Structural Modeling with Weak Instrumentation and Covariance Restrictions

This paper considers estimating a panel data simultaneous equations model under both coefficient and covariance matrix restrictions in a scenario where one or the other set of identifying restrictions may be invalid or may hold only weakly. We study the limiting properties of various estimators in an asymptotic framework, which takes both the cross-sectional dimension N and the time dimension T to infinity. In this setting as in the pure cross-sectional setup, the performance of the 2SLS estimator depends on the strength of the identifying conditions imposed on the coefficients of the model, and it fails to be consistent once these conditions break down sufficiently resulting in instruments that are too weakly correlated with the endogenous regressors. On the other hand, the between-group (BG) estimator is consistent and asymptotically normal even when coefficient restrictions fail, but it has the shortcoming that its precision depends only on variations in the cross-sectional dimension; and, hence, it is less efficient and has slower rate of convergence than alternatives which make better use of the large time dimension. A GMM estimator which combines the moment conditions of the BG estimator with that of the withingroup IV estimator is more robust to instrument weakness than 2SLS and is more efficient than the BG estimator, but it has a second-order bias even under strong instruments if the assumed covariance restrictions do not hold. To remedy the deficiency of the aforementioned estimators, we propose in this paper two new model averaging estimators which are weighted averages of the GMM estimator and a bias-corrected GMM estimator. The two proposed estimators have weighting functions that depend on alternative transformations of the Bayesian Information Criterion (BIC), which is employed here to assess the validity of the covariance restrictions. We show that these new estimators have some nice robustness properties against possible failure of either the coefficient restrictions or the covariance restrictions. The author would like to thank the Guest Editor, Bruce Hansen, and two anonymous referees for their very helpful comments and suggestions. With great affection, this paper is dedicated to my mentor and friend, Peter C. B. Phillips. JEL classification: C13, C33.