An Integrated GMM Shrinkage Approach with Consistent Moment Selection from Multiple External Sources

Interest has grown in analyzing primary internal data by using some independent external aggregated statistics for efficiency gain. However, when population heterogeneity exists, inappropriate incorporation may lead to a biased estimator. With multiple external sources under generalized estimation equations and possibly heterogeneous populations, we propose an integrated generalized moment method that can perform a data-driven selection of valid moment equations from external sources and make efficient parameter estimation simultaneously. Moment equation selection consistency and asymptotic normality are established for the proposed estimator. Further, when the sample sizes of all external sources are large compared to the internal sample size, asymptotically the proposed estimator is more efficient than the estimator based on the internal data only and is oracle-efficient in the sense that it is as efficient as the oracle estimator based on all valid moment equations. Simulation studies confirm the theoretical results and the efficiency of the proposed method empirically. An example is also included for illustration. Supplementary materials for this article are available online.