SEMAINE D'ÉTUDE SUR LE ROLE DE L’ANALYSE ECONOMETRIQUE ETC. 14}
(1) Determunistic and stochastic models do not obey the
same operational rules. Since stochastic models cover deter-
ministic models as a special case, the general rules for operating
with eo ipso predictors are valid also for deterministic models,
but the converse is not always true. Operations that in a ge-
neral way extend from deterministic relationships to eo ipso
predictors include addition and substitution. Procedures that
never extend to eo ipso predictors include squaring and inver-
sion. The explicit solving of a system of implicit relationships
extends to eo ipso predictors only in the special case when the
solving can be performed by iterated substitutions (1). It is
this last restriction that lies behind the fact, noted in 1.2 (3)
and 1.4 (3), that the predictor specification (13) of the primary
form of CC-systems has no counterpart in ID-systems.
Causal relations ('?). If we compare the operative aspects
of cause-effect relationships and eo ipso predictors we note a
far-going isomorphism, and specifically so with regard to the
basic operations of inversion and substitution. Thus if y is in-
fluenced by a causal factor x, this does not imply that x is in-
fluenced by y; isomorphically, if f(x) is an eo ipso predictor
of y this does not imply that f~!(y) is an eo ipso predictor of x.
As regards substitution, if y is influenced by a causal factor x,
and x is influenced by a causal factor z, we say — and in
principle this is a piece of causal inference — that y is in-
fluenced by z via x. For linear eo ipso predictors we have
the corresponding theorem that if the variables x, - a”
interrelated by
E(ylx, 2Y=1(x, 2) and E(x|z)=e(2)
then (13)
(") Ref. 27; cf. also Refs. 28 and 29.
(') For a more detailed discussion of the causal aspects of model build
ing, see Refs. 16-19 and 30.
(7) See Ref. 12 for a detailed treatment of the linear case. The substi-
tutional theorem is in (53)-(54) quoted for three one-dimensional variables
x, y, z. It extends to the case when z is a vector variable, the kev feature
being that functions f and g involve the same vector »
Wold - pag. 27
