Before revision:  To improve the estimation efficiency, we propose a new method that combines experimental and observational data by leveraging covariates that satisfy the first-stage relevance condition in the observational data . We first use the observational data to derive a biased estimate of the causal effect and then correct this bias using the experimental data. We show thatthis bias-corrected estimator is uncorrelated with the benchmark estimator that only uses the experimental data, and can thus be combined with the benchmark estimator to improve efficiency.
After revision:  To improve the estimation efficiency, we propose a new method that combines experimental and observational data by leveraging covariates that satisfy the first-stage relevance condition in the observational data , they can be used to derive moment restrictions that, in combination with the experimental data, and can thus be combined with the benchmark estimator to improve efficiency.