  Randomized controlled trials generate experimental variation that can
credibly identify causal effects, but often suffer from limited scale, while
observational datasets 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 dataset from Expedia that studies the causal effect of search
rankings on clicks, and show that the method can substantially improve the
precision.
