This paper deals with several issues related to the identification of unconfounded (causal) effects of a treatment variable \(X\) on a response variable \(Y\). The starting point of our investigation is a paper by J. Pearl [Gabbay, Dov M. (ed.) et al., Handbook of defeasible reasoning and uncertainty management systems. Vol. 1: Quantified representation of uncertainty and imprecision. Dordrecht: Kluwer Academic Publishers, 367--389 (1998; Zbl 0933.03020)] where the term stable unconfoundedness has been introduced. We generalize this definition with respect to a set of variables \(T\) and state a necessary criterion for stable unconfoundedness in our more general situation, that is operational for real world applications. Under more specific assumptions, also a sufficient criterion is derived.