MHB Comparing R^2 from log and level models

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R^2 values from log and level models cannot be directly compared due to the differing transformations of the dependent variable, which fundamentally alter the scale and interpretation of the results. The log transformation compresses the range of the dependent variable, affecting the variance and the model's fit. This results in R^2 values that reflect different aspects of model performance, making them incompatible for direct comparison. Additionally, the underlying assumptions and distributions of the residuals differ between the two models, further complicating any comparative analysis. Understanding these differences is crucial for accurate model evaluation and interpretation.
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Say we have 2 models:

ln(y) = \beta_0 + \beta_1 x_1 + \cdots + \beta_nx_n with a known R^2

and

y = \beta_0 + \beta_1 x_1 + \cdots + \beta_nx_n with a known R^2

Now I know that we can not compare the R^2's from these 2 models to determine goodness-of-fit and I am also aware of how we can manipulate the log model so that we can compare, but my question is, what is the reason for which we can't compare? Obviously the dependent variable is the natural log for the first one and the second model is level in terms of y, but is there a deeper reason? Why is it that if the dependent variable's form is different, then we cannot compare the R^2's between the models?

Thanks
 
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Deep as you would like, I would think.

http://www.yale.edu/ciqle/Breen_Scaling%20effects.pdf
 
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