# Does OLS give the maximum likelihood estimation for a linear log model?

I'm fitting a model $$y=a\times \log(x)+ b$$ using standard scikit linear regression (wich uses OLS) and a transformation $$x'=\log(x)$$. My doubt is: the parameters I get for the model are the best one if I assume, for example, that my errors are normally distributed? I know that OLS will give the maximum likelihood estimator for the $$x'$$ but I have doubts if that is true for $$x$$. I would appreciate any references on the subject too.

• Are you ever using just the $x$ alone?
– Dave
Commented Apr 14 at 17:07
• Hi, from the business perspective my problem has a logarithm like curve ( involves saturation). That's why we need to use log(x). Commented Apr 14 at 17:55
• Then how does $x$ get involved with the maximum likelihood issue? (You seem to know the equivalence between minimizing square loss and maximizing Gaussian likelihood.)
– Dave
Commented Apr 14 at 18:12

A transformation of your independent variable $$x$$ does not affect the distribution of the dependent variable $$y$$ (or the error, by extension). Given your stated model setup, which goes by many names ("linear-log", "level-log", "semi-log", etc.), $$\hat{a}_{OLS}$$ is still the maximum likelihood estimate.

As for references, have a look at chapter 5.4 in Gujarati or chapter 4.2 in Dougherty.