I have to get a regression model for a dataset, and it seems that the best fit is obtained by log transforming the outcome, so that I simply applied the least squares on the trasformed response vector.
$ \hat{log}({Y}) = \hat{f}(\bf{X}) = \hat\beta_0 + \bf{\hat\beta}^TX$
A prediction with this model estimates $E[log(Y)|X]$, which is NOT the same as a generalized linear model (log-linear), that estimates $log(E[Y|X])$.
In the latter case, if I wanted to get the vector of the estimated outcomes, I would simply compute $\hat{Y} = exp(\hat{f}(X)) = exp(log(E[Y|X])$.
Is there a way to retrieve $\hat{Y}$ in my case?
SOLVED: The difference between the two cases is that in the latter you get $E(Y|X)$ simply by exponentiating the mean of the normal distribution, while in the former you need to consider the mean of the associated lognormal. Multipling $exp(\hat{f}(X))$ by $exp(\frac12\hat{\sigma}^2) = exp(\frac12MSE) $ corrects the bias.