Causal aggregation: estimation and inference of causal effects by constraint-based data fusion

Randomized experiments are the gold standard for causal inference. In experiments, usually one variable is manipulated and its effect is measured on an outcome. However, practitioners may also be interested in the effect on a fixed target variable of simultaneous interventions on multiple covariates. We propose a novel method that allows to estimate the effect of joint interventions using data from different experiments in which only very few variables are manipulated. If the joint causal effect is linear, the proposed method can be used for estimation and inference of joint causal effects, and we characterize conditions for identifiability. The proposed method allows to combine data sets arising from randomized experiments as well as observational data sets for which IV assumptions or unconfoundedness hold: we indicate how to leverage all the available causal information to efficiently estimate the causal effects in the overidentified setting. If the dimension of the covariate vector is large, we may have data from experiments on every covariate, but only a few samples per randomized covariate. Under a sparsity assumption, we derive an estimator of the causal effects in this high-dimensional scenario. In addition, we show how to deal with the case where a lack of experimental constraints prevents direct estimation of the causal effects. When the joint causal effects are non-linear, we characterize conditions under which identifiability holds, and propose a non-linear causal aggregation methodology for experimental data sets similar to the gradient boosting algorithm where in each iteration we combine weak learners trained on different datasets using only unconfounded samples. We demonstrate the effectiveness of the proposed method on simulated and semi-synthetic data.