I am trying to decide whether to use a square root transformed dependent variable in multiple linear regression. Transforming $y$ leads to more normally distributed residuals and also to less heteroskedasticity. However, coefficient of determination is reduced compared to the model using the non-transformed dependent variable.

What criterion is most important when deciding whether to use a transformation?

Edit (Background and goal of regression):

I have a large panel data set and I am running a pooled OLS regression. The main goal is to spot how the dependent variable (electricity consumption) differs with different independent variables, so I am mainly interested in the actual parameter estimates.

However, the model should also be useful for prediction, both on individual as well as aggregated level. In my context, aggregated means the sum of individual $y$'s for each day. I suspect modeling aggregated $y$ using the panel model might not be entirely correct. Back-transformation of $y$ when comparing aggregated metered and modeled values still causes some headache... But this is probably yet another question.

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    $\begingroup$ $R^2$ values aren't really comparable across transformations. $\endgroup$
    – Glen_b
    Jan 4 '15 at 18:12
  • $\begingroup$ What is your goal in fitting the model? $\endgroup$ Jan 4 '15 at 18:51
  • $\begingroup$ It might also be worth linking back to your other question, for coherence and posterity $\endgroup$ Jan 4 '15 at 18:53

Simply put, you should not use a model that violates its assumptions just because it yields a higher $R^2$. So, you should use the transformed variable for your model. However, bear in mind that the square root is a non-linear transformation. In other words, if a straight line was most appropriate before the transformation, necessarily a straight line will not be the most appropriate fit afterwards. You should probably add a squared term, $x^2$, or something similar to compensate for the transformation. It is hard to diagnose this in the abstract, but you should look at plots of your data and your model both with and without the transformed $y$ to ensure the assumptions are met and the functional form is appropriate.

  • $\begingroup$ Very good point about introducing nonlinearity $\endgroup$ Jan 4 '15 at 18:52

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