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June 2015 Covariance matrix estimation and linear process bootstrap for multivariate time series of possibly increasing dimension
Carsten Jentsch, Dimitris N. Politis
Ann. Statist. 43(3): 1117-1140 (June 2015). DOI: 10.1214/14-AOS1301


Multivariate time series present many challenges, especially when they are high dimensional. The paper’s focus is twofold. First, we address the subject of consistently estimating the autocovariance sequence; this is a sequence of matrices that we conveniently stack into one huge matrix. We are then able to show consistency of an estimator based on the so-called flat-top tapers; most importantly, the consistency holds true even when the time series dimension is allowed to increase with the sample size. Second, we revisit the linear process bootstrap (LPB) procedure proposed by McMurry and Politis [ J. Time Series Anal. 31 (2010) 471–482] for univariate time series. Based on the aforementioned stacked autocovariance matrix estimator, we are able to define a version of the LPB that is valid for multivariate time series. Under rather general assumptions, we show that our multivariate linear process bootstrap (MLPB) has asymptotic validity for the sample mean in two important cases: (a) when the time series dimension is fixed and (b) when it is allowed to increase with sample size. As an aside, in case (a) we show that the MLPB works also for spectral density estimators which is a novel result even in the univariate case. We conclude with a simulation study that demonstrates the superiority of the MLPB in some important cases.


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Carsten Jentsch. Dimitris N. Politis. "Covariance matrix estimation and linear process bootstrap for multivariate time series of possibly increasing dimension." Ann. Statist. 43 (3) 1117 - 1140, June 2015.


Received: 1 January 2014; Revised: 1 September 2014; Published: June 2015
First available in Project Euclid: 15 May 2015

zbMATH: 1320.62099
MathSciNet: MR3346699
Digital Object Identifier: 10.1214/14-AOS1301

Primary: 62G09
Secondary: 62M10

Keywords: asymptotics , bootstrap , Covariance matrix , High-dimensional data , multivariate time series , sample mean , Spectral density

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 3 • June 2015
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