# Counterexamples to parsimony and BIC

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## Abstract

Suppose that the log-likelihood-ratio sequence of two models with different numbers of estimated parameters is bounded in probability, without necessarily having a chi-square limiting distribution. Then BIC and all other related “consistent” model selection criteria, meaning those which penalize the number of estimated parameters with a weight which becomes infinite with the sample size, will, with asymptotic probability 1, select the model having fewer parameters. This note presents examples of nested and non-nested regression model pairs for which the likelihood-ratio sequence is bounded in probability and which have the property that the model in each pair with *more* estimated parameters has better predictive properties, for an independent replicate of the observed data, than the model with fewer parameters. Our second example also shows how a one-dimensional regressor can overfit the data used for estimation in comparison to the fit of a two-dimensional regressor.

## Key words and phrases

Model selection linear regression misspecified models AIC BIC MDL Hannan-Quinn criterion overfitting## Preview

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