Causal Linear Stochastic Dependencies: The Formal Theory

Abstract

The formal background of the theory of causal linear stochastic dependence is provided, which was introduced by Steyer (1984). The theory presented is concerned with those kinds of dependencies which can be described by snecifying the functional form of a conditional expectation E(Y|X). This includes also those situations in which X is a multidimensional random variable. The main concepts of the theory are causal and weak causal linear stochastic dependencies , the definition of which is based on the pre- and equiorderedness relations of sigma-fields and stochastic variables, on the notion of potential disturbing sigma-fields and variables , as well as on the invariance and on the average conditions. These concepts are formally defined and their properties are studied in some detail. Causal linear stochastic dependence is defined by the preorderedness condition that the influencing variable is antecedent to the influenced variable and by the invariance con-dition , whereas weak causal linear stochastic dependence is defined by the Dreorderedness and average conditions . Both, the invariance and the average conditions, and therefore both kinds of causal hypotheses, can empirically be tested in experimental as well as in nonexperimental observational studies.