A Curious Result on Exact FIML and Instrumental Variables

The iterative application of an instrumental variable method to a system of simultaneous equations may exactly produce FIML, upon convergence. Instruments achieving this target do not need to be either uncorrelated with the error terms or correlated as much as possible with the replaced explanatory variables. This curious mathematical result, which contradicts some common wisdom and intuition, is proved in our paper. Our proof also provides a unified scheme that covers the available traditional instrumental variable interpretations of FIML, whether the model is linear or nonlinear, and whether covariance restrictions are or are not imposed.