{20

PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28

include forecasting as-a main purpose in their applied use;
(c) make use of the device called forecasting by the chain principle.
 We say that a forecast that spans % periods ¢+1,
+2, ..., t+k is obtained by the chain principle if the calculations
 proceed step by step so that when the forecast for
the period #+i has been obtained (t=1, 2, ..., k- I) it is used
as an actual observation when calculating the forecast for the
period £+{+I.

t-4h t-3h t-2h t-h
Observations: O O e

Forecasts

t t+h t+2h
Q O O
“ENG

FiG. 1 — Forecasting by the chain brincible

All through sections 1-2 we shall be concerned with purely
theoretical aspects of the three types of model. For the specification
 of the models we shall make use of the notion of
go 1pso predictor, that is: If a random variable y allows the
representation

I)

y=f(x)+v

where f(x) is the conditional expectation of y for given x,

2)

E(vlx)= f(x)

then f(x) is called an eo ipso predictor of y. The notion extends
to vector variables y, x, v. As to the empirical treatment of
20 1pso predictors, we note that they can under very general
conditions be consistently estimated by least squares regression
 (3).

(*) For a detailed proof. see Ref. 14

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Wold - pag. 6