High-dimensional autocovariance matrices and optimal linear prediction

Abstract

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.