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2.5 Forecasting Stationary Time Series 75
Hence,
PnXn+h PnPn+h−1Xn+h
Pn ˆXn+h
Pn
n+h−1
j1
θn+h−1,j
Xn+h−j − ˆXn+h−j
.
Applying (2.5.29) again and using the linearity of Pn we find that
PnXn+h
n+h−1
jh
θn+h−1,j
Xn+h−j − ˆXn+h−j
, (2.5.30)
where the coefficients θnj are determined as before by the innovations algorithm.
Moreover, the mean squared error can be expressed as
E(Xn+h − PnXn+h) 2 EX 2
2
n+h − E(PnXn+h)
n+h−1
κ(n + h, n + h) −
jh
θ 2
n+h−1,j vn+h−j−1. (2.5.31)
2.5.3 Prediction of a Stationary Process in Terms of Infinitely Many Past
Values
It is often useful, when many past observations Xm,...,X0,X1,...,Xn (m