  Experimental and observational studies often lack validity due to untestable
assumptions. We propose a double machine learning approach to combine
experimental and observational studies, allowing practitioners to test for
assumption violations and estimate treatment effects consistently. Our
framework tests for violations of external validity and ignorability under
milder assumptions. When only one of these assumptions is violated, we provide
semiparametrically efficient treatment effect estimators. However, our
no-free-lunch theorem highlights the necessity of accurately identifying the
violated assumption for consistent treatment effect estimation. Through
comparative analyses, we show our framework's superiority over existing data
fusion methods. The practical utility of our approach is further exemplified by
three real-world case studies, underscoring its potential for widespread
application in empirical research.
