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

Estimating long-term causal effects from short-term experiments and long-term observational data with unobserved confounding