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Although model selection is routinely used in practice nowadays, little is known about its precise effects on any subsequent inference that is carried out. The same goes for the effects induced by the closely related technique of model averaging. This paper is concerned with the use of the same data first to select a model and then to carry out inference, in particular point estimation and point prediction. The properties of the resulting estimator, called a post-model-selection estimator (PMSE), are hard to derive. Using selection criteria such as hypothesis testing, AIC, BIC, HQ and Cp, we illustrate that, in terms of risk function, no single PMSE dominates the others. The same conclusion holds more generally for any penalised likelihood information criterion. We also compare various model averaging schemes and show that no single one dominates the others in terms of risk function. Since PMSEs can be regarded as a special case of model averaging, with 0-1 random-weights, we propose a connection between the two theories, in the frequentist approach, by taking account of the selection procedure when performing model averaging. We illustrate the point by simulating a simple linear regression model.
Model averaging model selection inference after model selection post-selection weights
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