  Conformal predictors, introduced by Vovk et al. (2005), serve to build
prediction intervals by exploiting a notion of conformity of the new data point
with previously observed data. In the present paper, we propose a novel method
for constructing prediction intervals for the response variable in multivariate
linear models. The main emphasis is on sparse linear models, where only few of
the covariates have significant influence on the response variable even if
their number is very large. Our approach is based on combining the principle of
conformal prediction with the $\ell_1$ penalized least squares estimator
(LASSO). The resulting confidence set depends on a parameter $\epsilon>0$ and
has a coverage probability larger than or equal to $1-\epsilon$. The numerical
experiments reported in the paper show that the length of the confidence set is
small. Furthermore, as a by-product of the proposed approach, we provide a
data-driven procedure for choosing the LASSO penalty. The selection power of
the method is illustrated on simulated data.
