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I have a relationship of two variables which is somehow log shaped. Now, I establish two models for this dataset, for one I log transform the dependent variable:

    y_ln=ln(y)
    fit=lm(y~x) #lm fits the linear model
    fit_ln=lm(y_ln~x)

If fit_ln has a higher R², is it valid to say that the fit_ln model is the better one?

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Nope, that will not work. Because the outcome is not the same, and the scaling is not linear.

There are many ways to get a measure, but the one suggested by Wooldridge is to calculate the square correlation af $y$ (in levels) and the fitted value $\hat{y}$ obtained from the regression - after a back transformation.

Now, of course, the question is how to obtained a prediction, when $log(y)$ is the dependent variable. Wooldridge suggests (but there are many ways to do this):

  1. Obtain the fitted values from $log(y)$ on $x_1, x_2, ... , x_n$, as well as the residuals $u_i$.

  2. Calculate: $a_0 = n^{-1} \sum_{i=1}^n exp(u_i)$

  3. Then for each $y_i$ calculate: $\hat{y}_i = a_0 \cdot exp(\widehat{log(y)})$

Now you can calculate an $R^2$ which is comparable.

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