Covariance matrix estimation and linear process bootstrap for multivariate time series of possibly increasing dimension

TL;DR

This paper develops a consistent method for estimating autocovariance matrices in high-dimensional multivariate time series and introduces a multivariate linear process bootstrap that remains valid as the dimension grows, demonstrated through simulations.

Contribution

It proposes a new estimator for autocovariance matrices that remains consistent with increasing dimension and extends the linear process bootstrap to multivariate series, ensuring validity in high-dimensional settings.

Findings

The autocovariance estimator is consistent even when dimension increases with sample size.

The multivariate linear process bootstrap (MLPB) is valid for the sample mean in high-dimensional scenarios.

Simulations show MLPB's superiority in certain cases.

Abstract

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.