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Many important modeling tasks in linear regression, including variableselection (in which slopes of some predictors are set equal to zero) andsimplified models based on sums or differences of predictors (in which slopesof those predictors are set equal to each other, or the negative of each other,respectively), can be viewed as being based on imposing linear restrictions onregression parameters. In this paper, we discuss how such models can becompared using information criteria designed to estimate predictive measureslike squared error and Kullback-Leibler (KL) discrepancy, in the presence ofeither deterministic predictors (fixed-X) or random predictors (random-X). Weextend the justifications for existing fixed-X criteria Cp, FPE and AICc, andrandom-X criteria Sp and RCp, to general linear restrictions. We furtherpropose and justify a KL-based criterion, RAICc, under random-X for variableselection and general linear restrictions. We show in simulations that the useof the KL-based criteria AICc and RAICc results in better predictiveperformance and sparser solutions than the use of squared error-based criteria,including cross-validation.

Selection of Regression Models under Linear Restrictions for Fixed and Random Designs