  In this paper we consider the conservative Lasso which we argue penalizes
more correctly than the Lasso and show how it may be desparsified in the sense
of van de Geer et al. (2014) in order to construct asymptotically honest
(uniform) confidence bands. In particular, we develop an oracle inequality for
the conservative Lasso only assuming the existence of a certain number of
moments. This is done by means of the Marcinkiewicz-Zygmund inequality. We
allow for heteroskedastic non-subgaussian error terms and covariates. Next, we
desparsify the conservative Lasso estimator and derive the asymptotic
distribution of tests involving an increasing number of parameters. Our
simulations reveal that the desparsified conservative Lasso estimates the
parameters more precisely than the desparsified Lasso, has better size
properties and produces confidence bands with superior coverage rates.
