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2.5 Forecasting Stationary Time Series 73<br />

coefficients and the mean squared errors vi E 2, Xi+1 − ˆXi+1 starting from the<br />

covariances κ(i,j).<br />

The Innovations Algorithm:<br />

The coefficients θn1,...,θnn can be computed recursively from the equations<br />

and<br />

v0 κ(1, 1),<br />

<br />

k−1<br />

κ(n + 1,k+ 1) −<br />

θn,n−k v −1<br />

k<br />

j0<br />

j0<br />

n−1<br />

vn κ(n + 1,n+ 1) − θ 2<br />

n,n−jvj. θk,k−jθn,n−jvj<br />

<br />

, 0 ≤ k 1, κ(i,i) σ 2 1 + θ 2 , and κ(i,i + 1) θσ 2 .<br />

Application of the innovations algorithm leads at once to the recursions<br />

θnj 0, 2 ≤ j ≤ n,<br />

θn1 v −1<br />

n−1 θσ2 ,<br />

v0 (1 + θ 2 )σ 2 ,

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