An Algebraic Estimator for Large Spectral Density Matrices

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 <i>l</i>₁ 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 <i>p</i> and the sample size <i>T</i> 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.

本文提出一种针对高维谱密度矩阵的新型估计量，命名为代数谱估计量（ALgebraic Spectral Estimator, ALSE），其构建基于动态因子模型中常用的低秩加稀疏结构假设。ALSE通过在核范数加L₁范数约束下最小化二次损失来计算，该约束用于控制潜在秩与残差稀疏模式。损失函数的输入为经典平滑周期图估计量与两个阈值参数，本文对这两个阈值的选取展开了详尽讨论。本文证明了当维度$p$与样本量$T$均趋于无穷时ALSE的相合性，同时证明其可以以概率1准确恢复潜在秩与残差稀疏模式。随后本文提出了非收缩代数谱估计量（UNshrunk ALgebraic Spectral Estimator, UNALSE），该估计量旨在在保留ALSE最优性的前提下，针对预估计量最小化弗罗贝尼乌斯损失。将UNALSE应用于标准美国季度宏观经济数据集后，本文发现存在两类主要的协同变动来源：一类是驱动经济周期频率下经济运行的实际因子，另一类是驱动更高频率动态的名义因子。本文还辅以一项大规模模拟实验。本文的补充材料可在线获取。