Combining Observational and Experimental Data to Improve Efficiency Using Imperfect Instruments

Randomized controlled trials generate experimental variation that can credibly identify causal effects, but often suffer from limited scale, whereas observational data sets are large, but often violate desired identification assumptions. To improve estimation efficiency, I propose a method that leverages imperfect instruments—pretreatment covariates that satisfy the relevance condition, but may violate the exclusion restriction. I show that these imperfect instruments can be used to derive moment restrictions that, in combination with the experimental data, improve estimation efficiency. I outline estimators for implementing this strategy and show that my methods can reduce variance by up to 50%; therefore, only half of the experimental sample is required to attain the same statistical precision. I apply my method to a search-listing data set from Expedia that studies the causal effect of search rankings on clicks and show that the method can substantially improve the precision.