  Understanding and quantifying cause and effect is an important problem in
many domains. The generally-agreed solution to this problem is to perform a
randomised controlled trial. However, even when randomised controlled trials
can be performed, they usually have relatively short duration's due to cost
considerations. This makes learning long-term causal effects a very challenging
task in practice, since the long-term outcome is only observed after a long
delay. In this paper, we study the identification and estimation of long-term
treatment effects when both experimental and observational data are available.
Previous work provided an estimation strategy to determine long-term causal
effects from such data regimes. However, this strategy only works if one
assumes there are no unobserved confounders in the observational data. In this
paper, we specifically address the challenging case where unmeasured
confounders are present in the observational data. Our long-term causal effect
estimator is obtained by combining regression residuals with short-term
experimental outcomes in a specific manner to create an instrumental variable,
which is then used to quantify the long-term causal effect through instrumental
variable regression. We prove this estimator is unbiased, and analytically
study its variance. In the context of the front-door causal structure, this
provides a new causal estimator, which may be of independent interest. Finally,
we empirically test our approach on synthetic-data, as well as real-data from
the International Stroke Trial.
