Goodness measure of time series prediciton IF I create daily weather prediction for next 30 days on out of sample data using various competing methods. How do I measure which one of them is the best time series prediction method?
 A: It's an entire field of research you're asking about: forecast evaluation.
Most practical approaches are based on a couple of things: forecast error and encompassing.
The popular forecast error measure is a root mean squared forecast error (RMSFE). You can straight compare RMSFEs from out of sample forecasts of different models. The smaller is better, obviously. 
If you have a single model, you can obtain the statistics of RMSFE and test it, that's what Chow tests do. You forecast 1 period ahead, and obtain the errors. You know the error variance from the model estimation, so you can see whether your errors are within the expected variance bounds.
The encompassing test idea is that one forecast may be encompassing another one. For instance, you could regress the actual values on the predicted values of two models. Then you can test whether the beta on one of the predicted models is zero while on another is not. This would mean that one model encompasses the other.
Both approaches are described in detail here:  A Framework for Economic Forecasting by Ericsson et al
