Approximations for Stationary Covariance Matrices and Their Inverses with Application to ARMA Models

Approximation of the covariance matrix Σ of T consecutive observations from a second-order stationary process with continuous positive spectral density f(λ) = σ²/(2π)²|∑^∞ⱼ₌₀δⱼeiλ j|² is considered. If Σ^ is the covariance matrix corresponding to a process with spectral density 1/(2π)²f(λ), then Σ^ - Σ⁻¹ 0. A matrix σ⁻²A'A with the property that Σ^ - σ⁻²A'A - Σ⁻¹ 0 is also considered. For autoregressive-moving average processes of order (p, q), Σ^ - σ⁻²A'A and σ⁻²A'A - Σ⁻¹ are shown to have rank min max (p, q), T and Σ^ - Σ⁻¹ to have rank min 2max (p, q), T. Some results concerning the covariance determinant are also discussed. If DT is σ-2T|Σ| for sample size T and D₀ = 1, then DT < DT+1, T = 0, 1, ⋯, unless the process is autoregressive of order p, in which case 1 < D₁ < ⋯ < Dₚ = Dₚ₊₁ = ⋯.