Robins' causal inference theory assumes existence of treatment specific counterfactual variables so that the observed data augmented by the counterfactual data will satisfy a consistency and a randomization assumption. Gill and Robins  show that the consistency and randomization assumptions do not add any restrictions to the observed data distribution. In particular, they provide a construction of counterfactuals as a function of the observed data distribution. In this paper we provide a construction of counterfactuals as a function of the observed data itself. Our construction provides a new statistical tool for estimation of counterfactual distributions. Robins [1987b] shows that the counterfactual distribution can be identified from the observed data distribution by a G-computation formula under an additional identifiability assumption. He proves this for discrete variables. Gill and Robins  prove the G-computation formula for continuous variable under some additional conditions and modifications of the consistency and the randomization assumptions. We prove that if treatment is discrete, then Robins' G-computation formula holds under the original consistency, randomization assumptions and a generalized version of identifiability assumption.
Longitudinal Data Analysis and Time Series | Statistical Methodology | Statistical Theory
Yu, Zhuo and van der Laan, Mark J., "Construction of Counterfactuals and the G-computation Formula" (December 2002). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 122.