  We revisit the adaptive Lasso as well as the thresholded Lasso with
refitting, in a high-dimensional linear model, and study prediction error,
$\ell_q$-error ($q \in \{1, 2 \} $), and number of false positive selections.
Our theoretical results for the two methods are, at a rather fine scale,
comparable. The differences only show up in terms of the (minimal) restricted
and sparse eigenvalues, favoring thresholding over the adaptive Lasso. As
regards prediction and estimation, the difference is virtually negligible, but
our bound for the number of false positives is larger for the adaptive Lasso
than for thresholding. Moreover, both these two-stage methods add value to the
one-stage Lasso in the sense that, under appropriate restricted and sparse
eigenvalue conditions, they have similar prediction and estimation error as the
one-stage Lasso, but substantially less false positives.
