Combining Instrumental Variable Estimators for a Panel Data Model with Factors

We address the estimation of factor-augmented panel data models using observed measurements to proxy for unobserved factors or loadings and explore the use of internal instruments to address the resulting endogeneity. The main challenge consists in that economic theory rarely provides insights into which measurements to choose as proxies when several are available. To overcome this problem, we propose a new class of estimators that are linear combinations of instrumental variable estimators and establish large sample results. We also show that an optimal weighting scheme exists, leading to efficiency gains relative to an instrumental variable estimator. Simulations show that the proposed approach performs better than existing methods. We illustrate the new method using data on test scores across U.S. school districts.