The instrumental variable (IV) approach is commonly used to infer causal effects in the presence of unmeasured confounding. Existing methods typically aim to estimate the mean causal effects, whereas a few other methods focus on quantile treatment effects. The aim of this work is to estimate the entire interventional distribution. We propose a method called Distributional Instrumental Variable (DIV), which uses generative modelling in a nonlinear IV setting. We establish identifiability of the interventional distribution under general assumptions and demonstrate an 'under-identified' case, where DIV can identify the causal effects while two-step least squares fails to. Our empirical results show that the DIV method performs well for a broad range of simulated data, exhibiting advantages over existing IV approaches in terms of the identifiability and estimation error of the mean or quantile treatment effects. Furthermore, we apply DIV to an economic data set to examine the causal relation between institutional quality and economic development and our results align well with the original study. We also apply DIV to a single-cell data set, where we study the generalizability and stability in predicting gene expression under unseen interventions. The software implementations of DIV are available in R and Python.

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