  Many important modeling tasks in linear regression, including variable
selection (in which slopes of some predictors are set equal to zero) and
simplified models based on sums or differences of predictors (in which slopes
of 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 on
regression parameters. In this paper, we discuss how such models can be
compared using information criteria designed to estimate predictive measures
like squared error and Kullback-Leibler (KL) discrepancy, in the presence of
either deterministic predictors (fixed-X) or random predictors (random-X). We
extend the justifications for existing fixed-X criteria Cp, FPE and AICc, and
random-X criteria Sp and RCp, to general linear restrictions. We further
propose and justify a KL-based criterion, RAICc, under random-X for variable
selection and general linear restrictions. We show in simulations that the use
of the KL-based criteria AICc and RAICc results in better predictive
performance and sparser solutions than the use of squared error-based criteria,
including cross-validation.
