S-curve in residuals plot: a problem? I am doing some linear regression and am predicting a absolutely non-normal dependent variable (for context: we are forecasting the amount of units sold for a shop). Therefore, we have transformed the variable to its natural log. The residuals do now have an s-curve.
I reckon that it means that the distribution is short-tailed.Should I do anything about this or what effects does it have on the validity of my forecast?
Edit:
I have two more pictures for you, I hope this will help in getting an answer.


 A: An S-shape P-P plot indicates that the distribution has the correct median. The "flattening" of the S means that the distribution has tails that are about as long as those of the normal distribution. So your tails aren't "short". Rather, the density decays faster to meet a tail of the same length as the Gaussian distribution.
This alone does not pose a problem for generating forecasts, because the symmetry in the plot (the S isn't badly lopsided) suggests that the residuals are distributed more or less symmetrically. That means that your forecasts, on the log scale, are overestimates about as often as they are underestimates. Depending on the kinds of forecasting you care about, this is as good as you'd hope for.
Making inferences about those forecasts is probably also safe, because of the large sample size and because the tails "match" those of the theoretical distribution.
The rapid decay implies that the bulk of your estimates are accurate (good!), but there is a relatively wide spread of inaccurate estimates. That could indicate nothing, or it could indicate there's some kind of problem with your model. This is why it would be good to see a Q-Q plot of the residuals, a plot of residuals vs fitted, and a plot of fitted vs actual on the original scale.
