High-dimensional autocovariance matrices and optimal linear prediction

A new methodology for optimal linear prediction of a stationary time series is introduced. Given a sample Xi,, the optimal linear predictor of Xn+1 is (X) over tilde (n+1) = phi(1)(n)X-n + phi(2)(n)Xn-1+...+phi(n)(n)X-1. In practice, the coefficient vector phi(n) equivalent to (phi(1)(n), phi(2)(n), ..., 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 x n auto-covariance 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.