I have previous experience with 'normal' K-fold cross-validation for model tuning and I am slightly confused by the application in time-series models.
It is my understanding that for time-series models the corollary for cross-validation is the 'rolling forward origin' procedure described by Hyndman. This makes plenty of sense to me and the code below demonstrates the use of the
tsCV function in R, from Hydman's blog, to show how the errors differ from CV vs. the entire dataset in one go.
library(fpp) e <- tsCV(dj, rwf, drift=TRUE, h=1) sqrt(mean(e^2, na.rm=TRUE)) ##  22.68249 sqrt(mean(residuals(rwf(dj, drift=TRUE))^2, na.rm=TRUE)) ##  22.49681
Now, in that link above it mentions that the drift parameter is re-estimated at each new forecast origin. In 'normal' CV I would have a grid of parameters I would be evaluating against each fold so I could get an average to determine the best parameters to use. I would then use those 'best' parameters to fit the full training set and use that as my final model to evaluate on my previously held out test set. Note, this is nested cross-validation so I am not training on my test set at any point.
This clearly is not the case with the 'rolling forward origin' procedure where the parameters are optimized for each fold (at least for the R methods like
auto.arima, etc.) . Am I mistaken to think about this method in terms of model parameter tuning or how would I chose the time series model parameters to set for the final model that would be used? Or is parameter tuning not consider an issue with time series models where optimization seems to be part of model fitting and the result of the CV is to just say how well each model performs overall? And that the final model built with the majority of the data at the end is the model I would use?
I realize this can be rephrased in an even simpler question. Following cross-validation ('rolling forward origin') do I just use the last model built (with the largest superset as the final fitted model? Or what is suggested?