An Algebraic Estimator for Large Spectral Density Matrices

Abstract

We propose a new estimator of high-dimensional spectral density matrices, called ALgebraic Spectral Estimator (ALSE), under the assumption of an underlying low rank plus sparse structure, as typically assumed in dynamic factor models. The ALSE is computed by minimizing a quadratic loss under a nuclear norm plus l₁ norm constraint to control the latent rank and the residual sparsity pattern. The loss function requires as input the classical smoothed periodogram estimator and two threshold parameters, the choice of which is thoroughly discussed. We prove consistency of ALSE as both the dimension p and the sample size T diverge to infinity, as well as the recovery of latent rank and residual sparsity pattern with probability one. We then propose the UNshrunk ALgebraic Spectral Estimator (UNALSE), which is designed to minimize the Frobenius loss with respect to the pre-estimator while retaining the optimality of the ALSE. When applying UNALSE to a standard U.S. quarterly macroeconomic dataset, we find evidence of two main sources of comovements: a real factor driving the economy at business cycle frequencies, and a nominal factor driving the higher frequency dynamics. The article is also complemented by an extensive simulation exercise. Supplementary materials for this article are available online.

Keywords:

• Dynamic rank
• Generalized dynamic factor model
• Large spectral density matrix
• Sparsity

Acknowledgments

Matteo Barigozzi gratefully acknowledges the support by MIUR (PRIN 2020) under grant 2020N9YFFE. Matteo Farnè gratefully acknowledges the support by the Accademia Nazionale dei Lincei under grant British Academy 2018.

Disclosure Statement

The authors report there are no competing interests to declare.