Abstract
Most treatments of the model selection problem are either restricted to special situations (lag selection in AR, MA or ARMA models, regression selection, selection of a model out of a nested sequence) or to special selection methods (selection through testing or penalisation). It is however unclear if the results obtained in these cases carry over to the general situation in which the models under scrutiny are non-necessarily nested and selection is performed using general criteria. Our aim is to provide some basic tools for the analysis of model selection as a statistical decision problem. In order to achieve this objective, we embed the model selection procedure in the theoretical framework offered by decision theory. First of all, we analyse what a ‘best’ model should be. In order to do so, we introduce a preference relation on the collection of models under scrutiny, defined in terms of their asymptotic goodness-of-fit and of their parsimony. Second, we consider what happens when replacing asymptotic goodness-of-fit with its finite-sample counterpart. We show that selection through the optimisation of an information criterion associated with each model is quite general, as it is the only technique ensuring the asymptotic selection of a model that is a majorant of the previous preference relation. Third, we show that the conditions under which the information criterion is given by a weighted sum of two components, one linked to goodness-of-fit and one to parsimony, are rather natural. This connects the approach in this paper to our recent work on model selection as a multiple-criteria optimization problem.
Raffaello Seri
Professor of Econometrics
My research interests include statistics, numerical analysis, operations research, psychology, economics and management.