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I have a question regarding the validity of transforming the predicted response from a regression model when the response has been log-transformed.

I'm completing an exercise geared towards fitting different model types to training data to predict a response variable (price) in the test data. The distribution of price in the training data made it obvious that a log-transform would help, so this was done. We then fit a variety of models (ordinary ols, glm, gam, pls) in R, computed the training RMSE, and the validation RMSE. I now must take the test data, predict the response, and submit it for the evaluator to check my test RMSE.

My question comes to the prediction itself - is it safe to assume that I can just exponentiate my predicted response since it will be in terms of $\log(x)$? Or do I need to perform some other kind of transformation?

Thanks!

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    $\begingroup$ You can probably answer this question yourself simply by considering how you would use the predicted response. Exactly how do you intend to interpret or use it? $\endgroup$ – whuber Apr 7 '17 at 15:41
  • $\begingroup$ Say the predicted log response is being used to predict log prices at some future point in time. My first-pass assumption is that a simple exponentiation is fine, but I have this nagging feeling that I'm increasing bias by simply taking the exponent of my predict log response and calling it the true predicted response? $\endgroup$ – Kyle Shank Apr 7 '17 at 15:47
  • $\begingroup$ If the log response predicts log prices, it seems there is no need for any transformation, because the two values would be directly comparable. $\endgroup$ – whuber Apr 7 '17 at 16:08
  • $\begingroup$ Even if I wanted to put the predicted response back into the original units? To have an external evaluator calculate the RMSE of my predictions, I need to back-transform. I just wanted to make sure my naive assumption (exponentiate) didn't somehow introduce unnecessary bias. Thanks! $\endgroup$ – Kyle Shank Apr 7 '17 at 16:28
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    $\begingroup$ Ahh... the RMSE is an issue. There will be no direct transformation from an RMSE on the log scale to an RMSE on the original scale. The trick is to use a fitting procedure that assures the best possible RMSE on the scale at which it will be evaluated. $\endgroup$ – whuber Apr 7 '17 at 16:34

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