  The paper considers linear regression problems where the number of predictor
variables is possibly larger than the sample size. The basic motivation of the
study is to combine the points of view of model selection and functional
regression by using a factor approach: it is assumed that the predictor vector
can be decomposed into a sum of two uncorrelated random components reflecting
common factors and specific variabilities of the explanatory variables. It is
shown that the traditional assumption of a sparse vector of parameters is
restrictive in this context. Common factors may possess a significant influence
on the response variable which cannot be captured by the specific effects of a
small number of individual variables. We therefore propose to include principal
components as additional explanatory variables in an augmented regression
model. We give finite sample inequalities for estimates of these components. It
is then shown that model selection procedures can be used to estimate the
parameters of the augmented model, and we derive theoretical properties of the
estimators. Finite sample performance is illustrated by a simulation study.
