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Using cross-validation ideas, two procedures are proposed for making a choice between different model structures used for (approximate) modelling of multivariable systems. The procedures are derived under fairly general conditions: the ‘true’ system does not need to be contained in the model set; model structures do not need to be nested and different criteria may be used for model estimation and validation. The proposed structure selection rules are shown to be invariant to parameter scaling. Under certain conditions (essentially requiring that the system belongs to the model set and that the maximum likelihood method is used for parameter estimation) they are shown to be asymptotically equivalent to the (generalized) Akaike structure selection criteria.

Two criteria for choosing between different model-structures are proposed. Their derivation is within a natural cross-validatory assessment context and is fairly assumption-free. In particular, the two criteria can be used for discriminating between non-nested model structures and, more importantly, the ‘true’ system is not required to belong to the considered set of models. Should the true system belong to the model set, the two proposed criteria will asymptotically reduce to some well-known structure selection criteria. This is believed to be a desirable feature of our proposals. On the other hand, it provides a nice cross-validation interpretation of some well-known model structure selection rules. Also, the cross-validation interpretation helps one to choose which criteria to use in a given application The paper also has a second purpose, somewhat decoupled from that mentioned above. It contains an extensive survey of the literature that is useful in its own right.