An Algebraic Estimator for Large Spectral Density Matrices
提出一种名为ALSE的代数谱估计量，用于高维谱密度矩阵的估计，假设其具有低秩加稀疏结构，并证明了一致性；进一步提出UNALSE估计量，应用于美国宏观经济数据发现两个主要共同波动来源。
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.