  We study the limitations of the well known LASSO regression as a variable
selector when there exists dependence structures among covariates. We analyze
both the classic situation with $n\geq p$ and the high dimensional framework
with $p>n$. Restrictive properties of this methodology to guarantee optimality,
as well as the inconveniences in practice, are analyzed. Examples of these
drawbacks are showed by means of a extensive simulation study, making use of
different dependence scenarios. In order to search for improvements, a broad
comparison with LASSO derivatives and alternatives is carried out. Eventually,
we give some guidance about what procedures are the best in terms of the data
nature.
