loss function?
I know that MLE applied to linear regression gives OLS, I'm just not sure what assumptions, if any, must hold for this to be the case.
 A: OLS relations with MLE
Maximum likelihood estimation (MLE) can be performed when the distribution of the error terms is known to belong to a certain parametric family ƒθ of probability distributions, and assuming randoms sampling. When ƒθ is a normal distribution with zero mean and variance θ, the resulting estimate is identical to the OLS estimate. GLS estimates are maximum likelihood estimates when ε follows a multivariate normal distribution with a known covariance matrix. Wikipedia
The goal of maximum likelihood estimation is to make inferences about the population that is most likely to have generated the sample, specifically the joint probability distribution of the random variables $\left\{ y_{1}, y_{2}, \ldots \right\}$, not necessarily independent and identically distributed.
OLS can be estimated without making full distributional assumption.  Instead, we focused primarily on zero-covariance and zero-conditional-mean
assumptions, and secondarily on assumptions about conditional variances and covariances. These assumptions were sufficient for obtaining consistent, asymptotically
normal estimators, some of which were shown to be efficient within certain classes of
estimators. As Wooldridge 2010 Page 385.
Here MLE of the OLS  in detail https://www.le.ac.uk/users/dsgp1/COURSES/MATHSTAT/13mlreg.pdf
Pros and Cons
Pros

*

*MLE as the unifying theme: most models are estimated by maximum likelihood. As Wooldridge 2010 Page 385.

*Efficient Properties: it is generally the most efficient estimation procedure in the class of
estimators that use information on the distribution of the endogenous variables given
the exogenous variables. As Wooldridge 2010 Page 385.

Cons

*

*Efficiency usually comes at the price of nonrobustness, and this
is certainly the case for maximum likelihood. Maximum likelihood estimators are
generally inconsistent if some part of the specified distribution is misspecified. As Wooldridge 2010 Page 385.

References

*

*Greene, William H. "Econometric analysis 4th edition." International edition, New Jersey: Prentice Hall (2000): 201-215. https://spu.fem.uniag.sk/cvicenia/ksov/obtulovic/Mana%C5%BE.%20%C5%A1tatistika%20a%20ekonometria/EconometricsGREENE.pdf

*Wooldridge, Jeffrey M. Econometric analysis of cross section and panel data. MIT press, 2010. https://jrvargas.files.wordpress.com/2011/01/wooldridge_j-_2002_econometric_analysis_of_cross_section_and_panel_data.pdf
A: Basically, five assumptions of linear regression must be satisfied:

*

*Linearity

*Little or no multicollinearity

*Homoscedasticity

*Independence

*Normality

Suppose $$Y = X\beta + \epsilon, \epsilon \sim N(0_n,\sigma^2I_n)$$
the probability density function of Y:
$$f(Y,\beta)=\frac{1}{\sqrt{(2\pi)^n|\Sigma|}}
\exp\left(-\frac{1}{2}({ Y}-{ X\beta})^T {\Sigma}^{-1}({ Y}-{X\beta})
\right)$$
The MLE for $\beta$:
$$\hat{\beta}_{MLE} = (X^TX)^{-1}X^TY$$
The residual sum of square is:
$$S(\beta) = (Y-X\beta)^T(Y-X\beta)$$
The OLS estimator for $\beta$: 
$$\hat{\beta}_{OLS}= (X^TX)^{-1}X^TY$$
If any assumption breaks, for example, $\mathbf{X}$  has perfect multicollinearity, then $\mathbf{X^TX}$ will not be full rank, which makes $\mathbf{X^TX}$ non-invertible.
