  Causal effect estimation relies on separating the variation in the outcome
into parts due to the treatment and due to the confounders. To achieve this
separation, practitioners often use external sources of randomness that only
influence the treatment called instrumental variables (IVs). We study variables
constructed from treatment and IV that help estimate effects, called control
functions. We characterize general control functions for effect estimation in a
meta-identification result. Then, we show that structural assumptions on the
treatment process allow the construction of general control functions, thereby
guaranteeing identification. To construct general control functions and
estimate effects, we develop the general control function method (GCFN). GCFN's
first stage called variational decoupling (VDE) constructs general control
functions by recovering the residual variation in the treatment given the IV.
Using VDE's control function, GCFN's second stage estimates effects via
regression. Further, we develop semi-supervised GCFN to construct general
control functions using subsets of data that have both IV and confounders
observed as supervision; this needs no structural treatment process
assumptions. We evaluate GCFN on low and high dimensional simulated data and on
recovering the causal effect of slave export on modern community trust.
