The autoregressive process with moving average residuals is a stationary process ₜ\ satisfying ∑ᵖs = 0 βₛyt - s = ∑qj = 0 αⱼνt - j, where the sequence \νₜ\ consists of independently identically distributed (unobservable) random variables. The distribution of y₁,⋯, yT can be approximated by the distribution of the T-component vector y satisfying ∑ᵖs = 0 βₛKₛy = ∑qj = 0 αⱼJⱼv, where v has covariance matrix σ²I, Kₛ = Jₛ = Lˢ, and L is the T × T matrix with 1's immediately below the main diagonal and 0's elsewhere. Maximum likelihood estimates are obtained when v has a normal distribution. The method of scoring is used to find estimates defined by linear equations which are consistent, asymptotically normal, and asymptotically efficient (as T→ ∞). Several special cases are treated. It is shown how to calculate the estimates.
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T. W. Anderson (1975) studied this question.