Almost Unbiased Estimation in Simultaneous Equations Models with Strong and / or Weak Instruments

We propose two simple bias reduction procedures that apply to estimators in a general static simultaneous equation model. Standard jackknife estimators, as applied to 2SLS, may not reduce the bias of the exogenous variable coefficient estimators since the estimator biases are not monotonically non-increasing with sample size (a necessary condition for successful bias reduction) and they have moments only up to the order of overidentification. Our proposed approaches do not have either of these drawbacks. In the first procedure, both endogenous and exogenous variable parameter estimators are unbiased to order T−2 and when implemented for k-class estimators for which k < 1, the higher order moments will exist. An alternative second approach is based on taking linear combinations of k-class estimators for k < 1. In general, this yields estimators which are unbiased to order T−1and which possess higher moments. We also prove theoretically how the combined k-class estimator produces a smaller mean squared error than 2SLS when the degree of overidentification of the system is larger than 7. Moreover, the combined k-class estimators remain unbiased to order T−1 even if there are redundant variables (including weak instruments) in any part of the simultaneous equation system, and we can allow for any number of endogenous variables. The performance of the two procedures is compared with 2SLS in a number of Monte Carlo experiments using a simple two equation model. ∗We are grateful for helpful comments from Bruce Hansen, Tim Vogelsang, Jeffrey Wooldridge and seminar participants at Harvard / MIT, Michigan State University, University of Montreal, Purdue and at SMU (Singapore); and also from participants at the 2006 Far Eastern Meeting of the Econometric Society, the 2006 European Meeting of the Econometric Society and the 2006 Midwest Econometrics Group. Financial support from the MSU Intramural Research Grants Program is gratefully acknowledged. †Corresponding author. Department of Economics, Michigan State University, 101 Marshall-Adams Hall, East Lansing, MI 48824-1038, USA. e-mail: [email protected].