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5.1 Preliminary Estimation 147
While ˆθ (3)
n is more efficient, ˆθ (2)
n has reasonably good efficiency, except when |θ1| is
close to 1, and can serve as initial value for the nonlinear optimization procedure in
computing the maximum likelihood estimator.
While the method of moments is an effective procedure for fitting autoregressive
models, it does not perform as well for ARMA models with q > 0. From a
computational point of view, it requires as much computing time as the more efficient
estimators based on either the innovations algorithm or the Hannan–Rissanen
procedure and is therefore rarely used except when q 0.
5.1.2 Burg’s Algorithm
The Yule–Walker coefficients ˆφp1,..., ˆφpp are precisely the coefficients of the best
linear predictor of Xp+1 in terms of {Xp,...,X1} under the assumption that the ACF
of {Xt} coincides with the sample ACF at lags 1,...,p.
Burg’s algorithm estimates the PACF {φ11,φ22,...} by successively minimizing
sums of squares of forward and backward one-step prediction errors with respect to the
coefficients φii. Given observations {x1,...,xn} of a stationary zero-mean time series
{Xt} we define ui(t), t i+1,...,n,0≤ i