Finite sample properties of ARMA order selection

The cost of order selection is defined as the loss in model quality due to selection. It is the difference between the quality of the best of all available candidate models that have been estimated from a finite sample of N observations and the quality of the model that is actually selected. The order selection criterion itself has an influence on the cost because of the penalty factor for each additionally selected parameter. Also, the number of competitive candidate models for the selection is important. The number of candidates is, of itself, small for the nested and hierarchical autoregressive/moving average (ARMA) models. However, intentionally reducing the number of selection candidates can be beneficial in combined ARMA(p,q) models, where two separate model orders are involved: the AR order p and the MA order q. The selection cost can be diminished by creating a nested sequence of ARMA(r,r-1) models. Moreover, not evaluating every combination (p,q) of the orders considerably reduces the required computation time. The disadvantage may be that the true ARMA(p,q) model is no longer among the nested candidate models. However, in finite samples, this disadvantage is largely compensated for by the reduction in the cost of order selection by considering fewer candidates. Thus, the quality of the selected model remains acceptable with only hierarchically nested ARMA(r,r-1) models as candidates.