Comparing error to predicted value is frequently done: the result is a relative error. Whether relative or absolute error is more important depends on your data/application/the task at hand.
However, relative errors are most useful if the denominator gives a good impression of what the encountered values are like. In some cases, the mean is suitable, sometimes the lower end of the absolute range is more suitable (you then get worst-case relative error). Sometimes also best case is used (maximum in denominator), particularly if values range from 0 to some maximum.
Ultimately, whether relative error is a good idea at all and which denominator to use should IMHO be decided in accordance with the application/task behind the data.
If your time series has positive and negative (and zero) values, then the mean is probably not a good summary of typically encountered values. max (abs (value)) may be an alternative if that makes sense for your data.