Is there in general a 'most appropriate' method to perform model selection in a time series context for forecasting purposes?
One can find in the literature a jungle of different information criteria (IC). To name a few: AIC, AICc, AICu, MAIC, TIC, FPE, FPEu, Mallow's Cp, SIC, HQ, HQc, EIC... Is anyone aware of a comprehensive review (or any good reason) to rank the performance of these criteria (either theoretically or empirically or just even qualitatively) in the context of time series model selection?
Moreover as far as I understand IC can be used to perform hyper parameter tuning/variable selection (example I can use AIC to find the best $p^*$ and $q^*$ for an ARMA$(p,q)$ model but I cannot compare the AIC values of an ARIMA$(1,1,1)$ with that of an ARIMA$(1,2,1)$ because of the order of differencing nor can I compare an ARMA with a ETS). Is it not incorrect than to refer to IC as to methods to perform model selection rather than variable selection? Am I then left only with rolling forecasting origin CV (or other modified CV) to actually do model selection in the strict sense?
Lastly if that is the case, it seems like IC and LASSO methods are in direct competition? How does the performance of the best IC compares to that of LASSO? And more specifically, to 'which' LASSO? I would in fact assume that it makes a significant difference the way in which we select the regularisation parameter $\lambda$: would it not be -for instance- redundant to select $\lambda$ through an IC (say AIC)?
An answer to any of the previous question or references would be greatly appreciated.