MLE to address multicollinearity in linear regression

OLS estimation assumes that the explanatory variables are independent in the linear regression model. There isn't such assumption when using the MLE estimation. So, my question is, can we use MLE to estimate the parameters in a linear regression model when the explanatory variables exhibit some sort of correlation? Thank you.

• The MLE of the coefficients is the same as the OLS estimate. Thus, both methods are subject to the same limitations and problems.
– whuber
Sep 17 at 12:59

OLS estimation assumes that the explanatory variables are independent in the linear regression model.

That statement is false. The absense of multicolinearity is not an assumption for Ordinary Least Squares. Multicolinearity has consequences, but it does not invalidate a model estimated with OLS. Moreover, the consequences of multicolinearity are the same if you estimate your models with ML.

One major caveat: identifiability.

Theorem: In presence of (exact) multicollinearity, one cannot deduce $${\boldsymbol\beta}$$ from the likelihood function, as it won't be identifiable.

Proof: Since $$r(\mathbf X) < K,$$ there exists $$\boldsymbol\gamma: ~\mathbf X\boldsymbol\gamma =\mathbf 0.$$ Therefore the likelihood function would be identical for $$\boldsymbol \beta$$ and $$\boldsymbol\beta^\star:=\boldsymbol\beta + \boldsymbol\gamma.$$ That is, the "observationally equivalent" $$\boldsymbol\beta$$ and $$\boldsymbol\beta^\star$$ cannot be distinguished from the likelihood function. $$\square$$

Resorting to generalized inverses would provide an unbiased estimator of $$\mathbf X\boldsymbol\beta:$$ it is identifiable. However problem still persists to uniquely estimate $$\boldsymbol\beta.$$

Reference:

$$[1]$$ Econometrics, Peter Schmidt, Taylor & Francis Group, $$1976.$$