Open Access
August 2015 Regularized estimation in sparse high-dimensional time series models
Sumanta Basu, George Michailidis
Ann. Statist. 43(4): 1535-1567 (August 2015). DOI: 10.1214/15-AOS1315


Many scientific and economic problems involve the analysis of high-dimensional time series datasets. However, theoretical studies in high-dimensional statistics to date rely primarily on the assumption of independent and identically distributed (i.i.d.) samples. In this work, we focus on stable Gaussian processes and investigate the theoretical properties of $\ell_{1}$-regularized estimates in two important statistical problems in the context of high-dimensional time series: (a) stochastic regression with serially correlated errors and (b) transition matrix estimation in vector autoregressive (VAR) models. We derive nonasymptotic upper bounds on the estimation errors of the regularized estimates and establish that consistent estimation under high-dimensional scaling is possible via $\ell_{1}$-regularization for a large class of stable processes under sparsity constraints. A key technical contribution of the work is to introduce a measure of stability for stationary processes using their spectral properties that provides insight into the effect of dependence on the accuracy of the regularized estimates. With this proposed stability measure, we establish some useful deviation bounds for dependent data, which can be used to study several important regularized estimates in a time series setting.


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Sumanta Basu. George Michailidis. "Regularized estimation in sparse high-dimensional time series models." Ann. Statist. 43 (4) 1535 - 1567, August 2015.


Received: 1 February 2014; Revised: 1 January 2015; Published: August 2015
First available in Project Euclid: 17 June 2015

zbMATH: 1317.62067
MathSciNet: MR3357870
Digital Object Identifier: 10.1214/15-AOS1315

Primary: 62J99 , 62M10
Secondary: 2M15

Keywords: Covariance estimation , high-dimensional time series , Lasso , stochastic regression , vector autoregression

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 4 • August 2015
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