Open Access
2015 High-dimensional autocovariance matrices and optimal linear prediction
Timothy L. McMurry, Dimitris N. Politis
Electron. J. Statist. 9(1): 753-788 (2015). DOI: 10.1214/15-EJS1000


A new methodology for optimal linear prediction of a stationary time series is introduced. Given a sample $X_{1},\ldots,X_{n}$, the optimal linear predictor of $X_{n+1}$ is $\tilde{X}_{n+1}=\phi_{1}(n)X_{n}+\phi_{2}(n)X_{n-1}+\cdots+\phi_{n}(n)X_{1}$. In practice, the coefficient vector $\phi(n)\equiv(\phi_{1}(n),\phi_{2}(n),\ldots,\phi_{n}(n))'$ is routinely truncated to its first $p$ components in order to be consistently estimated. By contrast, we employ a consistent estimator of the $n\times n$ autocovariance matrix $\Gamma_{n}$ in order to construct a consistent estimator of the optimal, full-length coefficient vector $\phi(n)$. Asymptotic convergence of the proposed predictor to the oracle is established, and finite sample simulations are provided to support the applicability of the new method. As a by-product, new insights are gained on the subject of estimating $\Gamma_{n}$ via a positive definite matrix, and four ways to impose positivity are introduced and compared. The closely related problem of spectral density estimation is also addressed.


Download Citation

Timothy L. McMurry. Dimitris N. Politis. "High-dimensional autocovariance matrices and optimal linear prediction." Electron. J. Statist. 9 (1) 753 - 788, 2015.


Received: 1 July 2014; Published: 2015
First available in Project Euclid: 2 April 2015

zbMATH: 1309.62155
MathSciNet: MR3331856
Digital Object Identifier: 10.1214/15-EJS1000

Primary: 62M20
Secondary: 62M10

Keywords: Autocovariance matrix , prediction , Spectral density , time series

Rights: Copyright © 2015 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.9 • No. 1 • 2015
Back to Top