A Remark on Bimodality and Weak Instrumentation in Structural Equation Estimation
In a simple model composed of a structural equation and identity, the ﬁnite sample distribution of the IV/LIML estimator is always bimodal and this is most apparent when the concentration parameter is small. Weak instrumentation is the energy that feeds the secondary mode and the coeﬀicient in the structural identity provides a point of compression in the density that gives rise to it. The IV limit distribution can be normal, bimodal, or inverse normal depending on the behavior of the concentration parameter and the weakness of the instruments. The limit distribution of the OLS estimator is normal in all cases and has a much faster rate of convergence under very weak instrumentation. The IV estimator is therefore more resistant to the attractive eﬀect of the identity than OLS. Some of these limit results diﬀer from conventional weak instrument asymptotics, including convergence to a constant in very weak instrument cases and limit distributions that are inverse normal.
Phillips, Peter C.B., "A Remark on Bimodality and Weak Instrumentation in Structural Equation Estimation" (2005). Cowles Foundation Discussion Papers. 1827.