  We propose a new estimator, the thresholded scaled Lasso, in high dimensional
threshold regressions. First, we establish an upper bound on the $\ell_\infty$
estimation error of the scaled Lasso estimator of Lee et al. (2012). This is a
non-trivial task as the literature on high-dimensional models has focused
almost exclusively on $\ell_1$ and $\ell_2$ estimation errors. We show that
this sup-norm bound can be used to distinguish between zero and non-zero
coefficients at a much finer scale than would have been possible using
classical oracle inequalities. Thus, our sup-norm bound is tailored to
consistent variable selection via thresholding.
  Our simulations show that thresholding the scaled Lasso yields substantial
improvements in terms of variable selection. Finally, we use our estimator to
shed further empirical light on the long running debate on the relationship
between the level of debt (public and private) and GDP growth.
