  In a linear regression model of fixed dimension $p \leq n$, we construct
confidence regions for the unknown parameter vector based on the Lasso
estimator that uniformly and exactly hold the prescribed in finite samples as
well as in an asymptotic setup. We thereby quantify estimation uncertainty as
well as the "post-model selection error" of this estimator. More concretely, in
finite samples with Gaussian errors and asymptotically in the case where the
Lasso estimator is tuned to perform conservative model selection, we derive
exact formulas for computing the minimal coverage probability over the entire
parameter space for a large class of shapes for the confidence sets, thus
enabling the construction of valid confidence regions based on the Lasso
estimator in these settings. The choice of shape for the confidence sets and
comparison with the confidence ellipse based on the least-squares estimator is
also discussed. Moreover, in the case where the Lasso estimator is tuned to
enable consistent model selection, we give a simple confidence region with
minimal coverage probability converging to one. Finally, we also treat the case
of unknown error variance and present some ideas for extensions.
