Causal Inference for Complex Longitudinal Data: the continuous case

We extend Robins’ theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the g-computation formula and a collection of powerful characterizations of the g-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past. We also establish in a precise mathematical sense the harmlessness of the assumptions concerning counterfactual variables.