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Combining information from multiple samples is often needed in biomedical andeconomic studies, but the differences between these samples must beappropriately taken into account in the analysis of the combined data. We studyestimation for moment restriction models with data combination from two samplesunder an ignorablility-type assumption but allowing for different marginaldistributions of common variables between the two samples. Suppose that anoutcome regression model and a propensity score model are specified. Byleveraging the semiparametric efficiency theory, we derive an augmented inverseprobability weighted (AIPW) estimator that is locally efficient and doublyrobust with respect to the outcome regression and propensity score models.Furthermore, we develop calibrated regression and likelihood estimators thatare not only locally efficient and doubly robust, but also intrinsicallyefficient in achieving smaller variances than the AIPW estimator when thepropensity score model is correctly specified but the outcome regression modelmay be misspecified. As an important application, we study the two-sampleinstrumental variable problem and derive the corresponding estimators whileallowing for incompatible distributions of common variables between the twosamples. Finally, we provide a simulation study and an econometric applicationon public housing projects to demonstrate the superior performance of ourimproved estimators.

Improved Methods for Moment Restriction Models with Marginally  Incompatible Data Combination and an Application to Two-sample Instrumental  Variable Estimation