An Approximate Inverse for the Covariance Matrix of Moving Average and Autoregressive Processes

Let Σ denote the convariance matrix of a vector x = (x₁, ⋯, xT)' of T successive observations from a stationary process \ xₜ\ with continuous positive spectral density f(λ). Let Γ be the T × T matrix with elements γ(s, t) = (2π)⁻² ∫^π-π eiλ (s-t) f⁻¹(λ) dλ. The properties of Γ considered as an approximate inverse of Σ are studied. When \ xₜ\ is a$(n)$ moving average (autoregressive) process of order q, rows (columns) q + 1, ⋯, T - q of ΣΓ - I are zero vectors. In this case ΣΓ - I has $2q$ positive characteristic roots which approach paired positive limiting values as T → ∞ if the roots of ∑qⱼ₌₀ βⱼ zq-j = 0 are less than 1 in absolute value, where β₁, ⋯, βq are the coefficients of the process. Statistical properties of x'Tx - x'Σ⁻¹ x and x'Γ x/ x'Σ⁻¹ x are also discussed.