I am comparing several OLS multivariate regression models of a dependent variable (we'll call it $Y$) using various transformations, some of which also involve one of the independent variables ($X_1$). These transformations include:
- no transformation
- $\frac{Y}{X_1}$
- $\ln(Y)$
- $\ln\left(\frac{Y}{X_1}\right)$
I want to be able to compare the AIC of several models using each of these transformations. I understand I must apply the Jacobian correction to the log-likelihood to compute an apples-to-apples AIC. I understand how to deal with $\ln(Y)$ since it's the most classical transformation, one that shows up everywhere this issue is raised (e.g. here), namely: add $2 \times \sum^N_{i=1} \ln(Y_i)$. If my math is correct, then the same correction applies for models based on $\ln\left(\frac{Y}{X_1}\right)$, since it is equivalent to $\ln(Y)-\ln(X_1)$ and the derivative of $-\ln(X_1)$ with respect to Y is zero.
But what exactly should I do with $\frac{Y}{X_1}$? Its derivative with respect to Y is ${X_1}^{-1}$. So its contribution to the log-likelihood is $-\ln(X_i)$ and the adjustment factor to the AIC should be to add $2 \times \sum^N_{i=1} \ln(X_i)$. However, after having run some simulations, this seems to be close but not quite right. Is my math off or am I going about this the wrong way?
