  Combining information from multiple samples is often needed in biomedical and
economic studies, but the differences between these samples must be
appropriately taken into account in the analysis of the combined data. We study
estimation for moment restriction models with data combination from two samples
under an ignorablility-type assumption but allowing for different marginal
distributions of common variables between the two samples. Suppose that an
outcome regression model and a propensity score model are specified. By
leveraging the semiparametric efficiency theory, we derive an augmented inverse
probability weighted (AIPW) estimator that is locally efficient and doubly
robust with respect to the outcome regression and propensity score models.
Furthermore, we develop calibrated regression and likelihood estimators that
are not only locally efficient and doubly robust, but also intrinsically
efficient in achieving smaller variances than the AIPW estimator when the
propensity score model is correctly specified but the outcome regression model
may be misspecified. As an important application, we study the two-sample
instrumental variable problem and derive the corresponding estimators while
allowing for incompatible distributions of common variables between the two
samples. Finally, we provide a simulation study and an econometric application
on public housing projects to demonstrate the superior performance of our
improved estimators.
