Almost Unbiased Estimation in Simultaneous Equation Models With Strong and/or Weak Instruments

Abstract

We propose two simple bias-reduction procedures that apply to estimators in a general static simultaneous equation model and that are valid under relatively weak distributional assumptions for the errors. 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 nonincreasing 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. (1) 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. (2) An alternative second approach is based on taking linear combinations of k-class estimators for k < 1. In general, this yields estimators that are unbiased to order T − 1 and that 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 0, 1, or at least 8. The performance of the two procedures is compared with 2SLS in a number of Monte Carlo experiments using a simple two-equation model. Finally, an application shows the usefulness of our new estimator in practice versus competitor estimators.