{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 prin- 
ciple. We say that a forecast that spans % periods ¢+1, 
+2, ..., t+k is obtained by the chain principle if the cal- 
culations 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 speci- 
fication 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 regres- 
sion (3). 
(*) For a detailed proof. see Ref. 14 
“21 
Wold - pag. 6