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PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28

with

(83)

EWdyi—1 Viras )=Bo+B1vi_1 +85 vy, 0+...

which is a special case of the predictive decomposition of y,
referred to earlier in this paper [section 1.4 (4)]. The forecasts
Yn+1> Yn+2 --- are obtained by the chain principle, making iterated
 use of (83). Thus when y,,; has been obtained, v,,;.,
is calculated from (83) in the basis of y,.;, v,.; 1, ... The
variance of the resulting forecasts is given by

84)

E(yrumT Ans) = (1 + a? + + a? _,) ok

showing that the accuracy of the forecast will decrease as the
forecast span me increases. This last feature is reflected also
in the JANUS quotient, inasmuch as (84) gives

I
(85) E(J)=1+(1- 2) o24 (1-5) a+ + — ab,

The predictive decomposition (82a-b) has the property that
the variance (84) is the smallest possible of all representations
of type (82b). This is the fundamental property of minimumdelay,
 established by E. RoBINsoN, Ref. 24, and referred to
earlier in this paper.
The approach (82)-(85) extends to the general stationarv
case when y, allows the predictive decomposition

‘86a-b)

Vi=Y +B ya th
=W+v,+%, v, +a,

‘À

where V, is the deterministic (also called singular) component
of y,. The procedure of forecasting first settles the prediction
of the deterministic component over the entire forecast range,

>]

Wold - pag. 44