Estimation of Dynamic Networks for High-Dimensional Nonstationary Time Series

Abstract

This paper is concerned with the estimation of time-varying networks for highdimensional nonstationary time series. Two types of dynamic behaviors are considered: structural breaks (i.e., abrupt change points) and smooth changes. To simultaneously handle these two types of time-varying features, a two-step approach is proposed: multiple change point locations are first identified on the basis of comparing the difference between the localized averages on sample covariance matrices, and then graph supports are recovered on the basis of a kernelized time-varying constrained L1-minimization for inverse matrix estimation (CLIME) estimator on each segment. We derive the rates of convergence for estimating the change points and precision matrices under mild moment and dependence conditions. In particular, we show that this twostep approach is consistent in estimating the change points and the piecewise smooth precision matrix function, under a certain high-dimensional scaling limit. The method is applied to the analysis of network structure of the S&P 500 index between 2003 and 2008.

Central limit theorems for high dimensional dependent data

“…Given d-dimensional dependent (possibly non-stationary) data {Y t } n t=1 with mean zero and instantaneous covariance Σ, i.e., E(Y t ) = 0 and Cov(Y t ) = Σ for any t ∈ [n], the instantaneous covariance matrix Σ and the precision matrix Ω = Σ −1 := (ω i,j ) d×d quantify the dependence among the d components of Y t . For the estimation of Σ and Ω, we refer to Bickel and Levina (2008a,b) and Cai, Liu and Luo (2011) for independent data, Chang et al (2018) and Xu, Chen and Wu (2020) for dependent data. Confidence regions for Σ and Ω can quantify the uncertainty in their estimates.…”

Estimating Time-Varying Networks for High-Dimensional Time Series

“…To address this problem, Ding, Qiu and Chen (2017) consider a time-varying VAR model for high-dimensional time series (allowing the number of variables to diverge at a sub-exponential rate of the sample size), and estimate the time-varying transition matrices by combining the kernel smoothing with 1 -regularisation, whereas Safikhani and Shojaie (2022) simultaneously detect breaks and estimate transition matrices in high-dimensional VAR via a three-stage procedure using the total variation penalty. Xu, Chen and Wu (2020) detect structural breaks and estimate smooth changes (between breaks) in the covariance and precision matrices of high-dimensional time series (covering VAR as a special case). In the present paper, we aim to jointly estimate the time-varying transition and precision matrices in the high-dimensional sparse VAR under the local stationarity framework.…”

Estimation of autocovariance matrices for high dimensional linear processes

“…The estimation of the inverse covariance matrix is used in the recovery of the true unknown structure of undirected B Konrad Furmańczyk konrad_furmanczyk@sggw.edu.pl 1 Institute of Information Technology, Warsaw University of Life Sciences (SGGW), Nowoursynowska 159, 02-776 Warsaw, Poland graphical models, especially in Gaussian graphical models, where a zero entry of the inverse covariance matrix is associated with a missing edge between two vertices in the graph. The recovery of undirected graphs on the basis of the estimation of the precision matrices for a general class of nonstationary time series is considered in Xu et al (2020).…”