Sharp Threshold Detection Based on Sup-Norm Error Rates in High-Dimensional Models

Abstract

We propose a new estimator, the thresholded scaled Lasso, in high-dimensional threshold regressions. First, we establish an upper bound on the ℓ_∞ estimation error of the scaled Lasso estimator of Lee, Seo, and Shin. This is a nontrivial task as the literature on high-dimensional models has focused almost exclusively on ℓ₁ and ℓ₂ estimation errors. We show that this sup-norm bound can be used to distinguish between zero and nonzero 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. Supplementary materials for this article are available online.

KEYWORDS:

• Debt effect on GDP growth
• Oracle inequality
• Sup-norm bound
• Threshold model
• Thresholded scaled Lasso

Notes

The notation suppresses that we are really dealing with a triangular array. Thus, more precisely, we assume uniform sub-Gaussianity across the rows of this triangular array.

Available at https://github.com/lcallot/ttlas

The original data are available at http://www.bis.org/publ/work352.htm, and can also be found in the replication material for this section.

U.S., Japan, Germany, the United Kingdom, France, Italy, Canada, Australia, Austria, Belgium, Denmark, Finland, Greece, the Netherlands, Norway, Portugal, Spain, and Sweden.

1984, 1989, 1994, 1999, 2004.

Here $K_1=\sqrt{7C_1{C}_2}$ where C₂ is the constants proven to exist in Lemma A.2 in the Appendix ensuring that ${sup}_{\tau \in T}{max}_{1\le j\le 2m}{∥X^{\left(j\right)}\left(\tau \right)∥}_n\le C_2$ with arbitrarily large probability (more precisely, for any ε > 0 there exists a C₂ such that ${sup}_{\tau \in T}{max}_{1\le j\le 2m}{∥X^{\left(j\right)}\left(\tau \right)∥}_n\le C_2$ with probability at least 1 − ε).

Alternatively, the arguments on pp. A4–A6 in Lee, Seo, and Shin (2015) yield a uniform (in τ) upper bound on ${∥\left(\Sigma \left(\tau \right)-\frac{1}{n}{X}^{\text{'}}\left(\tau \right)X\left(\tau \right)\right)∥}_{\infty }$ of the order $O_p\left(\sqrt{\frac{log\left(mn\right)}{n}}\right)$ which could also be used resulting in only slightly worse rates.