There has been an increase in interest in experimental evaluations to estimate causal effects, partly because their internal validity tends to be high. At the same time, as part of the big data revolution, large, detailed, and representative, administrative data sets have become more widely available. However, the credibility of estimates of causal effects based on such data sets alone can be low. In this paper, we develop statistical methods for systematically combining experimental and observational data to obtain credible estimates of the causal effect of a binary treatment on a primary outcome that we only observe in the observational sample. Both the observational and experimental samples contain data about a treatment, observable individual characteristics, and a secondary (often short term) outcome. To estimate the effect of a treatment on the primary outcome while addressing the potential confounding in the observational sample, we propose a method that makes use of estimates of the relationship between the treatment and the secondary outcome from the experimental sample. If assignment to the treatment in the observational sample were unconfounded, we would expect the treatment effects on the secondary outcome in the two samples to be similar. We interpret differences in the estimated causal effects on the secondary outcome between the two samples as evidence of unobserved confounders in the observational sample, and develop control function methods for using those differences to adjust the estimates of the treatment effects on the primary outcome. We illustrate these ideas by combining data on class size and third grade test scores from the Project STAR experiment with observational data on class size and both third and eighth grade test scores from the New York school system.
