# All else equal, should an MLE estimation have a lower standard error than OLS?

If I have a model with Y$$\in$${0,1}, and am estimating y= $$\beta$$x+$$\eta$$

My understanding is if I use a probit model say, I am imposing structure on the DGP by assuming Y|x=$$\eta$$ is distributed normally, which should achieve the cramer-rao lower bound.

So is it the case that estimating the equation by ols vs probit, the latter (or even a logit?) should yield lower standard errors?

• Why is the conditional variable distributed normally? – Dave May 6 at 4:52
• I was just saying for a probit specification, you can interpret it as the error being normally distributed, which is the same as the condtional variable being normal? or is that incorrect – Steve May 6 at 15:11
• Probit is for when the response variable is binomial. – Dave May 6 at 16:28
• yes the reponse is binomial, but then the error is normal, no? – Steve May 6 at 17:36
• No, that’s not how it works. How would you end up with a residual of 2, for instance? – Dave May 7 at 2:08