Sharp Threshold Detection Based on Sup-norm Error rates in High-dimensional Models

TL;DR

This paper introduces a new thresholded scaled Lasso estimator for high-dimensional threshold regressions, providing sharper variable selection through sup-norm error bounds, with applications to economic data analysis.

Contribution

It establishes a novel sup-norm error bound for the scaled Lasso, enabling more precise variable selection in high-dimensional models.

Findings

Sup-norm bounds improve variable selection accuracy.

Thresholded scaled Lasso outperforms classical methods in simulations.

Application to economic data offers new insights on debt and GDP growth.

Abstract

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.