Costs of order selection in time series analysis

The costs of order selection are defined as the loss in model quality due to selection. It is the difference between the quality of the best of the candidate models that have been estimated from N given observations of the process and the model that is actually selected for the data. This definition creates a distinction between the quality of parameter estimation and the quality of order selection. The order selection criterion itself has an influence on the costs, but also the number of competitive candidate models for the selection. The AIC criterion of Akaike has a penalty factor 2 for every additional estimated parameter in the model. Taking a larger penalty factor for each estimated parameter may improve the selection quality. The number of candidates for order selection is of itself limited for the nested and hierarchical autoregressive (AR) or moving average (MA) models. However, intentionally reducing the number of selection candidates can be beneficial in combined ARHA(p,q) processes, where two separate model orders are involved. the AR order p and the MA order q.