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 instrumental variable/limited information maximum likelihood (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 ordinary least squares (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.My thanks to Richard Smith and two referees for comments on an earlier version. Section 2 of the paper is based on lectures given to students over the 1970s and 1980s at Essex, Birmingham, and Yale. Partial support is acknowledged from a Kelly Fellowship at the University of Auckland School of Business and the NSF under Grant SES 04-142254.