Speaker: Matteo Farnè (Visiting Assistant Professor, Statistics, UC Davis)
Title: "Covariance matrix and factor model estimation by composite minimization"
Abstract: In this talk, we address the problem of covariance matrix and factor model estimation in large dimensions under the low rank plus sparse assumption. Existing estimators based on principal component analysis (like POET, Fan et al. 2013) fail to catch low rank components characterized by non-spiked eigenvalues, as in that case the asymptotic consistency of principal components established in (Bai, 2003) defaults. For this reason, UNALCE (UNshrunk ALgebraic Covariance Estimator), an alternative estimator based on the solution of a low rank plus sparse decomposition problem, has been developed in (Farn´e and Montanari, 2020). Given the finite sample, the UNALCE approach is shown to return the covariance matrix estimate with the least possible dispersed eigenvalues among all the matrix pairs having the same rank of the low rank component and the same support of the sparse component respectively. In addition, parametric consistency in various norms and the exact recovery of the latent rank and the residual sparsity pattern are guaranteed until p α+δ ≪ n, where p is the dimension and n is the sample size, provided that latent eigenvalues scale to p α, α ∈ [0, 1], and the maximum number of nonzero residual covariances per row scales to p δ , δ ∈ [0, 1/2]. Consequently, if p and n are fixed, exploiting the eigenvalue dispersion lemma in (Ledoit and Wolf, 2004) we can prove that Bartlett’s and Thompson’s loadings and factor scores estimated via UNALCE provide the tightest possible uniform error bound in Euclidean norm. Simulation results show that UNALCE is particularly effective with respect to POET for recovering the proportion of latent variance, as well as the proportion of residual covariance and the number of residual non-zeros. Unlike POET, UNALCE exactly recovers the latent rank and the residual sparsity pattern, showing also better fitting properties. Moreover, UNALCE factor model estimates are proved to be particularly effective if latent eigenvalues are not so spiked and the residual component is very sparse. Two real data-sets, regarding UK market data and ECB supervisory data respectively, provide further insights about the usefulness of UNALCE in practical applications.
Seminar Date/Time: Thursday April 2nd, 4:10pm
This seminar will be delivered remotely via Zoom. To access the Zoom meeting for this seminar, please contact the instructor Shizhe Chen (email@example.com) or Pete Scully (firstname.lastname@example.org) for the meeting ID.