How to show that the goodness of fit (significantly) differs? I have a certain model that I want to do some out-of-sample testing on. I allow it to do predictions and then receive an estimate of the variable "RETURNS".
I then apply the following regression:
TRUE_RETURNS = c + b*ESTIMATED_RETURNS + e
Under perfect estimations you'd have: c = 0 and b = 1
But I find it a bit silly to run the regression and simply say: look, c is not close to 0 and neither is b close to 1, so the out-of-sample predictions are not good. Or some senseless talk about the R^2..
I would like to be able to compare the out-of-sample predictions of multiple different sets of predictions. That is, I want to have sets predictions_1, predictions_2, predictions_3 and predictions_4 and then be able to conclude that predictions_1 are much better fit than predictions_2 for example.
I therefore would like to kindly ask if anyone can please provide me with some inspiration on how to properly evaluate the goodness of fit? Perhaps a likelihood test or something (or am I completely wrong?)
Thank you very much in advance!

There are standard loss functions for this, such as mean of absolute value (or square) of prediction minus truth. For a more in-depth treatment you may read chapter 7 of this book (Elements of Statistical Learning), freely available on that website.
