# Notation: Is the model linear or loglinear? $E(\ln Y|X)=X\beta$

I'm given a model of the form $E(\ln Y|X)=X\beta$.

Should I call it a linear model or loglinear model?

I'm assuming $Y$ is log-normally distributed.

• How would you call $E(Z|X) = X\beta$ where $Z=\ln Y$..? – Tim Nov 24 '16 at 10:15
• This would be linear, so $Z$ has a linear form $\rightarrow E(\ln Y|X) = X\beta$ is linear? – GRS Nov 24 '16 at 10:24

$E(\ln Y|X)=X\beta$ is a linear model of transformed variable $\ln Y$, more precisely, it is linear in parameters. As I noted in the comment, you can introduce additional variable $Z = \ln Y$ and then $E(Z|X)=X\beta$ is just an ordinary regression model. You can make all kinds of transformations to your variables, but when talking about linearity of the model, we have in mind the relationship between variables.
• I've got 2 models, one of the form $E(Y|X)$ and the other $E(\ln Y|X)$ where in both models I investigate $Y$. I was refering to $E(\ln Y|X)$ as the loglinear model, is this incorrect? – GRS Nov 24 '16 at 12:01
• Both are linear for the reasons given above. "$\ln$" is not a part of your model, it's a transformation of the variable. – Tim Nov 24 '16 at 12:13