Prediction of Time Series

Abstract

The covariance matrix of a stacked data vector (y₁′, y₂′, ... y_n′)′ of a meanzero weakly stationary process {y_t} has a special structure: A submatrix Λ₀ =Ey₁y₁′ is located along the main diagonal, the matrix ${\Lambda _\ell } = E{y_{\ell + 1}}{y'_1}$ along the $\ell$-th diagonal below the main diagonal, and ${\Lambda _{ - \ell }} = E{y_1}{y'_{\ell + 1}} = {\Lambda '_\ell }$ along the $\ell$-th diagonal above the main diagonal. This covariance matrix is a block Toeplitz matrix because the same submatrices are arranged in the same way that elements are arranged in Toeplitz matrices.