Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains

The autoregressive moving average model is a stationary stochastic process ₜ\ satisfying ∑ᵖₖ₌₀ βₖyₜ₋ₖ = ∑qg=0 αgνt-g, where the (unobservable) process ₜ\ consists of independently identically distributed random variables. The coefficients in this equation and the variance of vₜ are to be estimated from an observed sequence y₁, ⋯, yT. To apply the method of maximum likelihood under normality the model is modified (i) by setting y₀ = y₋₁ = ⋯ = y₁₋ₚ = 0 and ν₀ = v₋₁ = ⋯ = v1-q = 0 and alternatively (ii) by setting y₀ ≡ yT, ⋯, y₁₋ₚ ≡ yT+1-p and v₀ ≡ vT, ⋯, v1-q ≡ vT+1-q; the former lead to procedures in the time domain and the latter to procedures in the frequency domain. Matrix methods are used for a unified development of the Newton-Raphson and scoring iterative procedures; most of the procedures have been obtained previously by different methods. Estimation of the covariances of the moving average part is also treated.